[
    {
        "index": 1,
        "problem": "Let $\\triangle ABC$ have area \\$240$. Points $X$, $Y$, and $Z$ are located on sides $AB$, $BC$, and $CA$, respectively, such that $\\frac{AX}{BX} = 3$, $\\frac{BY}{CY} = 4$, and $\\frac{CZ}{AZ} = 5$. Find the area of $\\triangle XYZ$.",
        "geo_code": "[asy]\nsize(175);\ndefaultpen(linewidth(0.8));\npair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;\ndraw(A--B--C--cycle^^X--Y--Z--cycle);\nlabel(\" $A$ \",A,N);\nlabel(\" $B$ \",B,S);\nlabel(\" $C$ \",C,E);\nlabel(\" $X$ \",X,W);\nlabel(\" $Y$ \",Y,S);\nlabel(\" $Z$ \",Z,NE);[/asy]",
        "answer": "$122$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 2,
        "problem": "In circle $O$, chord $AB$ is extended beyond $B$ to a point $C$ such that $BC$ equals the radius of the circle. $CO$ is drawn and extended to $D$, and $AO$ is also drawn. If $y = 1$, find the value of $x$.",
        "geo_code": "[asy]size(200);defaultpen(linewidth(0.7)+fontsize(10));\npair O=origin, D=dir(195), A=dir(150), B=dir(30), C=B+1*dir(0);\ndraw(O--A--C--D);\ndot(A^^B^^C^^D^^O);\npair point=O;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $O$ \", O, dir(285));\nlabel(\" $x$ \", O+0.1*dir(172.5), dir(172.5));\nlabel(\" $y$ \", C+0.4*dir(187.5), dir(187.5));\ndraw(Circle(O,1));\n[/asy]",
        "answer": "$3$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 3,
        "problem": "In $\\triangle ABC$, suppose $\\angle C = 3\\angle A$, $a = 27$, and $c = 48$. What is the value of $b$?",
        "geo_code": "[asy]size(200);\ndefaultpen(linewidth(0.7)+fontsize(10));\npair A=origin, B=(14,0), C=(10,6);\ndraw(A--B--C--cycle);\nlabel(\" $A$ \", A, SW);\nlabel(\" $B$ \", B, SE);\nlabel(\" $C$ \", C, N);\nlabel(\" $a$ \", B--C, dir(B--C)*dir(-90));\nlabel(\" $b$ \", A--C, dir(C--A)*dir(-90));\nlabel(\" $c$ \", A--B, dir(A--B)*dir(-90));\n[/asy]",
        "answer": "$35$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 4,
        "problem": "Square $ABCD$ has side length \\$4$, and $M$ is the midpoint of $\\overline{CD}$. A circle with radius \\$2$ and center $M$ intersects a circle with radius \\$4$ and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\\overline{AD}$?",
        "geo_code": "[asy]unitsize(8mm);\ndefaultpen(linewidth(.8pt));\ndotfactor=4;\n\ndraw(Circle((2,0),2));\ndraw(Circle((0,4),4));\nclip(scale(4)*unitsquare);\ndraw(scale(4)*unitsquare);\nfilldraw(Circle((2,0),0.07));\nlabel(\" $A$ \",(0,4),NW);\nlabel(\" $B$ \",(4,4),NE);\nlabel(\" $C$ \",(4,0),SE);\nlabel(\" $D$ \",(0,0),SW);\nlabel(\" $M$ \",(2,0),S);\nlabel(\" $P$ \",(3.6,1.4),N);[/asy]",
        "answer": "$\\frac{16}{5}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 5,
        "problem": "A circle with center $O$ and radius \\$1$ contains point $A$. Segment $AB$ is tangent to the circle at $A$, and $\\angle AOB = 30^\\circ$. If point $C$ lies on $\\overline{OA}$ and $\\overline{BC}$ bisects $\\angle ABO$, find the length $OC$.",
        "geo_code": "[asy]import olympiad;\nunitsize(2cm);\ndefaultpen(fontsize(8pt)+linewidth(.8pt));\nlabelmargin=0.2;\ndotfactor=3;\n\npair O=(0,0);\npair A=(1,0);\npair B=(1,1.5);\npair D=bisectorpoint(A,B,O);\npair C=extension(B,D,O,A);\n\ndraw(Circle(O,1));\ndraw(O--A--B--cycle);\ndraw(B--C);\n\nlabel(\" $O$ \",O,SW);\ndot(O);\nlabel(\" $\\theta$ \",(0.1,0.05),ENE);\n\ndot(C);\nlabel(\" $C$ \",C,S);\n\ndot(A);\nlabel(\" $A$ \",A,E);\n\ndot(B);\nlabel(\" $B$ \",B,E);[/asy]",
        "answer": "$\\frac{2}{3}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 6,
        "problem": "In right triangle $\\triangle ABC$ with legs measuring \\$5$ and \\$12$, two arcs are drawn: one with center $A$ and radius \\$12$, and one with center $B$ and radius \\$5$. These arcs intersect the hypotenuse at points $M$ and $N$. Find the length of $MN$.",
        "geo_code": "[asy]size(200);\ndefaultpen(linewidth(0.7)+fontsize(10));\npair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A);\nreal r=degrees(B);\ndraw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270));\npair point=incenter(A,B,C);\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $M$ \", M, dir(point--M));\nlabel(\" $N$ \", N, dir(point--N));\n\nlabel(\" $12$ \", (6,0), S);\nlabel(\" $5$ \", (12,3.5), E);[/asy]",
        "answer": "$4$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 7,
        "problem": "On square $ABCD$, points $E, F, G,$ and $H$ lie on sides $\\overline{AB},\\,\\overline{BC},\\,\\overline{CD},$ and $\\overline{DA}$, respectively, such that $\\overline{EG} \\perp \\overline{FH}$ and $EG = FH = 34$. The segments $\\overline{EG}$ and $\\overline{FH}$ intersect at point $P$, and the areas of the quadrilaterals $AEPH$, $BFPE$, $CGPF$, and $DHPG$ are in the ratio \\$269:275:405:411$. Find the area of square $ABCD$.",
        "geo_code": "[asy]\nsize(200);\ndefaultpen(linewidth(0.8)+fontsize(10.6));\npair A = (0,sqrt(850));\npair B = (0,0);\npair C = (sqrt(850),0);\npair D = (sqrt(850),sqrt(850));\ndraw(A--B--C--D--cycle);\ndotfactor = 3;\ndot(\" $A$ \",A,dir(135));\ndot(\" $B$ \",B,dir(215));\ndot(\" $C$ \",C,dir(305));\ndot(\" $D$ \",D,dir(45));\npair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850));\npair F = ((2sqrt(850)+sqrt(306)+7)/6,0);\ndot(\" $H$ \",H,dir(90));\ndot(\" $F$ \",F,dir(270));\ndraw(H--F);\npair E = (0,(sqrt(850)-6)/2);\npair G = (sqrt(850),(sqrt(850)+sqrt(100))/2);\ndot(\" $E$ \",E,dir(180));\ndot(\" $G$ \",G,dir(0));\ndraw(E--G);\npair P = extension(H,F,E,G);\ndot(\" $P$ \",P,dir(60));\nlabel(\" $w$ \", (H+E)/2,fontsize(15));\nlabel(\" $x$ \", (E+F)/2,fontsize(15));\nlabel(\" $y$ \", (G+F)/2,fontsize(15));\nlabel(\" $z$ \", (H+G)/2,fontsize(15));\nlabel(\" $w:x:y:z=269:275:405:411$ \",(sqrt(850)/2,-4.5),fontsize(11));\n[/asy]",
        "answer": "$850$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 8,
        "problem": "Let  $XYZ$  be an acute-angled triangle. Let  $s$  be the side-length of the square which has two adjacent vertices on side  $YZ$ , one vertex on side  $XY$  and one vertex on side  $XZ$ . Let  $h$  be the distance from  $X$  to the side  $YZ$  and let  $b$  be the distance from  $Y$  to  $Z$ .\n\n\n\nIf the vertices have coordinates  $X = (2, 4)$ ,  $Y = (0, 0)$  and  $Z = (4, 0)$ , find  $s$ .",
        "geo_code": "[asy]\ndraw((4, 8)--(0, 0)--(14, 0)--cycle);\ndraw((2.54,0)--(2.54+5.1,0)--(2.54+5.1,5.1)--(2.54,5.1)--cycle);\nlabel(\" $X$ \",(4,8),N);\nlabel(\" $Y$ \",(0,0),W);\nlabel(\" $Z$ \",(14,0),E);\n[/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 9,
        "problem": "Let  $XYZ$  be an acute-angled triangle. Let  $s$  be the side-length of the square which has two adjacent vertices on side  $YZ$ , one vertex on side  $XY$  and one vertex on side  $XZ$ . Let  $h$  be the distance from  $X$  to the side  $YZ$  and let  $b$  be the distance from  $Y$  to  $Z$ .\n\n\n\nGiven the height  $h = 3$  and  $s = 2$ , find the base  $b$ .",
        "geo_code": "[asy]\npair S, D;\nD = 1.27;\nS = 2.55;\ndraw((2, 4)--(0, 0)--(7, 0)--cycle);\ndraw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle);\nlabel(\" $X$ \",(2,4),N);\nlabel(\" $Y$ \",(0,0),W);\nlabel(\" $Z$ \",(7,0),E);\n[/asy]",
        "answer": "$6$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 10,
        "problem": "Let $XYZ$ be an acute-angled triangle. Let $s$ denote the side length of a square with two adjacent vertices on side $YZ$, one vertex on side $XY$, and one vertex on side $XZ$. Let $h$ be the distance from $X$ to side $YZ$, and let $b$ be the distance from $Y$ to $Z$.\n\nIf the area of the square is \\$2017$, determine the minimum possible area of triangle $XYZ$.",
        "geo_code": "[asy]\npair S, D;\nD = 1.27;\nS = 2.55;\ndraw((2, 4)--(0, 0)--(7, 0)--cycle);\ndraw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle);\nlabel(\" $X$ \",(2,4),N);\nlabel(\" $Y$ \",(0,0),W);\nlabel(\" $Z$ \",(7,0),E);\n[/asy]",
        "answer": "$4034$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 11,
        "problem": "Two circles $A$ and $B$, each with radius \\$1$, are externally tangent to one another. Four circles $P$, $Q$, $R$, and $S$, each with the same radius $r$, are arranged such that $P$ is externally tangent to $A$, $B$, $Q$, and $S$; $Q$ is externally tangent to $P$, $B$, and $R$; $R$ is externally tangent to $A$, $B$, $Q$, and $S$; and $S$ is externally tangent to $P$, $A$, and $R$. Find the value of $r$.",
        "geo_code": "[asy]\nunitsize(0.3 cm);\n\npair A, B, P, Q, R, S;\nreal r = (3 + sqrt(17))/2;\n\nA = (-1,0);\nB = (1,0);\nP = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180));\nR = -P;\nQ = (r + 2,0);\nS = (-r - 2,0);\n\ndraw(Circle(A,1));\ndraw(Circle(B,1));\ndraw(Circle(P,r));\ndraw(Circle(Q,r));\ndraw(Circle(R,r));\ndraw(Circle(S,r));\n\nlabel(\" $A$ \", A);\nlabel(\" $B$ \", B);\nlabel(\" $P$ \", P);\nlabel(\" $Q$ \", Q);\nlabel(\" $R$ \", R);\nlabel(\" $S$ \", S);\n[/asy]",
        "answer": "$2$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 12,
        "problem": "In a circle with center $O$, $AD$ is a diameter and $ABC$ is a chord. If $BO = 5$ and $\\angle ABO = \\overset{\\frown}{CD} = 60^\\circ$, find the length of $BC$.",
        "geo_code": "[asy]size(200);\ndefaultpen(linewidth(0.7)+fontsize(10));\npair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C);\ndraw(C--A--D^^B--O^^Circle(O,1));\npair point=O;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $O$ \", O, dir(305));\n\nlabel(\" $5$ \", B--O, dir(O--B)*dir(90));\nlabel(\" $60^\\circ$ \", dir(185), dir(185));\nlabel(\" $60^\\circ$ \", B+0.05*dir(-25), dir(-25));[/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 13,
        "problem": "A rectangle $HOMF$ has sides $HO = 11$ and $OM = 5$. Let $\\triangle ABC$ be a triangle with $H$ as the orthocenter, $O$ as the circumcenter, $M$ as the midpoint of $BC$, and $F$ as the foot of the altitude from $A$. Find the length of $BC$.",
        "geo_code": "[asy]\nunitsize(0.3 cm);\npair F, H, M, O;\nF = (0,0);\nH = (0,5);\nO = (11,5);\nM = (11,0);\ndraw(H--O--M--F--cycle);\nlabel(\" $F$ \", F, SW);\nlabel(\" $H$ \", H, NW);\nlabel(\" $M$ \", M, SE);\nlabel(\" $O$ \", O, NE);\n[/asy]",
        "answer": "$28$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 14,
        "problem": "A circle with radius \\$1$ is surrounded by \\$4$ circles of radius $r$, as shown. What is the value of $r$?",
        "geo_code": "[asy]defaultpen(linewidth(.9pt));\n\nreal r = 2 + sqrt(2);\npair A = dir(45)*(r + sqrt(2));\npair B = dir(135)*(r + sqrt(2));\npair C = dir(-135)*(r + sqrt(2));\npair D = dir(-45)*(r + sqrt(2));\n\ndraw(Circle(origin,sqrt(2)));\ndraw(Circle(A,r));draw(Circle(B,r));draw(Circle(C,r));draw(Circle(D,r));\n\ndraw(A--(dir(45)*r + A));\ndraw(B--(dir(45)*r + B));\ndraw(C--(dir(45)*r + C));\ndraw(D--(dir(45)*r + D));\ndraw(origin--(dir(25)*sqrt(2)));\n\nlabel(\" $r$ \",midpoint(A--(dir(45)*r + A)), SE);\nlabel(\" $r$ \",midpoint(B--(dir(45)*r + B)), SE);\nlabel(\" $r$ \",midpoint(C--(dir(45)*r + C)), SE);\nlabel(\" $r$ \",midpoint(D--(dir(45)*r + D)), SE);\nlabel(\" $1$ \",origin,W);[/asy]",
        "answer": "$1 + \\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 15,
        "problem": "Older television screens have an aspect ratio of \\$4:3$, meaning the width is to the height as \\$4$ is to \\$3$. Many movies have a different aspect ratio, so when shown on such a screen, black bars of equal height (called 'letterboxing') appear at the top and bottom. Suppose a movie with an aspect ratio of \\$2:1$ is displayed on a television screen with a \\$27$-inch diagonal and \\$4:3$ aspect ratio. What is the height, in inches, of each black bar?",
        "geo_code": "[asy]unitsize(1mm);\ndefaultpen(linewidth(.8pt));\nfilldraw((0,0)--(8,0)--(8,1)--(0,1)--cycle,grey,black);\nfilldraw((0,5)--(8,5)--(8,6)--(0,6)--cycle,grey,black);\ndraw((0,1)--(0,5));\ndraw((8,1)--(8,5));[/asy]",
        "answer": "$2.7$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 16,
        "problem": "In right triangle $ABC$, $BC = 5$, $AC = 12$, and $AM = 5$. Let $\\overline{MN} \\perp \\overline{AC}$ and $\\overline{NP} \\perp \\overline{BC}$, with $N$ lying on $AB$. If $y = MN + NP$, which is one-half the perimeter of rectangle $MCPN$, find the value of $y$.",
        "geo_code": "[asy]defaultpen(linewidth(.8pt));\nunitsize(2cm);\npair A = origin;\npair M = (1,0);\npair C = (2,0);\npair P = (2,0.5);\npair B = (2,1);\npair Q = (1,0.5);\n\ndraw(A--B--C--cycle);\ndraw(M--Q--P);\n\nlabel(\" $A$ \",A,SW);\nlabel(\" $M$ \",M,S);\nlabel(\" $C$ \",C,SE);\nlabel(\" $P$ \",P,E);\nlabel(\" $B$ \",B,NE);\nlabel(\" $N$ \",Q,NW);[/asy]",
        "answer": "$\\frac{109}{12}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 17,
        "problem": "A rectangle intersects a circle as shown such that $AB = 4$, $BC = 5$, and $DE = 3$. Find the length of $EF$.",
        "geo_code": "[asy]size(200);\ndefaultpen(linewidth(0.7)+fontsize(10));\npair D=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F);\ndraw(D--G--H--A--cycle);\ndraw(Circle(O, abs(O-C)));\nlabel(\" $A$ \", A, NW);\nlabel(\" $B$ \", B, NW);\nlabel(\" $C$ \", C, NE);\nlabel(\" $D$ \", D, SW);\nlabel(\" $E$ \", E, SE);\nlabel(\" $F$ \", F, SW);\n\nlabel(\"4\", (2,0.85), N);\nlabel(\"3\", D--E, S);\nlabel(\"5\", (6.5,0.85), N);\n[/asy]",
        "answer": "$7$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 18,
        "problem": "Two concentric circles have radii \\$1$ and \\$4$. Between them, six congruent circles are arranged in a ring so that each is tangent to its two neighboring circles. Of these six, three (shown lightly shaded) are tangent on the inside to the circle of radius \\$4$, and the other three (shown darkly shaded) are tangent on the outside to the circle of radius \\$1$. The radius of each of the six congruent circles can be expressed as $\\frac{k + \\sqrt{m}}{n}$, where $k$, $m$, and $n$ are integers with $k$ and $n$ relatively prime. Compute the value of $k + m + n$.",
        "geo_code": "[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\nreal r = (sqrt(133)-9)/2;\ndraw(circle(origin,1)^^circle(origin,4));\nfor(int i=0;i<=2;i=i+1)\n{\nfilldraw(circle(dir(90 + i*120)*(4-r),r),gray);\n}\nfor(int j=0;j<=2;j=j+1)\n{\nfilldraw(circle(dir(30+j*120)*(1+r),r),darkgray);\n}\n[/asy]",
        "answer": "$126$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 19,
        "problem": "If $\\angle A = 20^\\circ$ and $\\angle AFG = \\angle AGF$, what is the value of $\\angle B + \\angle D$?",
        "geo_code": "[asy]\npair A,B,C,D,EE,F,G;\nA = (0,0);\nB = (9,4);\nC = (21,0);\nD = (13,-12);\nEE = (4,-16);\nF = (13/2,-6);\nG = (8,0);\n\ndraw(A--C--EE--B--D--cycle);\n\nlabel(\" $A$ \",A,W);\nlabel(\" $B$ \",B,N);\nlabel(\" $C$ \",C,E);\nlabel(\" $D$ \",D,SE);\nlabel(\" $E$ \",EE,SW);\nlabel(\" $F$ \",F,WSW);\nlabel(\" $G$ \",G,NW);\n[/asy]",
        "answer": "$80$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 20,
        "problem": "The pattern in the figure below continues inward infinitely. The base of the largest triangle is \\$1$, and all triangles are equilateral. Find the total shaded area.",
        "geo_code": "[asy]\ndefaultpen(linewidth(0.8));\npen blu = rgb(0,112,191);\nreal r=sqrt(3);\nfill((8,0)--(0,8r)--(-8,0)--cycle, blu);\nfill(origin--(4,4r)--(-4,4r)--cycle, white);\nfill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);\nfill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]",
        "answer": "$\\frac{\\sqrt{3}}{5}$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 21,
        "problem": "Points $K$, $L$, $M$, and $N$ are located in the plane of square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are all equilateral triangles. If the area of $ABCD$ is \\$16$, find the area of quadrilateral $KLMN$.",
        "geo_code": "[asy]unitsize(2cm);\ndefaultpen(fontsize(8)+linewidth(0.8));\npair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);\npair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0);\ndraw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A);\nlabel(\" $A$ \",A,SE);\nlabel(\" $B$ \",B,SW);\nlabel(\" $C$ \",C,NW);\nlabel(\" $D$ \",D,NE);\nlabel(\" $K$ \",K,NNW);\nlabel(\" $L$ \",L,E);\nlabel(\" $M$ \",M,S);\nlabel(\" $N$ \",N,W);[/asy]",
        "answer": "$32 + 16\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 22,
        "problem": "The area of the dark gray triangle depicted below is  $35$ , and a segment is divided into lengths  $14$  and  $10$  as shown below. What is the area of the light gray triangle?",
        "geo_code": "[asy] \nsize(150);\nfilldraw((0,0)--(0,12)--(24,-60/7)--cycle, lightgray);\nfilldraw((14,0)--(14,5)--(0,12)--cycle, gray);\ndraw((0,0)--(24,0)--(0,12)--cycle);\ndraw((0,0)--(24,0)--(24,-60/7)--cycle);\ndraw((0,12)--(24,-60/7));\ndraw((14,5)--(14,0));\ndot((0,0));\ndot((0,12));\ndot((14,5));\ndot((24,0));\ndot((14,0));\ndot((24,-60/7));\nlabel(\" $14$ \", (7,0), S);\nlabel(\" $10$ \", (19,0), S);\ndraw((0,2/3)--(2/3,2/3)--(2/3,0));\ndraw((14,2/3)--(14+2/3,2/3)--(14+2/3,0));\ndraw((24-2/3,0)--(24-2/3,-2/3)--(24,-2/3));\n[/asy]",
        "answer": "$144$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 23,
        "problem": "The centers of the faces of the right rectangular prism shown below are connected to form an octahedron. What is the volume of this octahedron?",
        "geo_code": "[asy]\nimport three; size(2inch);\ncurrentprojection=orthographic(4,2,2);\ndraw((0,0,0)--(0,0,3),dashed);\ndraw((0,0,0)--(0,4,0),dashed);\ndraw((0,0,0)--(5,0,0),dashed);\ndraw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3));\ndraw((0,4,3)--(5,4,3)--(5,4,0));\nlabel(\"3\",(5,0,3)--(5,0,0),W);\nlabel(\"4\",(5,0,0)--(5,4,0),S);\nlabel(\"5\",(5,4,0)--(0,4,0),SE);\n[/asy]",
        "answer": "$10$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "volume"
    },
    {
        "index": 24,
        "problem": "Six cubes, each with edge length \\$1$ inch, are joined together as shown. Find the total surface area, in square inches, including the top, bottom, and sides.",
        "geo_code": "[asy]\ndraw((0,0)--(0,1)--(1,1)--(1,0)--cycle);\ndraw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));\ndraw((1,0)--(1.5,0.5)--(1.5,1.5));\ndraw((0.5,1.5)--(1,2)--(1.5,2));\ndraw((1.5,1.5)--(1.5,3.5)--(2,4)--(3,4)--(2.5,3.5)--(2.5,0.5)--(1.5,.5));\ndraw((1.5,3.5)--(2.5,3.5));\ndraw((1.5,1.5)--(3.5,1.5)--(3.5,2.5)--(1.5,2.5));\ndraw((3,4)--(3,3)--(2.5,2.5));\ndraw((3,3)--(4,3)--(4,2)--(3.5,1.5));\ndraw((4,3)--(3.5,2.5));\ndraw((2.5,.5)--(3,1)--(3,1.5));[/asy]",
        "answer": "$26$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 25,
        "problem": "Points $A$ and $B$ lie on a circle centered at $O$ such that $\\angle AOB$ is a right angle. Points $C$ and $D$ are located on radii $OA$ and $OB$, respectively, such that $AC = T - 3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.",
        "geo_code": "[asy]\ndraw(circle((0,0),10));\ndraw((0,10)--(0,0)--(10,0)--(0,10));\ndraw((0,3)--(4,0));\nlabel(\"O\",(0,0),SW);\nlabel(\"C\",(0,3),W);\nlabel(\"A\",(0,10),N);\nlabel(\"D\",(4,0),S);\nlabel(\"B\",(10,0),E);\n[/asy]",
        "answer": "$44$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 26,
        "problem": "In an isosceles right triangle $AOB$, points $P$, $Q$, and $S$ are chosen on sides $OB$, $OA$, and $AB$, respectively, so that a square $PQRS$ is constructed as shown. If $OP = a$ and $OQ = b$, and the area of $PQRS$ is $\\dfrac{2}{5}$ the area of triangle $AOB$, determine the ratio $a : b$.",
        "geo_code": "[asy]\n\npair A = (0,3);\npair B = (0,0);\npair C = (3,0);\npair D = (0,1.5);\npair E = (0.35,0);\npair F = (1.2,1.8);\npair J = (0.17,0);\npair Y = (0.17,0.75);\n\npair Z = (1.6,0.2);\ndraw(A--B);\ndraw(B--C);\ndraw(C--A);\ndraw(D--F--Z--E--D);\ndraw(\" $O$ \", B, dir(180));\ndraw(\" $B$ \", A, dir(45));\ndraw(\" $A$ \", C, dir(45));\ndraw(\" $Q$ \", E, dir(45));\ndraw(\" $P$ \", D, dir(45));\ndraw(\" $R$ \", Z, dir(45));\ndraw(\" $S$ \", F, dir(45));\ndraw(\" $a$ \", Y, dir(210));\ndraw(\" $b$ \", J, dir(100));\n[/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 27,
        "problem": "In circle $O$, let $Q$ be the midpoint of radius $OX$. At $Q$, $\\overline{AB} \\perp \\overline{XY}$. The semicircle with diameter $\\overline{AB}$ intersects $\\overline{XY}$ at $M$. Line $\\overline{AM}$ meets circle $O$ at $C$, and line $\\overline{BM}$ meets circle $O$ at $D$. Line $\\overline{AD}$ is drawn. If the radius of circle $O$ is \\$1$, what is the length of $AD$?",
        "geo_code": "[asy]defaultpen(linewidth(.8pt));\nunitsize(2.5cm);\n\nreal m = 0;\nreal b = 0;\n\npair O = origin;\npair X = (-1,0);\npair Y = (1,0);\npair Q = midpoint(O--X);\npair A = (Q.x, -1*sqrt(3)/2);\npair B = (Q.x, -1*A.y);\npair M = (Q.x + sqrt(3)/2,0);\n\nm = (B.y - M.y)/(B.x - M.x);\nb = (B.y - m*B.x);\n\npair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));\n\nm = (A.y - M.y)/(A.x - M.x);\nb = (A.y - m*A.x);\n\npair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));\n\ndraw(Circle(O,1));\ndraw(Arc(Q,sqrt(3)/2,-90,90));\ndraw(A--B);\ndraw(X--Y);\ndraw(B--D);\ndraw(A--C);\ndraw(A--D);\ndot(O);dot(M);\n\nlabel(\" $B$ \",B,NW);\nlabel(\" $C$ \",C,NE);\nlabel(\" $Y$ \",Y,E);\nlabel(\" $D$ \",D,SE);\nlabel(\" $A$ \",A,SW);\nlabel(\" $X$ \",X,W);\nlabel(\" $Q$ \",Q,SW);\nlabel(\" $O$ \",O,SW);\nlabel(\" $M$ \",M,NE+2N);[/asy]",
        "answer": "$\\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 28,
        "problem": "In $\\triangle ABC$, points $D$ and $E$ are located on $\\overline{BC}$ and $\\overline{AC}$, respectively. Suppose $\\overline{AD}$ and $\\overline{BE}$ intersect at $T$ such that $AT/DT = 3$ and $BT/ET = 4$. Find the value of $CD/BD$.",
        "geo_code": "[asy]unitsize(2cm);\ndefaultpen(linewidth(.8pt));\n\npair A = (0,0);\npair C = (2,0);\npair B = dir(57.5)*2;\npair E = waypoint(C--A,0.25);\npair D = waypoint(C--B,0.25);\npair T = intersectionpoint(D--A,E--B);\n\nlabel(\" $B$ \",B,NW);label(\" $A$ \",A,SW);label(\" $C$ \",C,SE);label(\" $D$ \",D,NE);label(\" $E$ \",E,S);label(\" $T$ \",T,2*W+N);\n\ndraw(A--B--C--cycle);\ndraw(A--D);\ndraw(B--E);[/asy]",
        "answer": "$\\frac{4}{11}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 29,
        "problem": "Two squares, each with side length \\$12$, are positioned so that one lies exactly on top of the other. One square is then rotated about a vertex by \\$30^\\circ$ with respect to the other. Determine the area of the region common to both squares.",
        "geo_code": "[asy]\nunitsize (2 cm);\n\npair A, B, C, D, Bp, Cp, Dp, P;\n\nA = (0,0);\nB = (-1,0);\nC = (-1,1);\nD = (0,1);\nBp = rotate(-30)*(B);\nCp = rotate(-30)*(C);\nDp = rotate(-30)*(D);\nP = extension(C, D, Bp, Cp);\n\nfill(A--Bp--P--D--cycle, gray(0.8));\ndraw(A--B--C--D--cycle);\ndraw(A--Bp--Cp--Dp--cycle);\n\nlabel(\" $30^\\circ$ \", (-0.5,0.1), fontsize(10));\n[/asy]",
        "answer": "$48\\sqrt{3}$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 30,
        "problem": "In the figure, $\\angle C = 90^\\circ$, $AD = DB$, $DE \\perp AB$, $AB = 20$, and $AC = 12$. What is the area of quadrilateral $ADEC$?",
        "geo_code": "[asy] unitsize(7); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E; A=(0,0); B=(20,0); C=(36/5,48/5); D=(10,0); E=(10,75/10); draw(A--B--C--cycle); draw(D--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,S); label(\"$E$\",E,NE); draw(rightanglemark(B,D,E,30)); [/asy]",
        "answer": "$58\\frac{1}{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 31,
        "problem": "Given the diagram, where $A$, $B$, and $C$ are the points of tangency, determine the value of $x$.",
        "geo_code": "[asy] unitsize(5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16); pair O=(0,3/8); draw((-2/3,9/16)--(2/3,9/16)); draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); draw(Circle(O,3/16)); draw((-2/3,0)--(2/3,0)); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$\\frac{3}{8}$\",O); draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3)); draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3)); label(\"$\\frac{1}{2}$\",(.5,.25)); draw((.5,.33)--(.5,.5),EndArrow(3)); draw((.5,.17)--(.5,0),EndArrow(3)); label(\"$x$\",midpoint((.5,.5)--(.5,9/16))); draw((.5,5/8)--(.5,9/16),EndArrow(3)); label(\"$60^{\\circ}$\",(0.01,0.12)); dot(A); dot(B); dot(C);[/asy]",
        "answer": "$\\frac{1}{16}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 32,
        "problem": "Point $F$ is taken on side $AD$ of square $ABCD$. At $C$, a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is \\$256$ square inches and the area of $\\triangle CEF$ is \\$200$ square inches. Find the number of inches in $BE$.",
        "geo_code": "[asy] size(6cm); pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7); draw(A--B--C--D--cycle); draw(F--C--E--B); label(\"$A$\", A, SW); label(\"$B$\", B, S); label(\"$C$\", C, N); label(\"$D$\", D, NW); label(\"$E$\", E, SE); label(\"$F$\", F, W); [/asy]",
        "answer": "$12$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 33,
        "problem": "Inside square $ABCD$ (see figure) with side length \\$12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ that is \\$5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and meets $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$, respectively. Find $\\dfrac{PM}{MQ}$.",
        "geo_code": "[asy] draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2)); label(\"D\",(0,1),NW);label(\"E\",(.4,1),N);label(\"C\",(1,1),NE); label(\"P\",(0,.6),W);label(\"M\",(.25,.55),E);label(\"Q\",(1,.2),E); label(\"A\",(0,0),SW);label(\"B\",(1,0),SE); [/asy]",
        "answer": "$\\frac{5}{19}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 34,
        "problem": "In parallelogram $ABCD$ shown in the diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ intersect at $O$. If the area of parallelogram $ABCD$ is \\$1$, then the area of triangle $QPO$ is equal to",
        "geo_code": "[asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label(\"A\", (0,0), S); label(\"B\", (5,0), S); label(\"C\", (6,3), NE); label(\"D\", (1,3), NW); label(\"P\", (10,0), E); label(\"Q\", (-5,0), W); label(\"M\", (.5,1.5), NW); label(\"N\", (5.65, 1.5), NE); label(\"O\", (3.4,1.75));[/asy]",
        "answer": "$\\frac{9}{8}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 35,
        "problem": "In $\\triangle ABC$ shown in the figure, $M$ is the midpoint of side $BC$, $AB = 12$, and $AC = 16$. Points $E$ and $F$ are located on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE = 2AF$, then $\\dfrac{EG}{GF}$ equals",
        "geo_code": "[asy] draw((0,0)--(12,0)--(14,7.75)--(0,0)); draw((0,0)--(13,3.875)); draw((5,0)--(8.75,4.84)); label(\"A\", (0,0), S); label(\"B\", (12,0), S); label(\"C\", (14,7.75), E); label(\"E\", (8.75,4.84), N); label(\"F\", (5,0), S); label(\"M\", (13,3.875), E); label(\"G\", (7,1)); [/asy]",
        "answer": "$\\frac{3}{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 36,
        "problem": "In the figure shown, $\\measuredangle E = 40^\\circ$ and arcs $AB$, $BC$, and $CD$ all have equal length. Find the measure of $\\measuredangle ACD$.",
        "geo_code": "[asy] size(120); path c = Circle((0, 0), 1); pair A = dir(20), B = dir(130), C = dir(240), D = dir(330); draw(c); pair F = 3(A-B) + B; pair G = 3(D-C) + C; pair E = intersectionpoints(B--F, C--G)[0]; draw(B--E--C--A); label(\"$A$\", A, NE); label(\"$B$\", B, NW); label(\"$C$\", C, SW); label(\"$D$\", D, SE); label(\"$E$\", E, E); [/asy]",
        "answer": "$15$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 37,
        "problem": "Each of the three circles in the figure shown is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius $3$, what is the perimeter of the triangle?",
        "geo_code": "[asy] size(120); real t = 2/sqrt(3); real x = 1 + sqrt(3); pair A = t*dir(90), D = x*A; pair B = t*dir(210), E = x*B; pair C = t*dir(330), F = x*C; draw(D--E--F--cycle); draw(Circle(A, 1)); draw(Circle(B, 1)); draw(Circle(C, 1)); [/asy]",
        "answer": "$18+18\\sqrt{3}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 38,
        "problem": "In $\triangle ABC$, $AB = 10$, $AC = 8$, and $BC = 6$. Circle $P$ is the circle with the smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$, of circle $P$ with sides $AC$ and $BC$, respectively. What is the length of segment $QR$?",
        "geo_code": "[asy] size(100); real a=4, b=3; // import cse5; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP(\"A\",A)--MP(\"B\",B,N)--MP(\"C\",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP(\"R\",IP(CP((X+C)/2,C),B--C),NW)--MP(\"Q\",IP(CP((X+C)/2,C),A--C+(0.1,0)))); [/asy]",
        "answer": "$4.8$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 39,
        "problem": "In the figure below, let $ABCDE$ be a regular pentagon such that $AG = 1$. Find the value of $FG + JH + CD$.",
        "geo_code": "[asy]\nimport cse5;pathpen=black;pointpen=black;\nsize(2inch);\npair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162);\nD(MP(\"A\",A,A)--MP(\"B\",B,B)--MP(\"C\",C,C)--MP(\"D\",D,D)--MP(\"E\",E,E)--cycle,linewidth(1.5));\nD(A--C--E--B--D--cycle);\npair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A);\nD(MP(\"F\",F,dir(126))--MP(\"I\",I,dir(270))--MP(\"G\",G,dir(54))--MP(\"J\",J,dir(198))--MP(\"H\",H,dir(342))--cycle);\n[/asy]",
        "answer": "$1 + \\sqrt{5}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 40,
        "problem": "The diagram below shows four regular hexagons, each with side length \\$1$ meter, attached to the sides of a square. This shape is drawn onto a thin sheet of metal and cut out. The hexagons are then folded upward along the sides of the square so that $A_1$ meets $A_2$, $B_1$ meets $B_2$, $C_1$ meets $C_2$, and $D_1$ meets $D_2$. If water is poured into the resulting dish, it fills to the height of the corner where $A_1$ and $A_2$ meet. There exist relatively prime positive integers $m$ and $n$ such that the number of cubic meters of water the dish will hold is $\\sqrt{\\frac{m}{n}}$. Find $m+n$.\n\n",
        "geo_code": "[asy]\nimport graph; size(7cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = -4.3, xmax = 14.52, ymin = -8.3, ymax = 6.3; \ndraw((0,1)--(0,0)--(1,0)--(1,1)--cycle); \ndraw((1,1)--(1,0)--(1.87,-0.5)--(2.73,0)--(2.73,1)--(1.87,1.5)--cycle); \ndraw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle); \ndraw((0,0)--(1,0)--(1.5,-0.87)--(1,-1.73)--(0,-1.73)--(-0.5,-0.87)--cycle); \ndraw((0,1)--(0,0)--(-0.87,-0.5)--(-1.73,0)--(-1.73,1)--(-0.87,1.5)--cycle); \n\ndraw((0,1)--(0,0)); \ndraw((0,0)--(1,0)); \ndraw((1,0)--(1,1)); \ndraw((1,1)--(0,1)); \ndraw((1,1)--(1,0)); \ndraw((1,0)--(1.87,-0.5)); \ndraw((1.87,-0.5)--(2.73,0)); \ndraw((2.73,0)--(2.73,1)); \ndraw((2.73,1)--(1.87,1.5)); \ndraw((1.87,1.5)--(1,1)); \ndraw((0,1)--(1,1)); \ndraw((1,1)--(1.5,1.87)); \ndraw((1.5,1.87)--(1,2.73)); \ndraw((1,2.73)--(0,2.73)); \ndraw((0,2.73)--(-0.5,1.87)); \ndraw((-0.5,1.87)--(0,1)); \n\ndot((1.87,-0.5),dotstyle); \nlabel(\" $C_1$ \", (1.72,-0.1), NE * labelscalefactor); \ndot((1.87,1.5),dotstyle); \nlabel(\" $B_2$ \", (1.76,1.04), NE * labelscalefactor); \ndot((1.5,1.87),dotstyle); \nlabel(\" $B_1$ \", (0.96,1.8), NE * labelscalefactor); \ndot((-0.5,1.87),dotstyle); \nlabel(\" $A_2$ \", (-0.26,1.78), NE * labelscalefactor); \ndot((-0.87,1.5),dotstyle); \nlabel(\" $A_1$ \", (-0.96,1.08), NE * labelscalefactor); \ndot((-0.87,-0.5),dotstyle); \nlabel(\" $D_2$ \", (-1.02,-0.18), NE * labelscalefactor); \ndot((-0.5,-0.87),dotstyle); \nlabel(\" $D_1$ \", (-0.22,-0.96), NE * labelscalefactor); \ndot((1.5,-0.87),dotstyle); \nlabel(\" $C_2$ \", (0.9,-0.94), NE * labelscalefactor); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n[/asy]",
        "answer": "$67$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 41,
        "problem": "In the figure below, $E$ is the midpoint of arc $ABEC$ and segment $ED$ is perpendicular to chord $BC$ at $D$. If the length of chord $AB$ is \\$5$ and the length of segment $BD$ is \\$3$, find the length of $DC$.\n\n",
        "geo_code": "[asy]\nunitsize(1 cm);\npair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C);\ndraw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8));\nlabel(\" $A$ \",A,.5A); label(\" $B$ \",B,.5B); label(\" $C$ \",C,.5C); label(\" $E$ \",E,.5E); label(\" $D$ \",D,dir(-60));\n[/asy]",
        "answer": "$8$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 42,
        "problem": "What is the radius of the smallest circle that can contain the symmetric figure formed by the \\$3$ unit squares shown above?\n\n",
        "geo_code": "[asy]\ndraw((0,0)--(0,1)--(2,1)--(2,0)--cycle^^(.5,1)--(.5,2)--(1.5,2)--(1.5,1)--(.5,2)^^(.5,1)--(1.5,2)^^(1,2)--(1,0));\n[/asy]",
        "answer": "$\\frac{5\\sqrt{17}}{16} $",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 43,
        "problem": "In the figure below, $\\triangle ABC$, $\\triangle DEF$, and $\\triangle GHI$ are overlapping equilateral triangles. Points $C$ and $F$ lie on $\\overline{BD}$, $F$ and $I$ lie on $\\overline{EG}$, and $C$ and $I$ lie on $\\overline{AH}$. Suppose $AB = 2FC$, $DE = 3FC$, and $GH = 4FC$. Given that the area of $\\triangle FCI$ is \\$3$, find the area of hexagon $ABGHDE$.\n\n",
        "geo_code": "[asy]\nsize(5cm);\npen dps = fontsize(10);\ndefaultpen(dps);\npair A,B,C,D,E,F,G,H,I;\nG=origin;\nH=(4,0);\nI=(2,2*sqrt(3));\nF=(1.5,3*sqrt(3)/2);\nC=F+(1,0);\nB=F-(1,0);\nD=C+(2,0);\nA=F+(0,sqrt(3));\nE=C+(0.5,3*sqrt(3)/2);\ndraw(A--H--G--E--D--B--cycle);\nlabel(\" $A$ \",A,N*.5);\nlabel(\" $B$ \",B,S*.5);\nlabel(\" $C$ \",C,SW*.5);\nlabel(\" $D$ \",D,S*.5);\nlabel(\" $E$ \",E,N*.5);\nlabel(\" $F$ \",F,SE*.5);\nlabel(\" $G$ \",G,S*.5);\nlabel(\" $H$ \",H,S*.5);\nlabel(\" $I$ \",I,N*2);\n[/asy]",
        "answer": "$114$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 44,
        "problem": "If the sum of the measures in degrees of angles $A$, $B$, $C$, $D$, $E$, and $F$ in the diagram above is \\$90n$, what is the value of $n$?\n\n",
        "geo_code": "[asy]\ndraw((3,-13)--(21.5,-5)--(19,-18)--(9,-18)--(10,-6)--(23,-14.5)--cycle);\nlabel(\"A\",(3,-13),W);label(\"C\",(21.5,-5),N);label(\"E\",(19,-18),E);label(\"F\",(9,-18),W);label(\"B\",(10,-6),N);label(\"D\",(23,-14.5),E);\n[/asy]",
        "answer": "$4$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 45,
        "problem": "In the figure, circle $K$ has diameter $AB$. Circle $L$ is tangent to circle $K$ and to $AB$ at the center of circle $K$. Circle $M$ is tangent to circle $K$, to circle $L$, and to $AB$. What is the ratio of the area of circle $K$ to the area of circle $M$?\n\n",
        "geo_code": "[asy]\nsize(150);\npair A=(0,0),B=(1,0),C=(0,1),D=(-1,0),E=(0,.5),F=(sqrt(2)/2,.25);\ndraw(circle(A,1)^^D--B);\ndraw(circle(E,.5)^^circle( F ,.25));\nlabel(\" $A$ \", D, W);\nlabel(\" $K$ \", A, S);\nlabel(\" $B$ \", B, dir(0));\nlabel(\" $L$ \", E, N);\nlabel(\" $M$ \",shift(-.05,.05)*F);\n[/asy]",
        "answer": "$16$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 46,
        "problem": "Pentagon $ABCDE$ consists of a square $ACDE$ and an equilateral triangle $ABC$ sharing the side $\\overline{AC}$. A circle centered at $C$ has area \\$24$. The intersection of the circle and the pentagon is exactly half the area of the pentagon. Find the area of the pentagon.\n\n",
        "geo_code": "[asy]\nimport graph; size(4.26cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = -1.52, xmax = 2.74, ymin = -2.18, ymax = 6.72;\ndraw((0,1)--(2,1)--(2,3)--(0,3)--cycle); \ndraw((0,3)--(2,3)--(1,4.73)--cycle); \ndraw((0,1)--(2,1)); \ndraw((2,1)--(2,3)); \ndraw((2,3)--(0,3)); \ndraw((0,3)--(0,1)); \ndraw((0,3)--(2,3)); \ndraw((2,3)--(1,4.73)); \ndraw((1,4.73)--(0,3)); \ndraw(circle((0,3), 1.44)); \nlabel(\" $C$ \",(-0.4,3.14),SE*labelscalefactor); \nlabel(\" $A$ \",(2.1,3.1),SE*labelscalefactor); \nlabel(\" $B$ \",(0.86,5.18),SE*labelscalefactor); \nlabel(\" $D$ \",(-0.28,0.88),SE*labelscalefactor); \nlabel(\" $E$ \",(2.1,0.8),SE*labelscalefactor); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]",
        "answer": "$20$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 47,
        "problem": "In the figure, $AB$ and $BC$ are adjacent sides of square $ABCD$. Let $M$ be the midpoint of $AB$, and $N$ the midpoint of $BC$. The lines $AN$ and $CM$ intersect at $O$. What is the ratio of the area of $AOCD$ to the area of $ABCD$?",
        "geo_code": "[asy]\ndraw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0));\nlabel(\"A\", (0,0), S);\nlabel(\"B\", (2,0), S);\nlabel(\"C\", (2,2), N);\nlabel(\"D\", (0,2), N);\nlabel(\"M\", (1,0), S);\nlabel(\"N\", (2,1), E);\nlabel(\"O\", (1.2, .8));\n[/asy]",
        "answer": "$\\frac{2}{3}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 48,
        "problem": "Calculate  $\\frac{area(CDF)}{area(CEF)}$  in the figure.\n\n",
        "geo_code": "[asy]\nimport graph; size(5.75cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = -2, xmax = 21, ymin = -2, ymax = 16;  \ndraw((0,0)--(20,0)); \ndraw((13.48,14.62)--(7,0)); \ndraw((0,0)--(15.93,9.12)); \ndraw((13.48,14.62)--(20,0)); \ndraw((13.48,14.62)--(0,0)); \nlabel(\"6\",(15.16,12.72),SE*labelscalefactor); \nlabel(\"10\",(18.56,5.1),SE*labelscalefactor); \nlabel(\"7\",(3.26,-0.6),SE*labelscalefactor); \nlabel(\"13\",(13.18,-0.71),SE*labelscalefactor); \nlabel(\"20\",(5.07,8.33),SE*labelscalefactor); \ndot((0,0),dotstyle); \nlabel(\" $B$ \", (-1.23,-1.48), NE * labelscalefactor); \ndot((20,0),dotstyle); \nlabel(\" $C$ \", (19.71,-1.59), NE * labelscalefactor); \ndot((7,0),dotstyle); \nlabel(\" $D$ \", (6.77,-1.64), NE * labelscalefactor); \ndot((13.48,14.62),dotstyle); \nlabel(\" $A$ \", (12.36,14.91), NE * labelscalefactor); \ndot((15.93,9.12),dotstyle); \nlabel(\" $E$ \", (16.42,9.21), NE * labelscalefactor); \ndot((9.38,5.37),dotstyle); \nlabel(\" $F$ \", (9.68,4.5), NE * labelscalefactor); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n[/asy]",
        "answer": "$\\frac{14}{15}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 49,
        "problem": "In the figure, a circle intersects the sides of an equilateral triangle at six points. Given that $AG = 2$, $GF = 13$, $FC = 1$, and $HJ = 7$, find the length of $DE$.\n\n",
        "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=sqrt(22);\npair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J);\ndot(A^^B^^C^^D^^E^^F^^G^^H^^J);\ndraw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7));\nlabel(\" $A$ \", A, N);\nlabel(\" $B$ \", B, dir(210));\nlabel(\" $C$ \", C, dir(330));\nlabel(\" $D$ \", D, SW);\nlabel(\" $E$ \", E, SE);\nlabel(\" $F$ \", F, dir(170));\nlabel(\" $G$ \", G, dir(250));\nlabel(\" $H$ \", H, SE);\nlabel(\" $J$ \", J, dir(0));\nlabel(\"2\", A--G, dir(30));\nlabel(\"13\", F--G, dir(180+30));\nlabel(\"1\", F--C, dir(30));\nlabel(\"7\", H--J, dir(-30));[/asy]",
        "answer": "$2\\sqrt{22}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 50,
        "problem": "In the diagram below, points $A$, $B$, and $C$ are each \\$6$ units apart. Define a point $X$ to be \\emph{reachable} if there exists a path (not necessarily straight) from $A$ to $X$ with length at most \\$8$ that does not pass through the interior of segment $\\overline{BC}$. (Both $X$ and the path must lie in the plane determined by $A$, $B$, and $C$.) Let $R$ denote the set of all reachable points. What is the area of $R$?\n\n",
        "geo_code": "[asy]\n  unitsize(40);\n  pair A = dir(90);\n  pair B = dir(210);\n  pair C = dir(330);\n  dot(A); \n  dot(B);\n  dot(C);\n  draw(B -- C);\n  label(\" $A$ \", A, N);\n  label(\" $B$ \", B, W);\n  label(\" $C$ \", C, E);\n[/asy]",
        "answer": "$56\\pi+9\\sqrt{3}$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 51,
        "problem": "A net is shown below consisting of \\$3$ squares, \\$4$ equilateral triangles, and \\$1$ regular hexagon, each with side length \\$1$. If this net is folded to create a polyhedron, what is the volume of the resulting solid?\n\nNet:\n\n\n\n",
        "geo_code": "[asy]\npair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P;\nA = origin;\nB = (0,3);\nC = 3*dir(150);\nD = (0,1);\nE = (0,2);\nF = C+2*dir(30);\nG = C+dir(30);\nH = 2*dir(150);\nI = dir(150);\nJ = (1,1);\nK = J+dir(30);\nL = (1,2);\nM = F+dir(120);\nN = G+dir(120);\nO = H+dir(240);\nP = I+dir(240);\ndraw(A--B--C--cycle);\ndraw(D--E--F--G--H--I--cycle);\ndraw(D--E--L--J--cycle);\ndraw(F--G--N--M--cycle);\ndraw(H--I--P--O--cycle);\ndraw(J--K--L--cycle);\n[/asy]",
        "answer": "$\\frac{5}{3\\sqrt{2}}$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "volume"
    },
    {
        "index": 52,
        "problem": "In the diagram, $BC$ is the diameter of a circle with $BC = \\sqrt{257}$, $BD = 1$, and $DA = 12$. Determine the length of the altitude from $A$ to $BC$.\n\n",
        "geo_code": "[asy]\nimport cse5;\nsize(200);\npair O=(2, 0), B=(0, 0), C=(4, 0), A=(1, 3), D, E;\nD=MP(\"D\",D(IP(D(CP(O,B)),D(MP(\"A\",D(A),N)--MP(\"B\",D(B),W)))),NW);\nE=MP(\"E\",D(IP(CP(O,B),D(MP(\"C\",D(C),NE)--A),1)),NE);\nD(B--C);\n[/asy]",
        "answer": "$7.18$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 53,
        "problem": "In the figure to the right, every point on circle $O'$ lies outside of circle $O$. Let $P$ and $Q$ be the intersection points of an internal common tangent and the two external common tangents. If the length of each external common tangent is \\$6$, what is the length of $PQ$?\n\n",
        "geo_code": "[asy]\nsize(150);\ndotfactor=4;\ndraw(circle((0,0),4));\ndraw(circle((10,-6),3));\npair O,A,P,Q;\nO = (0,0);\nA = (10,-6);\nP = (-.55, -4.12);\nQ = (10.7, -2.86);\ndot(\" $O$ \", O, NE);\ndot(\" $O'$ \", A, SW);\ndot(\" $P$ \", P, SW);\ndot(\" $Q$ \", Q, NE);\ndraw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle);\ndraw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle);\ndraw(P--Q--cycle);\n[/asy]",
        "answer": "$6$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 54,
        "problem": "In the figure below, each vertex of a unit square is connected by a line segment to the midpoint of the opposite side. Determine the area of the quadrilateral $ABCD$ formed by the intersection of these segments.\n\n",
        "geo_code": "[asy]\ndraw((0,0)--(1,0)--(1,1)--(0,1)--cycle);\ndraw((0,0)--(1,0.5));\ndraw((1,0)--(0.5,1));\ndraw((1,1)--(0,0.5));\ndraw((0,1)--(0.5,0));\nlabel(\" $A$ \",(0.21,0.6),N);\nlabel(\" $B$ \",(0.4,0.21),W);\nlabel(\" $C$ \",(0.8,0.4),S);\nlabel(\" $D$ \",(0.6,0.8),E);\n[/asy]",
        "answer": "$\\frac{1}{5}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 55,
        "problem": "The diagram below shows two parallel lines, $\\ell$ and $m$, which are \\$12$ units apart. A circle is tangent to line $\\ell$ at point $A$, and a congruent circle is tangent to line $m$ at point $B$. The two circles are also tangent to each other. If the distance between $A$ and $B$ is \\$13$, what is the radius of each circle?",
        "geo_code": "[asy]unitsize(7);\n  draw((-7, 0) -- (12, 0));\n  draw((-7, 12) -- (12, 12));\n  real r = 169 / 48;\n  draw(circle((0, r), r));\n  draw(circle((5, 12 - r), r));\n  pair A = (0, 0);\n  pair B = (5, 12);\n  dot(A);\n  dot(B);\n  label(\" $A$ \", A, plain.S);\n  label(\" $B$ \", B, plain.N);\n  label(\" $\\ell$ \", (12, 0), plain.E);\n  label(\" $m$ \", (12, 12), plain.E);[/asy]",
        "answer": "$\\frac{169}{48}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 56,
        "problem": "The figure below shows two congruent right triangles, each with angles \\$40^\\circ$, \\$50^\\circ$, and \\$90^\\circ$. What is the measure of the obtuse angle $\\alpha$ created by the intersection of their hypotenuses?\n\n",
        "geo_code": "[asy]\nimport olympiad;\n\tsize(80);\ndefaultpen(linewidth(0.8));\ndraw((0,0)--(3,0)--(0,4.25)--(0,0)^^(0,3)--(4.25,0)--(3,0)^^rightanglemark((0,3),(0,0),(3,0),10));\npair P = intersectionpoint((3,0)--(0,4.25),(0,3)--(4.25,0));\ndraw(anglemark((4.25,0),P,(0,4.25),10));\nlabel(\" $\\alpha$ \",P,2 * NE);\n[/asy]",
        "answer": "$170$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 57,
        "problem": "In rectangle $ADEH$, points $B$ and $C$ divide $\\overline{AD}$ into three equal segments, and points $G$ and $F$ divide $\\overline{HE}$ into three equal parts. Additionally, $AH = AC = 2$. What is the area of quadrilateral $WXYZ$ shown in the diagram?",
        "geo_code": "[asy]defaultpen(linewidth(0.7));pointpen=black; pathpen=black;\nsize(7cm);\npair A,B,C,D,E,F,G,H,W,X,Y,Z;\nA=(0,2); B=(1,2); C=(2,2); D=(3,2);\nH=(0,0); G=(1,0); F=(2,0); E=(3,0);\nD('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,NE); D('F',F,NE); D('G',G,NW); D('H',H,NW);\nD(A--F); D(B--E); D(D--G); D(C--H);\nZ=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H);\nD('W',W,N); D('X',X,plain.E); D('Y',Y,S); D('Z',Z,plain.W);\nD(A--D--E--H--cycle);[/asy]",
        "answer": "$\\frac{1}{2}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 58,
        "problem": "In the figure, BC\u00a0\u00a0is a diameter of the circle, where  $BC=\\sqrt{901}, BD=1$ , and  $DA=16$ . If  $EC=x$ , what is the value of x?\n\n\n",
        "geo_code": "[asy]size(2inch);\npair O,A,B,C,D,E;\nB=(0,0);\nO=(2,0);\nC=(4,0);\nD=(.333,1.333);\nA=(.75,2.67);\nE=(1.8,2);\ndraw(Arc(O,2,0,360));\ndraw(B--C--A--B);\nlabel(\" $A$ \",A,N);\nlabel(\" $B$ \",B,W);\nlabel(\" $C$ \",C,E);\nlabel(\" $D$ \",D,W);\nlabel(\" $E$ \",E,N);\nlabel(\"Figure not drawn to scale\",(2,-2.5),S);\n[/asy]",
        "answer": "$26$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 59,
        "problem": "Region $ABCDEFGHIJ$ is composed of 13 congruent squares and is inscribed within rectangle $PQRS$ such that $A$ lies on $\\overline{PQ}$, $B$ on $\\overline{QR}$, $E$ on $\\overline{RS}$, and $H$ on $\\overline{SP}$, as shown in the figure to the right. If $PQ = 28$ and $QR = 26$, find the area of region $ABCDEFGHIJ$.\n\n",
        "geo_code": "[asy]\nsize(200);\ndefaultpen(linewidth(0.7)+fontsize(12)); pair P=(0,0), Q=(0,28), R=(26,28), S=(26,0), B=(3,28);\ndraw(P--Q--R--S--cycle);\npicture p = new picture;\ndraw(p, (0,0)--(3,0)^^(0,-1)--(3,-1)^^(0,-2)--(5,-2)^^(0,-3)--(5,-3)^^(2,-4)--(3,-4)^^(2,-5)--(3,-5));\ndraw(p, (0,0)--(0,-3)^^(1,0)--(1,-3)^^(2,0)--(2,-5)^^(3,0)--(3,-5)^^(4,-2)--(4,-3)^^(5,-2)--(5,-3));\ntransform t = shift(B) * rotate(-aSin(1/26^.5)) * scale(26^.5);\nadd(t*p);\nlabel(\" $P$ \",P,SW); label(\" $Q$ \",Q,NW); label(\" $R$ \",R,NE); label(\" $S$ \",S,SE); label(\" $A$ \",t*(0,-3),W); label(\" $B$ \",B,N); label(\" $C$ \",t*(3,0),plain.ENE); label(\" $D$ \",t*(3,-2),NE); label(\" $E$ \",t*(5,-2),plain.E); label(\" $F$ \",t*(5,-3),plain.SW); label(\" $G$ \",t*(3,-3),(0.81,-1.3)); label(\" $H$ \",t*(3,-5),plain.S); label(\" $I$ \",t*(2,-5),NW); label(\" $J$ \",t*(2,-3),SW);[/asy]",
        "answer": "$338$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 60,
        "problem": "In the figure, $\\overline{CE}$ and $\\overline{DE}$ are equal chords of a circle with center $O$. The arc $AB$ represents a quarter of the circle. By what factor is the area of triangle $CED$ greater than the area of triangle $AOB$?",
        "geo_code": "[asy]defaultpen(linewidth(.8pt));\nunitsize(2cm);\n\npair O = origin;\npair C = (-1,0);\npair D = (1,0);\npair E = (0,1);\npair A = dir(-135);\npair B = dir(-60);\n\ndraw(Circle(O,1));\ndraw(C--E--D--cycle);\ndraw(A--O--B--cycle);\n\nlabel(\" $A$ \",A,SW);\nlabel(\" $C$ \",C,W);\nlabel(\" $E$ \",E,N);\nlabel(\" $D$ \",D,NE);\nlabel(\" $B$ \",B,SE);\nlabel(\" $O$ \",O,N);[/asy]",
        "answer": "$2$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 61,
        "problem": "In the given diagram, suppose that points $A$, $B$, and $C$ are points of tangency. Find the value of $x$.",
        "geo_code": "[asy]unitsize(5cm);\ndefaultpen(linewidth(.8pt)+fontsize(8pt));\ndotfactor=3;\n\npair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);\npair O=(0,3/8);\n\ndraw((-2/3,9/16)--(2/3,9/16));\ndraw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2));\ndraw(Circle(O,3/16));\ndraw((-2/3,0)--(2/3,0));\n\nlabel(\" $A$ \",A,SW);\nlabel(\" $B$ \",B,SE);\nlabel(\" $C$ \",C,N);\nlabel(\" $\\frac{3}{8}$ \",O);\ndraw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3));\ndraw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3));\nlabel(\" $\\frac{1}{2}$ \",(.5,.25));\ndraw((.5,.33)--(.5,.5),EndArrow(3));\ndraw((.5,.17)--(.5,0),EndArrow(3));\nlabel(\" $x$ \",midpoint((.5,.5)--(.5,9/16)));\ndraw((.5,5/8)--(.5,9/16),EndArrow(3));\nlabel(\" $60^{\\circ}$ \",(0.01,0.12));\ndot(A);\ndot(B);\ndot(C);[/asy]",
        "answer": "$\\frac{1}{16}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 62,
        "problem": "Unit square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$, where $EF$ is parallel to $AB$ and $GH$ is parallel to $BC$. Given that $BF = \\frac{1}{4}$ and point $P$ is the intersection of $EF$ and $GH$, suppose $BF + DH = FH$. Find the nearest integer to the measure, in degrees, of angle $\\angle FAH$.\n\n",
        "geo_code": "[asy]\nsize(100); defaultpen(linewidth(0.7)+fontsize(10)); \npair D2(pair P) {\ndot(P,linewidth(3)); return P;\n}\npair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];\ndraw(A--B--C--D--cycle);\ndraw(F--A--H); draw(E--F); draw(G--H); \nlabel(\" $A$ \",D2(A),NW);\nlabel(\" $B$ \",D2(B),SW);\nlabel(\" $C$ \",D2(C),SE);\nlabel(\" $D$ \",D2(D),NE);\nlabel(\" $E$ \",D2(E),plain.N);\nlabel(\" $F$ \",D2(F),S);\nlabel(\" $G$ \",D2(G),W);\nlabel(\" $H$ \",D2(H),plain.E);\nlabel(\" $P$ \",D2(P),SE);\n[/asy]",
        "answer": "$45$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 63,
        "problem": "In the figure below, square $ABDE$ is shown along with certain angles and lengths. The length $\\overline{CF}$ can be expressed in the form $a\\sqrt{b}$, where $a$ and $b$ are integers, and $b$ is not divisible by the square of any integer greater than \\$1$. Find the value of $a + b$.\n\n\n",
        "geo_code": "[asy]\nmarkscalefactor=0.15;\nsize(8cm);\npair A = (0,0);\npair B = (17,0);\npair E = (0,17);\npair D = (17,17);\npair F = (-120/17,225/17);\npair C = (17+120/17, 64/17);\ndraw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle);  \nlabel(\" $A$ \", A, S);\nlabel(\" $B$ \", B, S);\nlabel(\" $C$ \", C, dir(0));\nlabel(\" $D$ \", D, N);\nlabel(\" $E$ \", E, N);\nlabel(\" $F$ \", F, W);\nlabel(\" $8$ \", (F+E)/2, NW);\nlabel(\" $15$ \", (F+A)/2, SW);\nlabel(\" $8$ \", (C+B)/2, SE);\nlabel(\" $15$ \", (D+C)/2, NE);\ndraw(rightanglemark(E,F,A));\ndraw(rightanglemark(D,C,B));\n[/asy]",
        "answer": "$25$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 64,
        "problem": "In the figure below, $\\angle CAB$, $\\angle CBD$, and $\\angle CDE$ are all right angles, and the given side lengths are $AC = 3$, $BC = 5$, $BD = 12$, and $DE = 84$. The distance from point $E$ to line $AB$ can be written as a reduced fraction $\\frac{m}{n}$, where $m$ and $n$ are positive integers with no common factors. Find $m+n$.\n\n",
        "geo_code": "[asy]\nsize(300);\ndefaultpen(linewidth(0.8));\ndraw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8));\nlabel(\" $A$ \",origin,SW);\nlabel(\" $B$ \",(0,4),dir(160));\nlabel(\" $C$ \",(3,0),S);\nlabel(\" $D$ \",(6,8),dir(80));\nlabel(\" $E$ \",(30,-4),E);[/asy]",
        "answer": "$5393$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 65,
        "problem": "Let $O$ be the centroid of an equilateral triangle $ABC$ with area $\\frac{1}{\\pi}$. As illustrated in the diagram below, a circle centered at $O$ intersects the triangle at points $D$, $E$, $F$, $G$, $H$, and $I$, which trisect the sides of the triangle. Find the total area of the six shaded regions.\n\n\n\n\n",
        "geo_code": "[asy]\nunitsize(90);\npair A = dir(0);\npair B = dir(120);\npair C = dir(240);\ndraw(A -- B -- C -- cycle);\npair D = (2*A + B)/3;\npair E = (A + 2*B)/3;\npair F = (2*B + C)/3;\npair G = (B + 2*C)/3;\npair H = (2*C + A)/3;\npair I = (C + 2*A)/3;\ndraw(E -- F);\ndraw(G -- H);\ndraw(I -- D);\ndraw(D -- G);\ndraw(E -- H);\ndraw(F -- I);\npair O = (0, 0);\nreal r = 1/sqrt(3);\ndraw(circle(O, r));\nfill(O -- D -- E -- cycle, gray);\nfill(O -- F -- G -- cycle, gray);\nfill(O -- H -- I -- cycle, gray);\nfill(arc(O, r, -30, 30) -- cycle, gray);\nfill(arc(0, r, 90, 150) -- cycle, gray);\nfill(arc(0, r, 210, 270) -- cycle, gray);\nlabel(\" $A$ \", A, A);\nlabel(\" $B$ \", B, B);\nlabel(\" $C$ \", C, C);\nlabel(\" $D$ \", D, unit(D));\nlabel(\" $E$ \", E, unit(E));\nlabel(\" $F$ \", F, unit(F));\nlabel(\" $G$ \", G, unit(G));\nlabel(\" $H$ \", H, unit(H));\nlabel(\" $I$ \", I, unit(I));\nlabel(\" $O$ \", O, C);\n[/asy]",
        "answer": "$\\frac{2\\sqrt{3}}{27}$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 66,
        "problem": "In the diagram, $ABCD$ and $EFGH$ are similar rectangles. Given that $DK:KC = 3:2$, by what factor is the area of rectangle $ABCD$ greater than the area of rectangle $EFGH$?\n\n",
        "geo_code": "[asy]draw((75,0)--(0,0)--(0,50)--(75,50)--(75,0)--(55,0)--(55,20)--(100,20)--(100,0)--cycle);\ndraw((55,5)--(60,5)--(60,0));\ndraw((75,5)--(80,5)--(80,0));\nlabel(\"A\",(0,50),NW);\nlabel(\"B\",(0,0),SW);\nlabel(\"C\",(75,0),SE);\nlabel(\"D\",(75,50),NE);\nlabel(\"E\",(55,20),NW);\nlabel(\"F\",(55,0),SW);\nlabel(\"G\",(100,0),SE);\nlabel(\"H\",(100,20),NE);\nlabel(\"K\",(75,20),NE);[/asy]",
        "answer": "$6.25$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 67,
        "problem": "In the figure below, square $ABCD$ with side length \\$23$ is divided into nine rectangles by two lines parallel to $\\overline{AB}$ and two lines parallel to $\\overline{BC}$. The areas of four of these rectangles are given in the diagram. Determine the maximum possible area of the central rectangle.",
        "geo_code": "[asy]\nsize(250);\n defaultpen (linewidth (0.7) + fontsize (10));\n\t\t draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);\n\t\t label(\" $A$ \", (0,23), NW);\n\t\t label(\" $B$ \", (23, 23), NE);\n\t\t label(\" $C$ \", (23,0), SE);\n\t\t label(\" $D$ \", (0,0), SW);\n\t\t draw((0,6)--(23,6));\n\t\t draw((0,19)--(23,19));\n\t\t draw((5,0)--(5,23));\n\t\t draw((12,0)--(12,23));\n\t\t label(\"13\", (17/2, 21));\n\t\t label(\"111\",(35/2,25/2));\n\t\t label(\"37\",(17/2,3));\n\t\t label(\"123\",(2.5,12.5));[/asy]",
        "answer": "$180$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 68,
        "problem": "In the diagram below, the regular hexagon $BCEGHJ$ is inscribed within rectangle $ADFI$. Let $\\theta$ denote the acute angle between side $\\overline{EG}$ of the hexagon and the diagonal $\\overline{AF}$ of the rectangle. Suppose that $\\sin^2 \\theta = \\frac{m}{n}$, where $m$ and $n$ are positive integers with no common factors. Find the value of $m+n$.\n\n",
        "geo_code": "[asy]\nimport graph; size(3.2cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);\ndraw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle); \ndraw((-1,3)--(-1,2)); \ndraw((-1,2)--(-0.13,1.5)); \ndraw((-0.13,1.5)--(0.73,2)); \ndraw((0.73,2)--(0.73,3)); \ndraw((0.73,3)--(-0.13,3.5)); \ndraw((-0.13,3.5)--(-1,3)); \ndraw((-1,3.5)--(0.73,3.5)); \ndraw((0.73,3.5)--(0.73,1.5)); \ndraw((-1,1.5)--(0.73,1.5)); \ndraw((-1,3.5)--(-1,1.5)); \nlabel(\" $ A $ \",(-1.4,3.9),SE*labelscalefactor); \nlabel(\" $ B $ \",(-1.4,3.28),SE*labelscalefactor); \nlabel(\" $ C $ \",(-1.4,2.29),SE*labelscalefactor); \nlabel(\" $ D $ \",(-1.4,1.45),SE*labelscalefactor); \nlabel(\" $ E $ \",(-0.3,1.4),SE*labelscalefactor); \nlabel(\" $ F $ \",(0.8,1.45),SE*labelscalefactor); \nlabel(\" $ G $ \",(0.8,2.24),SE*labelscalefactor); \nlabel(\" $ H $ \",(0.8,3.26),SE*labelscalefactor); \nlabel(\" $ I $ \",(0.8,3.9),SE*labelscalefactor); \nlabel(\" $ J $ \",(-0.25,3.9),SE*labelscalefactor); [/asy]",
        "answer": "$55$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 69,
        "problem": "Let $ABC$ be an isosceles triangle with side lengths $AB = AC = 10\\sqrt{2}$ and $BC = 10\\sqrt{3}$. Semicircles $P$, $Q$, and $R$ are constructed with diameters $AB$, $AC$, and $BC$, respectively, so that the plane of each semicircle is perpendicular to the plane of $ABC$ and all semicircles are positioned on the same side of the plane $ABC$, as illustrated. There is a plane located above triangle $ABC$ that is tangent to each of the semicircles $P$, $Q$, and $R$ at points $D$, $E$, and $F$, respectively. Find the area of triangle $DEF$.\n\n",
        "geo_code": "[asy]\nsize(200);\nimport three; defaultpen(linewidth(0.7)+fontsize(10)); currentprojection = orthographic(0,4,2.5); \n// 1.15 x-scale distortion factor \ntriple A = (0,0,0), B = (75^.5/1.15,-125^.5,0), C = (-75^.5/1.15,-125^.5,0), D = (A+B)/2 + (0,0,abs((B-A)/2)), E = (A+C)/2 + (0,0,abs((C-A)/2)), F = (C+B)/2 + (0,0,abs((B-C)/2)); \ndraw(D--E--F--cycle); draw(B--A--C); \nreal r = 1.38; draw(B--(r*B+C)/(1+r)^^(B+r*C)/(1+r)--C,linetype(\"4 4\")); draw((B+r*C)/(1+r)--(r*B+C)/(1+r));\ndraw(arc((A+B)/2,A,D)); draw(arc((A+B)/2,D,B)); draw(arc((A+C)/2,E,A)); draw(arc((A+C)/2,E,C)); draw(arc((C+B)/2,F,B)); draw(arc((C+B)/2,F,C)); \nlabel(\" $A$ \",A,S); label(\" $B$ \",B,W); label(\" $C$ \",C,plain.E);\nlabel(\" $D$ \",D,SW); label(\" $E$ \",E,SE); label(\" $F$ \",F,N);[/asy]",
        "answer": "$24$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 70,
        "problem": "In the figure below, there are two congruent circles with centers $A$ and $B$, which are tangent to each other. A third circle is tangent to the circle centered at $A$ at point $C$, and passes through point $B$. Points $C$, $A$, and $B$ lie on a straight line. The line segment $\\overline{CDEFG}$ passes through the circles at the labeled points. Given that $DE = 6$ and $FG = 9$, determine the length $AG$.\n\n",
        "geo_code": "[asy]\nunitsize(5);\npair A = (-9 sqrt(3), 0);\npair B = (9 sqrt(3), 0);\npair C = (-18 sqrt(3), 0);\npair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3);\npair E = (2 sqrt(3), 4 sqrt(6));\npair F = (7 sqrt(3), 5 sqrt(6));\npair G = (12 sqrt(3), 6 sqrt(6));\nreal r = 9sqrt(3);\ndraw(circle(A, r));\ndraw(circle(B, r));\ndraw(circle((B + C) / 2, 3r / 2));\ndraw(C -- D);\ndraw(\" $6$ \", E -- D);\ndraw(E -- F);\ndraw(\" $9$ \", F -- G);\ndot(A);\ndot(B);\nlabel(\" $A$ \", A, plain.E);\nlabel(\" $B$ \", B, plain.E);\nlabel(\" $C$ \", C, W);\nlabel(\" $D$ \", D, dir(160));\nlabel(\" $E$ \", E, S);\nlabel(\" $F$ \", F, SSW);\nlabel(\" $G$ \", G, N);\n[/asy]",
        "answer": "$9\\sqrt{19}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 71,
        "problem": "A large kite is wrapped in gold foil, which is cut from a rectangular sheet that exactly covers the entire grid. How many square inches of foil are trimmed from the four corners as waste?\n\n",
        "geo_code": "[asy]\nfor (int a = 0; a < 7; ++a)\n{\nfor (int b = 0; b < 8; ++b)\n{\ndot((a,b));\n}\n}\ndraw((3,0)--(0,5)--(3,7)--(6,5)--cycle);[/asy]",
        "answer": "$189$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 72,
        "problem": "A rubber band is stretched around two cylindrical pipes, one with a radius of \\$3$ inches and the other with a radius of \\$9$ inches, as illustrated. The length of the rubber band can be expressed as $a\\pi + b\\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is square-free. Find the value of $a+b+c$.\n\n",
        "geo_code": "[asy]\nsize(4cm);\ndraw(circle((0,0),3));\ndraw(circle((12,0),9));\ndraw(3*dir(120)--(12,0)+9*dir(120));\ndraw(3*dir(240)--(12,0)+9*dir(240));\n[/asy]",
        "answer": "$29$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 73,
        "problem": "A trapezoid has parallel sides of lengths \\$3$ and \\$9$, and non-parallel sides of lengths \\$4$ and \\$6$. A line parallel to the bases divides the original trapezoid into two smaller trapezoids with equal perimeters. In what integer ratio does this line divide each of the non-parallel sides?",
        "geo_code": "[asy]defaultpen(linewidth(.8pt));\nunitsize(2cm);\n\npair A = origin;\npair B = (2.25,0);\npair C = (2,1);\npair D = (1,1);\npair E = waypoint(A--D,0.25);\npair F = waypoint(B--C,0.25);\n\ndraw(A--B--C--D--cycle);\ndraw(E--F);\n\nlabel(\"6\",midpoint(A--D),NW);\nlabel(\"3\",midpoint(C--D),N);\nlabel(\"4\",midpoint(C--B),NE);\nlabel(\"9\",midpoint(A--B),S);[/asy]",
        "answer": "$4$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 74,
        "problem": "A semicircle has a diameter of length \\$16$, and inside it there is a circle of radius \\$3$ that is tangent to both the diameter and the semicircle itself. Another, larger circle is also placed inside the semicircle such that it is tangent to the semicircle, tangent to the diameter, and tangent externally to the smaller circle. If the diameter of the larger circle can be written in the form $\\frac{n + k\\sqrt{2}}{m}$, where $m$, $n$, and $k$ are positive integers with $\\gcd(m, n) = 1$, find $m + n + k$.\n\n",
        "geo_code": "[asy]\nsize(200);\npair O=(0,0);\nreal R=10, r=4.7;\ndraw(arc(O,R,0,180)--cycle);\npair P=(sqrt((R-r)^2-r^2),r),Q;\ndraw(circle(P,r));\nreal a=0,b=r,c;\nfor(int k=0;k<20;++k)\n{\nc=(a+b)/2;\nQ=(-sqrt((R-c)^2-c^2),c);\nif(abs(P-Q)>c+r) a=c; else b=c;\n}\ndraw(circle(Q,c));[/asy]",
        "answer": "$361$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 75,
        "problem": "In triangle $CAT$, suppose that $\\angle ACT = \\angle ATC$ and $\\angle CAT = 36^\\circ$. If $\\overline{TR}$ is the angle bisector of $\\angle ATC$, what is the measure of angle $\\angle CRT$?",
        "geo_code": "[asy]\npair A,C,T,R;\nC = (0,0); T = (2,0); A = (1,sqrt(5+sqrt(20))); R = (3/2 - sqrt(5)/2,1.175570);\ndraw(C--A--T--cycle);\ndraw(T--R);\nlabel(\" $A$ \",A,N);\nlabel(\" $T$ \",T,SE);\nlabel(\" $C$ \",C,SW);\nlabel(\" $R$ \",R,NW);\n[/asy]",
        "answer": "$72$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 76,
        "problem": "In triangle $ABC$, suppose that $AB = AC$. If there exists a point $P$ located strictly between $A$ and $B$ such that $AP = PC = CB$, what is the value of $\\angle A$?",
        "geo_code": "[asy]\ndraw((0,0)--(8,0)--(4,12)--cycle);\ndraw((8,0)--(1.6,4.8));\nlabel(\"A\", (4,12), N);\nlabel(\"B\", (0,0), W);\nlabel(\"C\", (8,0), E);\nlabel(\"P\", (1.6,4.8), NW);\ndot((0,0));\ndot((4,12));\ndot((8,0));\ndot((1.6,4.8));\n[/asy]",
        "answer": "$36$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 77,
        "problem": "In $\\triangle ABC$, the medians $\\overline{AD}$ and $\\overline{CE}$ meet at point $P$. If $PE = 1.5$, $PD = 2$, and $DE = 2.5$, what is the area of quadrilateral $AEDC$?",
        "geo_code": "[asy]\nunitsize(75);\npathpen = black; pointpen=black;\npair A = MP(\"A\", D((0,0)), dir(200));\npair B = MP(\"B\", D((2,0)), dir(-20));\npair C = MP(\"C\", D((1/2,1)), dir(100));\npair D = MP(\"D\", D(midpoint(B--C)), dir(30));\npair E = MP(\"E\", D(midpoint(A--B)), dir(-90));\npair P = MP(\"P\", D(IP(A--D, C--E)), dir(150)*2.013);\ndraw(A--B--C--cycle);\ndraw(A--D--E--C);\n[/asy]",
        "answer": "$13.5$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 78,
        "problem": "Let $G$ be the centroid of triangle $ABC$, and let $D$ be the midpoint of side $BC$. Suppose triangle $BDG$ is equilateral with each side having length \\$1$. Find the length of $BC$ in triangle $ABC$.",
        "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=100.8933946;\npair A=sqrt(7)*dir(r), B=origin, C=(3,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);\ndraw(A--B--C--A--D^^B--E^^C--F);\npair point=G;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $E$ \", E, dir(point--E));\nlabel(\" $F$ \", F, dir(point--F));\nlabel(\" $G$ \", G, dir(20));\nlabel(\"1\", B--G, dir(150));\nlabel(\"1\", D--G, dir(30));\nlabel(\"1\", B--D, dir(270));[/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 79,
        "problem": "The area of trapezoid $ABCD$ is \\$164\\,\\text{cm}^2$. The height is \\$8\\,\\text{cm}$, with $AB = 10\\,\\text{cm}$ and $CD = 17\\,\\text{cm}$. Determine the length of $BC$ in centimeters.",
        "geo_code": "[asy]\nsize(4inch,2inch);\ndraw((0,0)--(31,0)--(16,8)--(6,8)--cycle);\ndraw((11,8)--(11,0), linetype(\"8 4\"));\ndraw((11,1)--(12,1)--(12,0));\nlabel(\" $A$ \", (0,0), SW);\nlabel(\" $D$ \", (31,0), SE);\nlabel(\" $B$ \", (6,8), NW);\nlabel(\" $C$ \", (16,8), NE);\nlabel(\"10\", (3,5), W);\nlabel(\"8\", (11,4), E);\nlabel(\"17\", (22.5,5), E);[/asy]",
        "answer": "$10$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 80,
        "problem": "In $\\triangle ABC$, suppose that $AX = XY = YB = BC$, and $m\\angle ABC = 120^\\circ$. Find the measure of angle $BAC$.",
        "geo_code": "[asy]\npair A, B, C, X, Y;\nA = origin;\nX = dir(30);\nY = X + dir(0);\nB = Y + dir(60);\nC = B + dir(330);\ndraw(A--B--C--cycle);\ndraw(X--Y--B);\nlabel(\" $A$ \",A,W);\nlabel(\" $B$ \",B,N);\nlabel(\" $C$ \",C,E);\nlabel(\" $X$ \",X,NW);\nlabel(\" $Y$ \",Y,SE);\n[/asy]",
        "answer": "$15$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 81,
        "problem": "In triangle $ABC$, the cevians $AD$, $BE$, and $CF$ meet at a common point $S$. Given that $AS : DS = 3 : 2$ and $BS : ES = 4 : 3$. By what factor is $CS$ greater than $FS$?",
        "geo_code": "[asy]\nunitsize (1 cm);\n\npair A, B, C, D, E, F, S;\n\nA = (0,0);\nB = (5,0);\nC = (1,4);\nS = (14*A + 15*B + 6*C)/35;\nD = extension(A,S,B,C);\nE = extension(B,S,C,A);\nF = extension(C,S,A,B);\n\ndraw(A--B--C--cycle);\ndraw(A--D);\ndraw(B--E);\ndraw(C--F);\n\ndot(\" $A$ \", A, SW);\ndot(\" $B$ \", B, SE);\ndot(\" $C$ \", C, N);\ndot(\" $D$ \", D, NE);\ndot(\" $E$ \", E, W);\ndot(\" $F$ \", F, dir(270));\ndot(\" $S$ \", S, NE);\n[/asy]",
        "answer": "$2.889$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 82,
        "problem": "In the diagram below, $ABDE$ is a square and certain angles and segment lengths are given. Find the length of $\\overline{CF}$. \n\n\n",
        "geo_code": "[asy]\nmarkscalefactor=0.15;\nsize(8cm);\npair A = (0,0);\npair B = (17,0);\npair E = (0,17);\npair D = (17,17);\npair F = (-120/17,225/17);\npair C = (17+120/17, 64/17);\ndraw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle);  \nlabel(\" $A$ \", A, S);\nlabel(\" $B$ \", B, S);\nlabel(\" $C$ \", C, dir(0));\nlabel(\" $D$ \", D, N);\nlabel(\" $E$ \", E, N);\nlabel(\" $F$ \", F, W);\nlabel(\" $8$ \", (F+E)/2, NW);\nlabel(\" $15$ \", (F+A)/2, SW);\nlabel(\" $8$ \", (C+B)/2, SE);\nlabel(\" $15$ \", (D+C)/2, NE);\ndraw(rightanglemark(E,F,A));\ndraw(rightanglemark(D,C,B));\n[/asy]",
        "answer": "$23\\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 83,
        "problem": "Let $G$ denote the point of intersection of the medians of triangle $ABC$ (the centroid), and let $D$ be the midpoint of side $BC$. If triangle $BDG$ is equilateral with all sides equal to \\$1$, find the length $AB$ of triangle $ABC$.\n\n",
        "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=100.8933946;\npair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);\ndraw(A--B--C--A--D^^B--E^^C--F);\npair point=G;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $E$ \", E, dir(point--E));\nlabel(\" $F$ \", F, dir(point--F));\nlabel(\" $G$ \", G, dir(20));\nlabel(\"1\", B--G, dir(150));\nlabel(\"1\", D--G, dir(30));\nlabel(\"1\", B--D, dir(270));[/asy]",
        "answer": "$\\sqrt{7}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 84,
        "problem": "Let $G$ be the centroid of triangle $ABC$, and let $D$ be the midpoint of side $BC$. Suppose that triangle $BDG$ is equilateral with each side of length \\$1$. Find the length $CA$ in triangle $ABC$.\n\n",
        "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=100.8933946;\npair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);\ndraw(A--B--C--A--D^^B--E^^C--F);\npair point=G;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $E$ \", E, dir(point--E));\nlabel(\" $F$ \", F, dir(point--F));\nlabel(\" $G$ \", G, dir(20));\nlabel(\"1\", B--G, dir(150));\nlabel(\"1\", D--G, dir(30));\nlabel(\"1\", B--D, dir(270));[/asy]",
        "answer": "$\\sqrt{13}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 85,
        "problem": "Let $A$, $B$, and $C$ be the centers of three circles, each with radius $r$ where \\$1 < r < 2$. The distance between every pair of centers is \\$2$. Define $B'$ as the intersection point of circles centered at $A$ and $C$ that lies outside the circle centered at $B$, and let $C'$ be the intersection point of circles centered at $A$ and $B$ that is outside the circle centered at $C$. Find the value of the length $B'C'$.",
        "geo_code": "[asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP(\"$A$\",A)); D(MP(\"$B$\",B)); D(MP(\"$C$\",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP(\"$B'$\",BB[0],NW);MP(\"$C'$\",CC[1],NE); [/asy]",
        "answer": "$1+\\sqrt{3(r^2-1)}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 86,
        "problem": "In $\\triangle ABC$, let $E$ denote the midpoint of $BC$ and let $D$ be a point on $AC$. Suppose $AC = 1$ and the angle measures are $\\angle BAC = 60^\\circ$, $\\angle ABC = 100^\\circ$, $\\angle ACB = 20^\\circ$, and $\\angle DEC = 80^\\circ$. Then, the sum of the area of $\\triangle ABC$ and twice the area of $\\triangle CDE$ is equal to",
        "geo_code": "[asy] size(200); import cse5; pathpen=black; anglefontpen=black; pointpen=black; anglepen=black; dotfactor=3; pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE; EE=(B+C)/2; D(MP(\"$A$\",A,W)--MP(\"$B$\",B,N)--MP(\"$C$\",C,E)--cycle); D(MP(\"$E$\",EE,N)--MP(\"$D$\",D,S)); D(D);D(EE); MA(\"80^\\circ\",8,D,EE,C,0.1); MA(\"20^\\circ\",8,EE,C,D,0.3,2,shift(1,3)*C); draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow()); MA(\"100^\\circ\",8,A,B,C,0.1,0); MA(\"60^\\circ\",8,C,A,B,0.1,0); [/asy]",
        "answer": "$\\frac{\\sqrt{3}}{8}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 87,
        "problem": "In the triangle $ABC$ shown in the adjacent diagram, the rays $AD$ and $AE$ divide $\\angle BAC$ into three equal parts. The segment lengths $BD$, $DE$, and $EC$ are \\$2$, \\$3$, and \\$6$, respectively. What is the length of the shortest side of $\\triangle ABC$? ",
        "geo_code": "[asy] defaultpen(linewidth(.8pt)); pair A = (0,11); pair B = (2,0); pair D = (4,0); pair E = (7,0); pair C = (13,0); label(\"$A$\",A,N); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,S); label(\"$E$\",E,S); label(\"$2$\",midpoint(B--D),N); label(\"$3$\",midpoint(D--E),NW); label(\"$6$\",midpoint(E--C),NW); draw(A--B--C--cycle); draw(A--D); draw(A--E); [/asy]",
        "answer": "$2\\sqrt{10}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 88,
        "problem": "The number of square units contained in polygon $ABCDEF$ is",
        "geo_code": "[asy] draw((1,9)--(6,9)--(6,0)--(2,0)--(2,4)--(1,4)--cycle); label(\"A\",(1,9),NW); label(\"B\",(6,9),NE); label(\"C\",(6,0),SE); label(\"D\",(2,0),SW); label(\"E\",(2,4),NE); label(\"F\",(1,4),SW); label(\"6\",(3,9),N); label(\"9\",(6,4.5),E); label(\"4\",(4,0),S); label(\"5\",(1,6.5),W); [/asy]",
        "answer": "$46$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 89,
        "problem": "The diagonal $DB$ of the rectangle $ABCD$ is divided into three segments of length \\$1$ each by two lines, $L$ and $L'$. These lines are perpendicular to $DB$ and pass through the vertices $A$ and $C$, respectively. Find the area of $ABCD$, rounded to one decimal place.",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B)^^rightanglemark(N1,E,B)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label(\"A\", A, NE); label(\"B\", B, NE); label(\"C\", C, dir(0)); label(\"D\", D, dir(180)); label(\"$L$\", (x/2,0), SW); label(\"$L^\\prime$\", C, SW); label(\"1\", D--F, NW); label(\"1\", F--E, SE); label(\"1\", E--B, SE); clip(W--X--Y--Z--cycle);[/asy]",
        "answer": "$4.2$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 90,
        "problem": "What is the perimeter of the polygon ?",
        "geo_code": "[asy] size(4cm); draw((0,0)--(0,6)--(7,6)--(7,3)--(2.7,3)--(2.7,0)--cycle); label(\"$6$\",(0,3),W); label(\"$8$\",(4,6),N); draw((0.5,0)--(0.5,0.5)--(0,0.5)); draw((0.5,6)--(0.5,5.5)--(0,5.5)); draw((6.5,6)--(6.5,5.5)--(7,5.5)); draw((6.5,3)--(6.5,3.5)--(7,3.5)); draw((2.2,0)--(2.2,0.5)--(2.7,0.5)); draw((2.7,2.5)--(3.2,2.5)--(3.2,3)) [/asy]",
        "answer": "$28$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 91,
        "problem": "The area of the rectangular region is ",
        "geo_code": "[asy] size(4cm);draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(.5 mm)); label(\"2\",(8,2),E); label(\"4\",(4,0),S); [/asy]",
        "answer": "$8$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 92,
        "problem": "What is area of the  shaded rectangular region?",
        "geo_code": "[asy] draw((0,0)--(18,0)--(18,12)--(0,12)--cycle); draw((0,6)--(18,6)); for(int a=6; a<12; ++a) { draw((1.5a,0)--(1.5a,6)); } fill((15,0)--(18,0)--(18,6)--(15,6)--cycle,black); label(\"0\",(0,0),W); label(\"9\",(9,0),S); label(\"18\",(18,0),S); label(\"6\",(0,6),W); label(\"12\",(0,12),W); [/asy]",
        "answer": "$18$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 93,
        "problem": "$\\text{ABCD}$ is a rectangle, $\\text{D}$ is the center of the circle, and $\\text{B}$ is on the circle. If $\\text{AD}=4$ and $\\text{CD}=3$, then the area of the shaded region is",
        "geo_code": "[asy] pair A,B,C,D; A=(0,4); B=(3,4); C=(3,0); D=origin; draw(circle(D,5)); fill((0,5)..(1.5,4.7697)..B--A--cycle,black); fill(B..(4,3)..(5,0)--C--cycle,black); draw((0,5)--D--(5,0)); label(\"A\",A,NW); label(\"B\",B,NE); label(\"C\",C,S); label(\"D\",D,SW); [/asy]",
        "answer": "$\\frac{25}{4}\\pi - 12$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 94,
        "problem": "In the figure, $\\triangle ABC$ has $\\angle A =45^{\\circ}$ and $\\angle B =30^{\\circ}$. A line $DE$, with $D$ on $AB$ and $\\angle ADE =60^{\\circ}$, divides $\\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$ The ratio $\\frac{AB}{AD}$ is",
        "geo_code": "[asy] size((220)); draw((0,0)--(20,0)--(7,6)--cycle); draw((6,6)--(10,-1)); label(\"A\", (0,0), W); label(\"B\", (20,0), E); label(\"C\", (7,6), NE); label(\"D\", (9.5,-1), W); label(\"E\", (5.9, 6.1), SW); label(\"$45^{\\circ}$\", (2.5,.5)); label(\"$60^{\\circ}$\", (7.8,.5)); label(\"$30^{\\circ}$\", (16.5,.5)); [/asy]",
        "answer": "$\\sqrt[4]{12}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 95,
        "problem": "If $\\angle \\text{CBD}$ is a right angle, then this protractor indicates that the measure of $\\angle \\text{ABC}$ is approximately",
        "geo_code": "[asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label(\"O\",(-2.5,0),W); label(\"A\",A,W); label(\"B\",B,S); label(\"C\",C,W); label(\"D\",D,E); label(\"0\",(-1.8,0),W); label(\"20\",(-1.7,.5),NW); label(\"160\",(1.6,.5),NE); label(\"180\",(1.7,0),E); [/asy]",
        "answer": "$50$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 96,
        "problem": "The shaded region formed by the two intersecting perpendicular rectangles, in square units, is",
        "geo_code": "[asy] fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4.5)--(6,4.5)--(6,2)--(0,2)--cycle,black); label(\"2\",(0,.9),W); label(\"3\",(7.3,4.5),N); draw((0,-3.3)--(0,-5.3),linewidth(1)); draw((0,-4.3)--(3.7,-4.3),linewidth(1)); label(\"10\",(4.7,-3.7),S); draw((5.7,-4.3)--(10,-4.3),linewidth(1)); draw((10,-3.3)--(10,-5.3),linewidth(1)); draw((11,4.5)--(13,4.5),linewidth(1)); draw((12,4.5)--(12,2),linewidth(1)); label(\"10\",(11.3,1),E); draw((12,0)--(12,-3.5),linewidth(1)); draw((11,-3.5)--(13,-3.5),linewidth(1)); [/asy]",
        "answer": "$44$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 97,
        "problem": "What is the ratio of the shaded part to the unshaded part in parallelogram $\\text{ABCD}$?",
        "geo_code": "[asy] unitsize(10); pair A,B,C,D,E; A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0); draw(A--B--C--D--cycle); fill(B--E--D--C--cycle,gray); label(\"A\",A,SW); label(\"B\",B,NW); label(\"C\",C,NE); label(\"D\",D,SE); label(\"E\",E,S); label(\"$10$\",(9,8),N); label(\"$6$\",(7,0),S); label(\"$8$\",(4,4),W); draw((3,0)--(3,1)--(4,1)); [/asy]",
        "answer": "$4$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 98,
        "problem": "All six sides of a rectangular solid were rectangles. A one-foot cube was cut out of the rectangular solid as shown. The total number of square feet in the surface of the new solid is how many more or less than that of the original solid?",
        "geo_code": "[asy] unitsize(24); draw((0,0)--(1,0)--(1,3)--(0,3)--cycle); draw((1,0)--(1+9*sqrt(3)/2,9/2)--(1+9*sqrt(3)/2,15/2)--(1+5*sqrt(3)/2,11/2)--(1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),4)--(1+2*sqrt(3),5)--(1,3)); draw((0,3)--(2*sqrt(3),5)--(1+2*sqrt(3),5)); draw((1+9*sqrt(3)/2,15/2)--(9*sqrt(3)/2,15/2)--(5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,5)); draw((1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),9/2)); draw((1+5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,11/2)); label(\"$1'$\",(.5,0),S); label(\"$3'$\",(1,1.5),E); label(\"$9'$\",(1+9*sqrt(3)/4,9/4),S); label(\"$1'$\",(1+9*sqrt(3)/4,17/4),S); label(\"$1'$\",(1+5*sqrt(3)/2,5),E);label(\"$1'$\",(1/2+5*sqrt(3)/2,11/2),S); [/asy]",
        "answer": "$0$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 99,
        "problem": "Four circles of radius $3$ are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is",
        "geo_code": "[asy] fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray); fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white); fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white); fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white); fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white); draw(circle((3,3),3)); draw(circle((3,-3),3)); draw(circle((-3,-3),3)); draw(circle((-3,3),3)); draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle); [/asy]",
        "answer": "$36-9\\pi$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 100,
        "problem": "Semicircle $\\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\\widehat{AB}$ and $\\overline{CD}\\perp\\overline{AB}$. Extend $\\overline{BD}$ and $\\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\\widehat{AE}$ and $\\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\\widehat{EF}$ has center $D$. The area of the shaded \"smile\" $AEFBDA$, is",
        "geo_code": "[asy] fill((1,0)--arc((1,0),2,180,225)--cycle,grey); fill((-1,0)--arc((-1,0),2,315,360)--cycle,grey); fill((0,-1)--arc((0,-1),2-sqrt(2),225,315)--cycle,grey); fill((0,0)--arc((0,0),1,180,360)--cycle,white); draw((1,0)--arc((1,0),2,180,225)--(1,0),black+linewidth(1)); draw((-1,0)--arc((-1,0),2,315,360)--(-1,0),black+linewidth(1)); draw((0,0)--arc((0,0),1,180,360)--(0,0),black+linewidth(1)); draw(arc((0,-1),2-sqrt(2),225,315),black+linewidth(1)); draw((0,0)--(0,-1),black+linewidth(1)); MP(\"C\",(0,0),N);MP(\"A\",(-1,0),N);MP(\"B\",(1,0),N); MP(\"D\",(0,-.8),NW);MP(\"E\",(1-sqrt(2),-sqrt(2)),SW);MP(\"F\",(-1+sqrt(2),-sqrt(2)),SE); [/asy]",
        "answer": "$2\\pi-\\pi \\sqrt{2}-1$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 101,
        "problem": "The convex pentagon $ABCDE$ has $\\angle{A}=\\angle{B}=120^\\circ,EA=AB=BC=4$ and $CD=DE=8$. What is the area of ABCDE?",
        "geo_code": "[asy] draw((-1,0)--(1,0)--(1+sqrt(2),sqrt(2))--(0,sqrt(2)+sqrt(13-2*sqrt(2)))--(-1-sqrt(2),sqrt(2))--cycle,black+linewidth(.75)); MP(\"A\",(-1,0),SW);MP(\"B\",(1,0),SE);MP(\"C\",(1+sqrt(2),sqrt(2)),E);MP(\"D\",(0,sqrt(2)+sqrt(13-2*sqrt(2))),N);MP(\"E\",(-1-sqrt(2),sqrt(2)),W); dot((-1,0));dot((1,0));dot((1+sqrt(2),sqrt(2)));dot((-1-sqrt(2),sqrt(2)));dot((0,sqrt(2)+sqrt(13-2*sqrt(2)))); [/asy]",
        "answer": "$28\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 102,
        "problem": "Amy painted a dartboard over a square clock face using the \"hour positions\" as boundaries.[See figure.] If $t$ is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then $\\frac{t}{q}=$",
        "geo_code": "[asy] draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75)); draw((0,-1)--(0,1), black+linewidth(.75)); draw((-1,0)--(1,0), black+linewidth(.75)); draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75)); draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75)); draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75)); draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75)); [/asy]",
        "answer": "$\\frac{\\sqrt{3}+1}{4}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 103,
        "problem": "A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the total area of the four rectangle?",
        "geo_code": "[asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label(\"6\", (1.5,6)); label(\"?\", (1.5,2.5)); label(\"14\", (6.5,6)); label(\"35\", (6.5,2.5)); [/asy]",
        "answer": "$70$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 104,
        "problem": "In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,3), (3,3), (3,1), (5,1)$ and $(5,0)$. Suppose there is a straight line that divides the area of the region into two equal parts. What is the tangent of the angle formed between this line and the y-axis?",
        "geo_code": "[asy] Label l; l.p=fontsize(6); xaxis(\"$x$\",0,6,Ticks(l,1.0,0.5),EndArrow); yaxis(\"$y$\",0,4,Ticks(l,1.0,0.5),EndArrow); draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));[/asy]",
        "answer": "$\\frac{9}{7}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 105,
        "problem": "Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $BC/AB=$",
        "geo_code": "[asy] draw(Circle((0,0), 13)); draw((-13,0)--(12,5)--(12,-5)--cycle); dot((-13,0)); dot((12,5)); dot((12,-5)); label(\"A\", (-13,0), W); label(\"B\", (12,5), NE); label(\"C\", (12,-5), SE); [/asy]",
        "answer": "$\\2\\sin\\left(\\frac{1}{4}\\right)$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 106,
        "problem": "Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\\overline{AM}$ with $AD=10$ and $\\angle BDC=3\\angle BAC.$ Then the perimeter of $\\triangle ABC$ is $",
        "geo_code": "[asy] import graph; size(5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label(\"$11$\",(1,1.63),W); dot((1,3),ds); label(\"$A$\",(1,3),N); dot((0,0),ds); label(\"$B$\",(0,0),SW); dot((2,0),ds); label(\"$C$\",(2,0),SE); dot((1,0),ds); label(\"$M$\",(1,0),S); dot((1,0.7),ds); label(\"$D$\",(1,0.7),NE); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]",
        "answer": "$11+\\sqrt{605}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 107,
        "problem": "Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates (2,2). For what value of the x-coordinate of point $T$ will the area of triangle $PQT$ be exactly twice the area of square $OPQR$?",
        "geo_code": "[asy] pair O,P,Q,R,T; O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0); draw((-5,0)--(3,0)); draw((0,-1)--(0,3)); draw(P--Q--R); draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8)); draw((-0.2,2.8)--(0,3)--(0.2,2.8)); draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2)); draw((2.8,-0.2)--(3,0)--(2.8,0.2)); draw(Q--T); label(\"$O$\",O,SW); label(\"$P$\",P,S); label(\"$Q$\",Q,NE); label(\"$R$\",R,W); label(\"$T$\",T,S); [/asy]",
        "answer": "$-6$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 108,
        "problem": "On a $4\\times 4\\times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is",
        "geo_code": "[asy] size(120); import three; currentprojection=orthographic(1, 4/5, 1/3); draw(box(O, (4,4,3))); triple A=(0,4,3), B=(0,0,0) , C=(4,4,0), D=(0,4,0); draw(A--B--C--cycle, linewidth(0.9)); label(\"$A$\", A, NE); label(\"$B$\", B, NW); label(\"$C$\", C, S); label(\"$D$\", D, E); label(\"$4$\", (4,2,0), SW); label(\"$4$\", (2,4,0), SE); label(\"$3$\", (0, 4, 1.5), E); [/asy]",
        "answer": "$\\frac{12}{\\sqrt{34}}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 109,
        "problem": "What fraction of this square region is unshaded? Stripes are equal in width, and the figure is drawn to scale.",
        "geo_code": "[asy] unitsize(8); fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black); fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white); fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black); fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white); fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white); draw((0,6)--(0,0)--(6,0)); [/asy]",
        "answer": "$\\frac{5}{12}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 110,
        "problem": "$\\angle 1 + \\angle 2 = 180^\\circ$ $\\angle 3 = \\angle 4$ Find $\\angle 4.$",
        "geo_code": "[asy] pair H,I,J,K,L; H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0); draw(H--I--J--cycle); draw(K--L--J); draw(arc((0,0),dir(70),(1,0),CW)); label(\"$70^\\circ$\",dir(30),NE); draw(arc(I,I+dir(250),I+dir(290),CCW)); label(\"$40^\\circ$\",I+1.25*dir(270),S); label(\"$1$\",J+0.25*dir(162.5),NW); label(\"$2$\",J+0.25*dir(17.5),NE); label(\"$3$\",L+dir(162.5),WNW); label(\"$4$\",K+dir(-52.5),SE); [/asy]",
        "answer": "$35$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 111,
        "problem": "Each corner cube is removed from this $6\\text{ cm}\\times 6\\text{ cm}\\times 6\\text{ cm}$ cube. The surface area of the remaining figure in sq.cm is",
        "geo_code": "[asy] draw((2.7,3.99)--(0,3)--(0,0)); draw((3.7,3.99)--(1,3)--(1,0)); draw((4.7,3.99)--(2,3)--(2,0)); draw((5.7,3.99)--(3,3)--(3,0)); draw((0,0)--(3,0)--(5.7,0.99)); draw((0,1)--(3,1)--(5.7,1.99)); draw((0,2)--(3,2)--(5.7,2.99)); draw((0,3)--(3,3)--(5.7,3.99)); draw((0,3)--(3,3)--(3,0)); draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33)); draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66)); draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99)); [/asy]",
        "answer": "$216$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 112,
        "problem": "Diameter $ACE$ is divided at $C$ in the ratio $1:2$. The two semicircles, $ABC$ and $CDE$, divide the circular region into an upper (shaded) region and a lower region. The area of the upper region divided by that of the lower region is",
        "geo_code": "[asy] pair A,B,C,D,EE; A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0); fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray); draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)); draw(circle((5,0),5)); dot(A); dot(B); dot(C); dot(D); dot(EE); label(\"$A$\",A,W); label(\"$B$\",B,N); label(\"$C$\",C,E); label(\"$D$\",D,N); label(\"$E$\",EE,W); [/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 113,
        "problem": "In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $2$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is",
        "geo_code": "[asy] size(6cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); draw((-1,1)--(2,1)); draw((-1,0)--(1,0)); draw((-1,1)--(-1,0)); draw((0,-1)--(0,3)); draw((1,2)--(1,0)); draw((-1,1)--(1,1)); draw((0,2)--(1,2)); draw((0,3)--(1,2)); draw((0,-1)--(2,1)); draw((0,-1)--((0,-1) + sqrt(2)*dir(-15))); draw(((0,-1) + sqrt(2)*dir(-15))--(1,0)); label(\"$\\textbf{A}$\",foot((0,2),(0,3),(1,2)),SW); label(\"$\\textbf{B}$\",midpoint((0,1)--(1,2))); label(\"$\\textbf{C}$\",midpoint((-1,0)--(0,1))); label(\"$\\textbf{D}$\",midpoint((0,0)--(1,1))); label(\"$\\textbf{E}$\",midpoint((1,0)--(2,1)),NW); label(\"$\\textbf{F}$\",midpoint((0,-1)--(1,0)),NW); label(\"$\\textbf{G}$\",midpoint((0,-1)--(1,0)),2SE);[/asy]",
        "answer": "$\\frac{20}{3}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "volume"
    },
    {
        "index": 114,
        "problem": "Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\\overline{AC}$ intersects $\\overline{BP}$ at point $D$. If $PB = 6$ and $PD= 4$, then $AD\\cdot CD =$",
        "geo_code": "[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (2,0); pair C = (3,1); pair P = (1,2.25); pair D = intersectionpoint(P--B,C--A); dot(A);dot(B);dot(C);dot(P);dot(D); label(\"$A$\",A,SW);label(\"$B$\",B,SE);label(\"$C$\",C,N);label(\"$D$\",D,NE + N);label(\"$P$\",P,N); draw(A--B--P--cycle); draw(A--C--B--cycle);[/asy]",
        "answer": "$20$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 115,
        "problem": "According to the figure, the degree measure of angle $A$ is",
        "geo_code": "[asy] unitsize(12); draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle); draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW)); draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW)); draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW)); label(rotate(30)*\"$40^\\circ$\",(2,-8.9),ENE); label(\"$100^\\circ$\",(21/3,-2/3),SE); label(\"$110^\\circ$\",(900/83,-317/83),NNW); label(\"$A$\",(0,0),NW); [/asy]",
        "answer": "$30$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 116,
        "problem": "If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\\overarc {AC}$ and $\\overarc{BC}$, and to $\\overline{AB}$. If the length of $\\overarc{BC}$ is $12$, then the circumference of the circle is",
        "geo_code": "[asy] label(\"A\", (0,0), W); label(\"B\", (64,0), E); label(\"C\", (32, 32*sqrt(3)), N); draw(arc((0,0),64,0,60)); draw(arc((64,0),64,120,180)); draw((0,0)--(64,0)); draw(circle((32, 24), 24)); [/asy]",
        "answer": "$27$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 117,
        "problem": "A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is not made of darker tiles?",
        "geo_code": "[asy] fill((0,2)--(1,3)--(2,3)--(2,4)--(3,5)--(4,4)--(4,3)--(5,3)--(6,2)--(5,1)--(4,1)--(4,0)--(2,0)--(2,1)--(1,1)--cycle, mediumgrey); fill((7,1)--(6,2)--(7,3)--(8,3)--(8,4)--(9,5)--(10,4)--(7,0)--cycle, mediumgrey); fill((3,5)--(2,6)--(2,7)--(1,7)--(0,8)--(1,9)--(2,9)--(2,10)--(3,11)--(4,10)--(4,9)--(5,9)--(6,8)--(5,7)--(4,7)--(4,6)--cycle, mediumgrey); fill((6,8)--(7,9)--(8,9)--(8,10)--(9,11)--(10,10)--(10,9)--(11,9)--(11,7)--(10,7)--(10,6)--(9,5)--(8,6)--(8,7)--(7,7)--cycle, mediumgrey); draw((0,0)--(0,11)--(11,11)); for ( int x = 1; x &lt; 11; ++x ) { draw((x,11)--(x,0), linetype(\"4 4\")); } for ( int y = 1; y &lt; 11; ++y ) { draw((0,y)--(11,y), linetype(\"4 4\")); } clip((0,0)--(0,11)--(11,11)--(11,5)--(4,1)--cycle);[/asy]",
        "answer": "$\\frac{5}{9}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 118,
        "problem": "Triangle $ABC$ is a right triangle with $\\angle ACB$ as its right angle, $m\\angle ABC = 60^\\circ$ , and $AB = 10$. Let $P$ be randomly chosen inside $ABC$ , and extend $\\overline{BP}$ to meet $\\overline{AC}$ at $D$. What is the probability that $BD < 5\\sqrt2$?",
        "geo_code": "[asy] import math; unitsize(4mm); defaultpen(fontsize(8pt)+linewidth(0.7)); dotfactor=4; pair A=(10,0); pair C=(0,0); pair B=(0,10.0/sqrt(3)); pair P=(2,2); pair D=extension(A,C,B,P); draw(A--C--B--cycle); draw(B--D); dot(P); label(\"A\",A,S); label(\"D\",D,S); label(\"C\",C,S); label(\"P\",P,NE); label(\"B\",B,N);[/asy]",
        "answer": "$\\frac{\\sqrt3}{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 119,
        "problem": "What is the area enclosed by the geoboard quadrilateral below?",
        "geo_code": "[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2; for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b) { dot((a,b)); }; draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy]",
        "answer": "$22\\frac{1}{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 120,
        "problem": "What is the area of trapezoid $ABCD$?",
        "geo_code": "[asy]size(3inch, 1.5inch); pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0); draw(a--b--c--d--cycle); draw(b--e); draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e); label(\"30\", (9,12), W); label(\"50\", (43,24), N); label(\"25\", (71.5, 12), E); label(\"24\", (18, 12), E); label(\"$A$\", a, SW); label(\"$B$\", b, N); label(\"$C$\", c, N); label(\"$D$\", d, SE); label(\"$E$\", e, S);[/asy]",
        "answer": "$1500$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 121,
        "problem": "A number of linked rings, each $2$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?",
        "geo_code": "[asy] size(7cm); pathpen = linewidth(0.7); D(CR((0,0),10)); D(CR((0,0),9.5)); D(CR((0,-18.5),9.5)); D(CR((0,-18.5),9)); MP(\"$\\vdots$\",(0,-31),(0,0)); D(CR((0,-39),3)); D(CR((0,-39),2.5)); D(CR((0,-43.5),2.5)); D(CR((0,-43.5),2)); D(CR((0,-47),2)); D(CR((0,-47),1.5)); D(CR((0,-49.5),1.5)); D(CR((0,-49.5),1.0)); D((12,-10)--(12,10)); MP('20',(12,0),E); D((12,-51)--(12,-48)); MP('3',(12,-49.5),E);[/asy]",
        "answer": "$346$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 122,
        "problem": "An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $B$ to the area of shaded region $C$ is 5/11. Find the ratio of shaded region $D$ to the area of shaded region $A.$",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); for(int i=0; i<4; i=i+1) { fill((2*i,0)--(2*i+1,0)--(2*i+1,6)--(2*i,6)--cycle, mediumgray); } pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10); draw(B--A--C); fill((7.3,0)--(7.8,0)--(7.8,6)--(7.3,6)--cycle, white); clip(B--A--C--cycle); for(int i=0; i<9; i=i+1) { draw((i,1)--(i,6)); } label(\"$\\mathcal{A}$\", A+0.2*dir(-17), S); label(\"$\\mathcal{B}$\", A+2.3*dir(-17), S); label(\"$\\mathcal{C}$\", A+4.4*dir(-17), S); label(\"$\\mathcal{D}$\", A+6.5*dir(-17), S);[/asy]",
        "answer": "$408$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 123,
        "problem": "Hexagon $ABCDEF$ is divided into five rhombuses, $\\mathcal{P, Q, R, S,}$ and $\\mathcal{T,}$ as shown. Rhombuses $\\mathcal{P, Q, R,}$ and $\\mathcal{S}$ are congruent, and each has area $\\sqrt{2006}.$ Let $K$ be the area of rhombus $\\mathcal{T}$. Given that $K$ is a positive integer, find the number of possible values for $K$.",
        "geo_code": "[asy] size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label(\"$\\mathcal{T}$\",(2.1,-1.6)); label(\"$\\mathcal{P}$\",(0,-1),NE); label(\"$\\mathcal{Q}$\",(4.2,-1),NW); label(\"$\\mathcal{R}$\",(0,-2.2),SE); label(\"$\\mathcal{S}$\",(4.2,-2.2),SW); [/asy]",
        "answer": "$89$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 124,
        "problem": "What is the area of the shaded pinwheel shown in the $5 \\times 5$ grid if each grid has side length $2$?",
        "geo_code": "[asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } [/asy]",
        "answer": "$24$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 125,
        "problem": "Two circles that share the same center have radii $5$ meters and $10$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run?",
        "geo_code": "[asy] size((150)); draw((10,0)..(0,10)..(-10,0)..(0,-10)..cycle); draw((20,0)..(0,20)..(-20,0)..(0,-20)..cycle); draw((20,0)--(-20,0)); draw((0,20)--(0,-20)); draw((-2,21.5)..(-15.4, 15.4)..(-22,0), EndArrow); draw((-18,1)--(-12, 1), EndArrow); draw((-12,0)..(-8.3,-8.3)..(0,-12), EndArrow); draw((1,-9)--(1,9), EndArrow); draw((0,12)..(8.3, 8.3)..(12,0), EndArrow); draw((12,-1)--(18,-1), EndArrow); label(\"$A$\", (0,20), N); label(\"$K$\", (20,0), E); [/asy]",
        "answer": "$10\\pi+20$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 126,
        "problem": "Jerry cuts a wedge from a $10$-cm cylinder of bologna as shown by the dashed curve. The volume of his wedge in cubic centimeters is ?",
        "geo_code": "[asy] defaultpen(linewidth(0.65)); real d=90-63.43494882; draw(ellipse((origin), 2, 4)); fill((0,4)--(0,-4)--(-8,-4)--(-8,4)--cycle, white); draw(ellipse((-4,0), 2, 4)); draw((0,4)--(-4,4)); draw((0,-4)--(-4,-4)); draw(shift(-2,0)*rotate(-d-5)*ellipse(origin, 1.82, 4.56), linetype(\"10 10\")); draw((-4,4)--(-8,4), dashed); draw((-4,-4)--(-8,-4), dashed); draw((-4,4.3)--(-4,5)); draw((0,4.3)--(0,5)); draw((-7,4)--(-7,-4), Arrows(5)); draw((-4,4.7)--(0,4.7), Arrows(5)); label(\"$8$ cm\", (-7,0), W); label(\"$6$ cm\", (-2,4.7), N);[/asy]",
        "answer": "$80\\pi$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 127,
        "problem": "A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?",
        "geo_code": "[asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label(\"\\(x\\)\",(-1.55,2.1),E); label(\"\\(1\\)\",(-0.5,3.8),S);[/asy]",
        "answer": "$\\frac{3\\sqrt{7}-\\sqrt{3}}{2}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 128,
        "problem": "The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ABD?",
        "geo_code": "[asy] size(250); defaultpen(linewidth(0.55)); pair A=(-6,0), B=origin, C=(0,6), D=(0,12); pair ac=C+2.828*dir(45), ca=A+2.828*dir(225), ad=D+2.828*dir(A--D), da=A+2.828*dir(D--A), ab=(2.828,0), ba=(-6-2.828, 0); fill(A--C--D--cycle, gray); draw(ba--ab); draw(ac--ca); draw(ad--da); draw((0,-1)--(0,15)); draw((1/3, -1)--(1/3, 15)); int i; for(i=1; i<15; i=i+1) { draw((-1/10, i)--(13/30, i)); } label(\"$A$\", A, SE); label(\"$B$\", B, SE); label(\"$C$\", C, SE); label(\"$D$\", D, SE); label(\"$3$\", (1/3,3), E); label(\"$3$\", (1/3,9), E); label(\"$3$\", (-3,0), S); label(\"Main\", (-3,0), N); label(rotate(45)*\"Aspen\", A--C, SE); label(rotate(63.43494882)*\"Brown\", A--D, NW); [/asy]",
        "answer": "$9$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 129,
        "problem": "Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?",
        "geo_code": "[asy] defaultpen(linewidth(0.6)); pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle); draw(O--D, dashed); draw(A--C, dashed);[/asy]",
        "answer": "$23$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 130,
        "problem": "A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\\frac{1}{2}$ foot from the top face. The second cut is $\\frac{1}{3}$ foot below the first cut, and the third cut is $\\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?The first diagram: ; The second diagram:",
        "geo_code": "[asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(2,8/15,7/15); int t=0; void f(real x) { path3 r=(t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--cycle; path3 f=(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)--cycle; path3 u=(t,1,x)--(t+1,1,x)--(t+1,0,x)--(t,0,x)--cycle; draw(surface(r), white, nolight); draw(surface(f), white, nolight); draw(surface(u), white, nolight); draw((t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--(t,1,x)--(t,0,x)--(t+1,0,x)--(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)); t=t+1; } f(d); f(1/2); f(1/3); f(1/17); label(\"D\", (1/2, 1, 0), SE); label(\"A\", (1+1/2, 1, 0), SE); label(\"B\", (2+1/2, 1, 0), SE); label(\"C\", (3+1/2, 1, 0), SE);[/asy][asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(1,8/15,7/15); draw(unitcube, white, thick(), nolight); void f(real x) { draw((0,1,x)--(1,1,x)--(1,0,x)); } f(d); f(1/6); f(1/2); label(\"A\", (1,0,3/4), W); label(\"B\", (1,0,1/3), W); label(\"C\", (1,0,1/6-d/4), W); label(\"D\", (1,0,d/2), W); label(\"1/2\", (1,1,3/4), E); label(\"1/3\", (1,1,1/3), E); label(\"1/17\", (0,1,1/6-d/4), E);[/asy]",
        "answer": "$11$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 131,
        "problem": "A cubical cake with edge length $2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $M$ is the midpoint of a top edge. The piece whose top is triangle $B$ contains $c$ cubic inches of cake and $s$ square inches of icing. What is $c+s$?",
        "geo_code": "[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); draw((1,1)--(-1,0)); pair P=foot((1,-1),(1,1),(-1,0)); draw((1,-1)--P); draw(rightanglemark((-1,0),P,(1,-1),4)); label(\"$M$\",(-1,0),W); label(\"$C$\",(-0.1,-0.3)); label(\"$A$\",(-0.4,0.7)); label(\"$B$\",(0.7,0.4)); [/asy]",
        "answer": "$\\frac{32}{5}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 132,
        "problem": "In $\\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\\overline{AC}$ and $\\overline{AB}$, respectively, so that $\\overline{BD}$ and $\\overline{CE}$ are angle bisectors, intersecting $\\overline{AH}$ at $Q$ and $P$, respectively. What is the value of the ratio: $PQ/AP$?",
        "geo_code": "[asy] import graph; size(9cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.381056062031275, xmax = 15.020004395092375, ymin = -4.051697595316909, ymax = 10.663513514111651; /* image dimensions */ draw((0.,0.)--(4.714285714285714,7.666518779999279)--(7.,0.)--cycle); /* draw figures */ draw((0.,0.)--(4.714285714285714,7.666518779999279)); draw((4.714285714285714,7.666518779999279)--(7.,0.)); draw((7.,0.)--(0.,0.)); label(\"7\",(3.2916797119724284,-0.07831656949355523),SE*labelscalefactor); label(\"9\",(2.0037562070503783,4.196493361737088),SE*labelscalefactor); label(\"8\",(6.114150371695219,3.785453945272603),SE*labelscalefactor); draw((0.,0.)--(6.428571428571427,1.9166296949998194)); draw((7.,0.)--(2.2,3.5777087639996634)); draw((4.714285714285714,7.666518779999279)--(3.7058823529411766,0.)); /* dots and labels */ dot((0.,0.),dotstyle); label(\"$A$\", (-0.2432592696221352,-0.5715638692509372), NE * labelscalefactor); dot((7.,0.),dotstyle); label(\"$B$\", (7.0458397156813835,-0.48935598595804014), NE * labelscalefactor); dot((3.7058823529411766,0.),dotstyle); label(\"$E$\", (3.8123296394941084,0.16830708038513573), NE * labelscalefactor); dot((4.714285714285714,7.666518779999279),dotstyle); label(\"$C$\", (4.579603216894479,7.895848109917452), NE * labelscalefactor); dot((2.2,3.5777087639996634),linewidth(3.pt) + dotstyle); label(\"$D$\", (2.1407693458718726,3.127790878929427), NE * labelscalefactor); dot((6.428571428571427,1.9166296949998194),linewidth(3.pt) + dotstyle); label(\"$H$\", (6.004539860638023,1.9494778850645704), NE * labelscalefactor); dot((5.,1.49071198499986),linewidth(3.pt) + dotstyle); label(\"$Q$\", (4.935837377830365,1.7302568629501784), NE * labelscalefactor); dot((3.857142857142857,1.1499778169998918),linewidth(3.pt) + dotstyle); label(\"$P$\", (3.538303361851119,1.2370095631927964), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]",
        "answer": "$\\frac{8}{27}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 133,
        "problem": "In $\\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\\overline{BC}$, and $\\overline{AD}$ bisects $\\angle BAC$. Point $E$ lies on $\\overline{AC}$, and $\\overline{BE}$ bisects $\\angle ABC$. The bisectors intersect at $F$. What is the value of the ratio $AF$ : $AD$?",
        "geo_code": "[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,NE); label(\"$E$\",E,NW); label(\"$F$\",F,1.5*N); [/asy]",
        "answer": "$\\frac{2}{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 134,
        "problem": "In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\\overline{AD}$. Points $F$ and $G$ lie on $\\overline{CE}$, and $H$ and $J$ lie on $\\overline{AB}$ and $\\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\\overline{GH}$, and $M$ and $N$ lie on $\\overline{AD}$ and $\\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $FGHJ$ is 539. Find the area of $KLMN$.",
        "geo_code": "[asy] pair A,B,C,D,E,F,G,H,J,K,L,M,N; B=(0,0); real m=7*sqrt(55)/5; J=(m,0); C=(7*m/2,0); A=(0,7*m/2); D=(7*m/2,7*m/2); E=(A+D)/2; H=(0,2m); N=(0,2m+3*sqrt(55)/2); G=foot(H,E,C); F=foot(J,E,C); draw(A--B--C--D--cycle); draw(C--E); draw(G--H--J--F); pair X=foot(N,E,C); M=extension(N,X,A,D); K=foot(N,H,G); L=foot(M,H,G); draw(K--N--M--L); label(\"$A$\",A,NW); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$E$\",E,dir(90)); label(\"$F$\",F,NE); label(\"$G$\",G,NE); label(\"$H$\",H,W); label(\"$J$\",J,S); label(\"$K$\",K,SE); label(\"$L$\",L,SE); label(\"$M$\",M,dir(90)); label(\"$N$\",N,dir(180)); [/asy]",
        "answer": "$99$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 135,
        "problem": "A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \\ge 1$, the circles in $\\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \\[\\sum_{C\\in S} \\frac{1}{\\sqrt{r(C)}}?\\]",
        "geo_code": "[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]",
        "answer": "$\\frac{143}{14}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 136,
        "problem": "The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\\tfrac{3}{2}$ and center $(0,\\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?",
        "geo_code": "[asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP(\"x\",(3.5,0),S)--(0,0)--MP(\"\\frac{3}{2}\",(0,3/2),W)--MP(\"y\",(0,3.5),W)); path P=(0,0)--MP(\"3\",(3,0),S)..(3*dir(45))..MP(\"3\",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy]",
        "answer": "$\\frac{9\\pi}{8}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 137,
        "problem": "The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what numerical point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?",
        "geo_code": "[asy]size(170);defaultpen(linewidth(0.9)+fontsize(13pt));draw(unitcircle^^circle((0,1.5),0.5)); path arrow = origin--(-0.13,-0.35)--(-0.06,-0.35)--(-0.06,-0.7)--(0.06,-0.7)--(0.06,-0.35)--(0.13,-0.35)--cycle; for(int i=1;i<=12;i=i+1){draw(0.9*dir(90-30*i)--dir(90-30*i));label(\"$\"+(string) i+\"$\",0.78*dir(90-30*i));} dot(origin);draw(shift((0,1.87))*arrow);draw(arc(origin,1.5,68,30),EndArrow(size=12));[/asy]",
        "answer": "$4$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 138,
        "problem": "Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\\overline{AB}.$ What fraction of the area of the octagon is shaded?",
        "geo_code": "[asy] pair A,B,C,D,E,F,G,H,O,X; A=dir(45); B=dir(90); C=dir(135); D=dir(180); E=dir(-135); F=dir(-90); G=dir(-45); H=dir(0); O=(0,0); X=midpoint(A--B); fill(X--B--C--D--E--O--cycle,rgb(0.75,0.75,0.75)); draw(A--B--C--D--E--F--G--H--cycle); dot(\"$A$\",A,dir(45)); dot(\"$B$\",B,dir(90)); dot(\"$C$\",C,dir(135)); dot(\"$D$\",D,dir(180)); dot(\"$E$\",E,dir(-135)); dot(\"$F$\",F,dir(-90)); dot(\"$G$\",G,dir(-45)); dot(\"$H$\",H,dir(0)); dot(\"$X$\",X,dir(135/2)); dot(\"$O$\",O,dir(0)); draw(E--O--X); [/asy]",
        "answer": "$\\frac{7}{16}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 139,
        "problem": "A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find $a$.",
        "geo_code": "[asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36); // don't look here D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(\" \",A,NW); dot(\" \",B,NW); dot(\" \",C,NW); dot(\" \",D,NW); dot(\" \",E,NW); dot(\" \",F,NW); //don't look here R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(\" \",R,NW); dot(\" \",S,NW); dot(\" \",T,NW); dot(\" \",X,NW); dot(\" \",Y,NW); dot(\" \",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236[/asy]",
        "answer": "$12\\sqrt{5}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 140,
        "problem": "In the figure, $ABCD$ is a square of side length $2$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $AE$?",
        "geo_code": "[asy] pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2); draw(A--B--C--D--cycle); draw(K--H--G--J--cycle); draw(F--E); label(\"$A$\",A,SE); label(\"$B$\",B,SW); label(\"$C$\",C,NW); label(\"$D$\",D,NE); label(\"$E$\",E,S); label(\"$F$\",F,N); label(\"$G$\",G,E); label(\"$H$\",H,N); label(\"$J$\",J,S); label(\"$K$\",K,W); [/asy]",
        "answer": "$2\\sqrt{3}-2$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 141,
        "problem": "Six regular hexagons surround a regular hexagon of side length $2$ as shown. What is the area of $\\triangle{ABC}$?",
        "geo_code": "[asy] draw((0,0)--(-5,8.66025404)--(0, 17.3205081)--(10, 17.3205081)--(15,8.66025404)--(10, 0)--(0, 0)); draw((30,0)--(25,8.66025404)--(30, 17.3205081)--(40, 17.3205081)--(45, 8.66025404)--(40, 0)--(30, 0)); draw((30,0)--(25,-8.66025404)--(30, -17.3205081)--(40, -17.3205081)--(45, -8.66025404)--(40, 0)--(30, 0)); draw((0,0)--(-5, -8.66025404)--(0, -17.3205081)--(10, -17.3205081)--(15, -8.66025404)--(10, 0)--(0, 0)); draw((15,8.66025404)--(10, 17.3205081)--(15, 25.9807621)--(25, 25.9807621)--(30, 17.3205081)--(25, 8.66025404)--(15, 8.66025404)); draw((15,-8.66025404)--(10, -17.3205081)--(15, -25.9807621)--(25, -25.9807621)--(30, -17.3205081)--(25, -8.66025404)--(15, -8.66025404)); label(\"A\", (0,0), W); label(\"B\", (30, 17.3205081), NE); label(\"C\", (30, -17.3205081), SE); draw((0,0)--(30, 17.3205081)--(30, -17.3205081)--(0, 0)); //(Diagram Creds-DivideBy0) [/asy]",
        "answer": "$12\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 142,
        "problem": "In triangle $ABC$, determine the measure of angle A. Note that all line segments marked with short ticks in the diagram are of equal length.",
        "geo_code": "[asy]\nimport graph; size(9.115122858763474cm); \nreal labelscalefactor = 0.5; /* changes label-to-point distance */\npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ \npen dotstyle = black; /* point style */ \nreal xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273;  /* image dimensions */\n\n /* draw figures */\ndraw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6)); \ndraw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6)); \ndraw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6)); \ndraw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6)); \ndraw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6)); \ndraw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6)); \ndraw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6)); \ndraw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6)); \nlabel(\"$A$\",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14)); \nlabel(\"$B$\",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14)); \nlabel(\"$C$\",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14)); \n /* dots and labels */\ndot((3.0842,-3.6348),linewidth(3.pt) + dotstyle); \ndot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle); \ndot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle); \ndot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle); \ndot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle); \ndot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle); \ndot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n\n[/asy]",
        "answer": "$72$",
        "category": "Global Abstract Integration",
        "source": "olympiads",
        "problem_type": "angle"
    },
    {
        "index": 143,
        "problem": "In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\\overline{BC}$ , and $\\overline{AD}$ bisects angle $CAB$ . Points $E$ and $F$ are on $\\overline{AB}$ and $\\overline{AC}$ , respectively, so that $AE=3$ and $AF=10$ . Given that $EB=9$ and $FC=27$ , find the area of quadrilateral $DCFG$. The final answer should discards the decimal part and retains only the integer.",
        "geo_code": "[asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$D$\", D, dir(point--D)); label(\"$E$\", E, dir(point--E)); label(\"$F$\", F, dir(point--F)); label(\"$G$\", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label(\"10\", A--F, dir(90)*dir(A--F)); label(\"27\", F--C, dir(90)*dir(F--C)); label(\"3\", (0,10), W); label(\"9\", (0,4), W); [/asy]",
        "answer": "$147$",
        "category": "Local Relation Composition",
        "source": "AIME-83-24",
        "problem_type": "area"
    },
    {
        "index": 144,
        "problem": "In $\\triangle{ABC}$, $AX=XY=YB=BC$, and $m\\angle{ABC}=120^{\\circ}$. What is the measure of $m\\angle{BCA}$?",
        "geo_code": "[asy]\npair A, B, C, X, Y;\nA = origin;\nX = dir(30);\nY = X + dir(0);\nB = Y + dir(60);\nC = B + dir(330);\ndraw(A--B--C--cycle);\ndraw(X--Y--B);\nlabel(\"$A$\",A,W);\nlabel(\"$B$\",B,N);\nlabel(\"$C$\",C,E);\nlabel(\"$X$\",X,NW);\nlabel(\"$Y$\",Y,SE);\n[/asy]",
        "answer": "$45$",
        "category": "Local Relation Composition",
        "source": "olympiads",
        "problem_type": "angle"
    },
    {
        "index": 145,
        "problem": "In triangle $ABC$, determine the measure of angle C. Note that all line segments marked with short ticks in the diagram are of equal length.",
        "geo_code": "[asy]\nimport graph; size(9.115122858763474cm); \nreal labelscalefactor = 0.5; /* changes label-to-point distance */\npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ \npen dotstyle = black; /* point style */ \nreal xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273;  /* image dimensions */\n\n /* draw figures */\ndraw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6)); \ndraw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6)); \ndraw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6)); \ndraw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6)); \ndraw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6)); \ndraw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6)); \ndraw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6)); \ndraw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6)); \nlabel(\"$A$\",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14)); \nlabel(\"$B$\",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14)); \nlabel(\"$C$\",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14)); \n /* dots and labels */\ndot((3.0842,-3.6348),linewidth(3.pt) + dotstyle); \ndot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle); \ndot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle); \ndot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle); \ndot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle); \ndot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle); \ndot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n /* end of picture */\n[/asy]",
        "answer": "$36$",
        "category": "Local Relation Composition",
        "source": "olympiads",
        "problem_type": "angle"
    },
    {
        "index": 146,
        "problem": "The diagram shows an octagon consisting of $10$ unit squares. The portion below $\\overline{PQ}$ is a unit square and a triangle with base $5$. If $\\overline{PQ}$ bisects the area of the octagon, what is the ratio $\\dfrac{XQ}{XY}$?",
        "geo_code": "[asy] import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((0,0)--(6,0),linewidth(1.2pt)); draw((0,0)--(0,1),linewidth(1.2pt)); draw((0,1)--(1,1),linewidth(1.2pt)); draw((1,1)--(1,2),linewidth(1.2pt)); draw((1,2)--(5,2),linewidth(1.2pt)); draw((5,2)--(5,1),linewidth(1.2pt)); draw((5,1)--(6,1),linewidth(1.2pt)); draw((6,1)--(6,0),linewidth(1.2pt)); draw((1,1)--(5,1),linewidth(1.2pt)); draw((1,1)--(1,0),linewidth(1.2pt)); draw((2,2)--(2,0),linewidth(1.2pt)); draw((3,2)--(3,0),linewidth(1.2pt)); draw((4,2)--(4,0),linewidth(1.2pt)); draw((5,1)--(5,0),linewidth(1.2pt)); draw((0,0)--(5,1.5),linewidth(1.2pt)); dot((0,0),ds); label(\"$P$\", (-0.23,-0.26),NE*lsf); dot((0,1),ds); dot((1,1),ds); dot((1,2),ds); dot((5,2),ds); label(\"$X$\", (5.14,2.02),NE*lsf); dot((5,1),ds); label(\"$Y$\", (5.12,1.14),NE*lsf); dot((6,1),ds); dot((6,0),ds); dot((1,0),ds); dot((2,0),ds); dot((3,0),ds); dot((4,0),ds); dot((5,0),ds); dot((2,2),ds); dot((3,2),ds); dot((4,2),ds); dot((5,1.5),ds); label(\"$Q$\", (5.14,1.51),NE*lsf); clip((-4.19,-5.52)--(-4.19,6.5)--(10.08,6.5)--(10.08,-5.52)--cycle); [/asy]",
        "answer": "$\\frac{2}{5}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 147,
        "problem": "In the accompanying figure, the outer square $S$ has side length $20$. A second square $S'$ of side length $7.5$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two closest vertices of $S'$. The result is a four-pointed starlike figure inscribed in $S$. The star figure is cut out and then folded to form a pyramid with base $S'$. Find the volume of this pyramid.",
        "geo_code": "[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]",
        "answer": "$\\frac{375}{4}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "volume"
    },
    {
        "index": 148,
        "problem": "Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the area of the two semicircles?",
        "geo_code": "[asy] import graph; size(7.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-6.27,xmax=10.01,ymin=-5.65,ymax=10.98; draw(circle((0,0),2)); draw((-3,0)--(3,0),EndArrow(6)); draw((0,-3)--(0,3),EndArrow(6)); draw(shift((0.01,1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,179.76,359.76)); draw(shift((-0.01,-1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,-0.38,179.62)); draw((-1.4,1.43)--(1.41,1.41)); draw((-1.42,-1.41)--(1.4,-1.42)); label(\"$ P(-1,1) $\",(-2.57,2.17),SE*lsf); label(\"$ Q(1,1) $\",(1.55,2.21),SE*lsf); label(\"$ R(-1,-1) $\",(-2.72,-1.45),SE*lsf); label(\"$S(1,-1)$\",(1.59,-1.49),SE*lsf); dot((0,0),ds); label(\"$O$\",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4,-1.42),ds); dot((-1.42,-1.41),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]",
        "answer": "$\\pi$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 149,
        "problem": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $20$ and $30$ meters. What fraction of the yard is occupied by the flower beds?\n\n\n$\\mathrm{(A)}\\frac {1}{8}\\qquad \\mathrm{(B)}\\frac {1}{6}\\qquad \\mathrm{(C)}\\frac {1}{5}\\qquad \\mathrm{(D)}\\frac {1}{4}\\qquad \\mathrm{(E)}\\frac {1}{3}$",
        "geo_code": "[asy] unitsize(2mm); defaultpen(linewidth(.8pt));  fill((0,0)--(0,5)--(5,5)--cycle,gray); fill((25,0)--(25,5)--(20,5)--cycle,gray); draw((0,0)--(0,5)--(25,5)--(25,0)--cycle); draw((0,0)--(5,5)); draw((20,5)--(25,0)); [/asy]",
        "answer": "$\\frac{1}{6}$",
        "category": "Local Relation Composition",
        "source": "olympiads",
        "problem_type": "ratio"
    },
    {
        "index": 150,
        "problem": "Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. How many times the diameter of $C_2$ is the diameter of $C_4$?",
        "geo_code": "[asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label(\"$C_4$\", D); label(\"$C_1$\", (-1.375, 0)); label(\"$C_2$\", (1.375,0)); label(\"$\\frac{1}{2}$\", (0, -.125)); label(\"$C_3$\", (-0.4, -0.4)); label(\"$1$\", (-.85, 0.70)); label(\"$1$\", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy]",
        "answer": "$\\frac{28}{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 151,
        "problem": "Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What is the measure of angle $\\theta = \\angle PAB=\\angle QPC=\\angle RQB=\\cdots$ will produce in a length that is $200$ meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.)",
        "geo_code": "[asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label(\"$\\theta$\", (-41.5,-10.5)); label(\"$\\theta$\", (-13,10.5)); label(\"$\\theta$\", (15.5,-10.5)); label(\"$\\theta$\", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label(\"$D$\",(-58,15)); label(\"$A$\",(-58,-15)); label(\"$C$\",(58,15)); label(\"$B$\",(58,-15)); label(\"$S$\",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); label(\"$P$\",(-22.5,23)); label(\"$Q$\",(5,-23)); label(\"$R$\",(32.5,23)); [/asy]",
        "answer": "$60$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 152,
        "problem": "Six regular hexagonal blocks of side length 2 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\\frac{3}{7}$ unit. What is the area of the region inside the frame not occupied by the blocks?",
        "geo_code": "[asy] unitsize(1cm); draw(scale(3)*polygon(6)); filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), lightgray); filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), lightgray); filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), lightgray); filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), lightgray); filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray); filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray); [/asy]",
        "answer": "$18\\sqrt{3}}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 153,
        "problem": "Isosceles $\\triangle ABC$ has equal side lengths $AB$ and $BC$. In the figure below, segments are drawn parallel to $\\overline{AC}$ so that the shaded portions of $\\triangle ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\\triangle ABC$?",
        "geo_code": "[asy] size(12cm); real h = 2.5; // height real g=4; //c2c space real s = 0.65; //Xcord of Hline real adj = 0.08; //adjust line diffs pair A,B,C; B=(0,h); C=(1,0); A=-conj(C); pair PONE=(s,h*(1-s)); //Endpoint of Hline ONE pair PTWO=(s+adj,h*(1-s-adj)); //Endpoint of Hline ONE path LONE=PONE--(-conj(PONE)); //Hline ONE path LTWO=PTWO--(-conj(PTWO)); path T=A--B--C--cycle; //Triangle fill (shift(g,0)*(LTWO--B--cycle),mediumgrey); fill (LONE--A--C--cycle,mediumgrey); draw(LONE); draw(T); label(\"$A$\",A,SW); label(\"$B$\",B,N); label(\"$C$\",C,SE); draw(shift(g,0)*LTWO); draw(shift(g,0)*T); label(\"$A$\",shift(g,0)*A,SW); label(\"$B$\",shift(g,0)*B,N); label(\"$C$\",shift(g,0)*C,SE); draw(B--shift(g,0)*B,dashed); draw(C--shift(g,0)*A,dashed); draw((g/2,0)--(g/2,h),dashed); draw((0,h*(1-s))--B,dashed); draw((g,h*(1-s-adj))--(g,0),dashed); label(\"$5$\", midpoint((g,h*(1-s-adj))--(g,0)),UnFill); label(\"$h$\", midpoint((g/2,0)--(g/2,h)),UnFill); label(\"$11$\", midpoint((0,h*(1-s))--B),UnFill); [/asy]",
        "answer": "$14.6$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 154,
        "problem": "A <i>regular octahedron</i> has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the left of $Q$?",
        "geo_code": "[asy] import graph; size(15cm); pair Fr, Lf, Rt, Tp, Bt, Bk; Lf=(0,0); Rt=(12,1); Fr=(7,-1); Bk=(5,2); Tp=(6,6.7); Bt=(6,-5.2); draw(Lf--Fr--Rt); draw(Lf--Tp--Rt); draw(Lf--Bt--Rt); draw(Tp--Fr--Bt); draw(Lf--Bk--Rt,dashed); draw(Tp--Bk--Bt,dashed); label(rotate(-8.13010235)*slant(0.1)*\"$Q$\", (4.2,1.6)); label(rotate(21.8014095)*slant(-0.2)*\"$?$\", (8.5,2.05)); pair g = (-8,0); real a = 8; draw(g+(-a/2,1)--g+(a/2,1), Arrow()); pair DA,DB,DC,CD,O; DA = (4*sqrt(3),0); DB = (2*sqrt(3),6); DC = (DA+DB)/3; CD = conj(DC); O=(0,0); transform trf=shift(3g+(0,3)); path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB); draw(trf*NET); label(\"$7$\",trf*DC); label(\"$Q$\",trf*DC+DA-DB); label(\"$5$\",trf*DC-DB); label(\"$3$\",trf*DC-DA-DB); label(\"$6$\",trf*CD); label(\"$4$\",trf*CD-DA); label(\"$2$\",trf*CD-DA-DB); label(\"$1$\",trf*CD-2DA); [/asy]",
        "answer": "$6$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 155,
        "problem": "The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is not shaded?",
        "geo_code": "[asy] size(6cm); draw(circle((3,3),3)); filldraw(circle((2,3),2),lightgrey); filldraw(circle((3,3),1),white); filldraw(circle((1,3),1),white); filldraw(circle((5.5,3),0.5),lightgrey); filldraw(circle((4.5,4.5),0.5),lightgrey); filldraw(circle((4.5,1.5),0.5),lightgrey); int i, j; for(i=0; i<7; i=i+1) { draw((0,i)--(6,i), dashed+grey); draw((i,0)--(i,6), dashed+grey); } [/asy]",
        "answer": "$\\frac{25}{36}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 156,
        "problem": "A bowl is formed by attaching four regular hexagons of side $3$ to a square of side $3$. The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?",
        "geo_code": "[asy] import three; size(225); currentprojection= orthographic(camera=(-5.52541796301147,-2.61548797564715,1.6545450372312), up=(0.00247902062334861,0.000877141782387748,0.00966536329192992), target=(0,0,0), zoom=0.570588560870951); currentpen = black+1.5bp; triple A = O; triple M = (X+Y)/2; triple B = (-1/2,-1/2,1/sqrt(2)); triple C = (-1,0,sqrt(2)); triple D = (0,-1,sqrt(2)); transform3 rho = rotate(90,M,M+Z); //arrays of vertices for the lower level (the square), the middle level, //and the interleaves vertices of the upper level (the octagon) triple[] lVs = {A}; triple[] mVs = {B}; triple[] uVsl = {C}; triple[] uVsr = {D}; for(int i = 0; i < 3; ++i){ lVs.push(rho*lVs[i]); mVs.push(rho*mVs[i]); uVsl.push(rho*uVsl[i]); uVsr.push(rho*uVsr[i]); } lVs.cyclic = true; uVsl.cyclic = true; for(int i : new int[] {0,1,2,3}){ draw(uVsl[i]--uVsr[i]); draw(uVsr[i]--uVsl[i+1]); } draw(lVs[0]--lVs[1]^^lVs[0]--lVs[3]); for(int i : new int[] {0,1,3}){ draw(lVs[0]--lVs[i]); draw(lVs[i]--mVs[i]); draw(mVs[i]--uVsl[i]); } for(int i : new int[] {0,3}){ draw(mVs[i]--uVsr[i]); } for(int i : new int[] {1,3}) draw(lVs[2]--lVs[i],dashed); draw(lVs[2]--mVs[2],dashed); draw(mVs[2]--uVsl[2]^^mVs[2]--uVsr[2],dashed); draw(mVs[1]--uVsr[1],dashed); //Comment two lines below to remove red edges //draw(lVs[1]--lVs[3],red+2bp); //draw(uVsl[0]--uVsr[0],red+2bp); [/asy]",
        "answer": "$63$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 157,
        "problem": "A ball with diameter 8 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 120$ inches, $R_2 = 80$ inches, and $R_3 = 100$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?",
        "geo_code": "[asy] pair A,B; size(8cm); A=(0,0); B=(480,0); draw((0,0)--(480,0),linetype(\"3 4\")); filldraw(circle((8,0),8),black); draw((0,0)..(100,-100)..(200,0)); draw((200,0)..(260,60)..(320,0)); draw((320,0)..(400,-80)..(480,0)); draw((100,0)--(150,-50sqrt(3)),Arrow(size=4)); draw((260,0)--(290,30sqrt(3)),Arrow(size=4)); draw((400,0)--(440,-40sqrt(3)),Arrow(size=4)); label(\"$A$\", A, SW); label(\"$B$\", B, SE); label(\"$R_1$\", (100,-40), W); label(\"$R_2$\", (260,40), SW); label(\"$R_3$\", (400,-40), W);[/asy]",
        "answer": "$296\\pi$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 158,
        "problem": "Four identical squares and one rectangle are placed together to form one large square as shown. The width of the rectangle is how many times as large as its length?",
        "geo_code": "[asy] unitsize(8mm); defaultpen(linewidth(.8pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((0,3)--(0,4)--(1,4)--(1,3)--cycle); draw((1,3)--(1,4)--(2,4)--(2,3)--cycle); draw((2,3)--(2,4)--(3,4)--(3,3)--cycle); draw((3,3)--(3,4)--(4,4)--(4,3)--cycle); [/asy]",
        "answer": "$\\frac{3}{4}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 159,
        "problem": "In triangle $ABC$, determine the measure of angle B. Note that all line segments marked with short ticks in the diagram are of equal length.",
        "geo_code": "[asy]\nimport graph; size(9.115122858763474cm); \nreal labelscalefactor = 0.5; /* changes label-to-point distance */\npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ \npen dotstyle = black; /* point style */ \nreal xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273;  /* image dimensions */\n\n /* draw figures */\ndraw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6)); \ndraw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6)); \ndraw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6)); \ndraw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6)); \ndraw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6)); \ndraw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6)); \ndraw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6)); \ndraw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6)); \nlabel(\"$A$\",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14)); \nlabel(\"$B$\",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14)); \nlabel(\"$C$\",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14)); \n /* dots and labels */\ndot((3.0842,-3.6348),linewidth(3.pt) + dotstyle); \ndot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle); \ndot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle); \ndot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle); \ndot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle); \ndot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle); \ndot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n\n[/asy]",
        "answer": "$72$",
        "category": "Primitive Recognition",
        "source": "olympiads",
        "problem_type": "angle"
    },
    {
        "index": 160,
        "problem": "Rectangle $ABCD$ is half the area of right triangle. They are joined to form a trapezoid, as shown. What is $DE$?",
        "geo_code": "[asy] size(250); defaultpen(linewidth(0.8)); pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0); draw(A--B--E--D--cycle^^C--D); draw(rightanglemark(D,C,E,30)); label(\"$A$\",A,NW); label(\"$B$\",B,SW); label(\"$C$\",C,S); label(\"$D$\",D,N); label(\"$E$\",E,S); label(\"$5$\",A/2,W); label(\"$6$\",(A+D)/2,N); [/asy]",
        "answer": "$\\sqrt{601}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 161,
        "problem": "A circle with radius $2$ is inscribed in a square and circumscribed about another square as shown. What is the ratio of the area between the two squares to circle's shaded area ?",
        "geo_code": "[asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,gray,black); filldraw(Circle((0,0),1), mediumgray,black); filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy]",
        "answer": "$\\frac{2}{\\pi-1}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 162,
        "problem": "The figure below shows a polygon $ABCDEFGH$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$. What is the volume of the prism?",
        "geo_code": "[asy] usepackage(\"mathptmx\"); size(275); defaultpen(linewidth(0.8)); real r = 2, s = 2.5, theta = 14; pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta); pair N = (B+G)/2, J = N + s/2 * dir(180+theta); pair E = F + r * dir(- 45 - theta/2), D = I+E-F; pair H = J + r * dir(135 + theta/2), A = B+H-J; draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B)); draw(J--B--G^^C--F--I,linetype (\"4 4\")); dot(\"$A$\",A,N); dot(\"$B$\",B,1.2*N); dot(\"$C$\",C,N); dot(\"$D$\",D,dir(0)); dot(\"$E$\",E,S); dot(\"$F$\",F,1.5*dir(-100)); dot(\"$G$\",G,S); dot(\"$H$\",H,W); dot(\"$I$\",I,NE); dot(\"$J$\",J,1.5*S); [/asy]",
        "answer": "$192$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 163,
        "problem": "The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the side length of the logo in square inches?",
        "geo_code": "[asy] defaultpen(linewidth(0.5)); size(5cm); defaultpen(fontsize(14pt)); label(\"$\\textbf{Math}$\", (2.1,3.7)--(3.9,3.7)); label(\"$\\textbf{Team}$\", (2.1,3)--(3.9,3)); filldraw((1,2)--(2,1)--(3,2)--(4,1)--(5,2)--(4,3)--(5,4)--(4,5)--(3,4)--(2,5)--(1,4)--(2,3)--(1,2)--cycle, mediumgray*0.5 + lightgray*0.5); draw((0,0)--(6,0), gray); draw((0,1)--(6,1), gray); draw((0,2)--(6,2), gray); draw((0,3)--(6,3), gray); draw((0,4)--(6,4), gray); draw((0,5)--(6,5), gray); draw((0,6)--(6,6), gray); draw((0,0)--(0,6), gray); draw((1,0)--(1,6), gray); draw((2,0)--(2,6), gray); draw((3,0)--(3,6), gray); draw((4,0)--(4,6), gray); draw((5,0)--(5,6), gray); draw((6,0)--(6,6), gray); [/asy]",
        "answer": "$12\\sqrt{2}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 164,
        "problem": "What is the area of the shaded figure shown below?",
        "geo_code": "[asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label(\"${\"+string(i)+\"}$\", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label(\"${\"+string(i)+\"}$\", (0,i), 2*W); } } label(\"$0$\", O, 2*SW); draw(O--X+(0.35,0), black+1.5, EndArrow(10)); draw(O--Y+(0,0.35), black+1.5, EndArrow(10)); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]",
        "answer": "$6$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 165,
        "problem": "Let $ABCD$ be an isosceles trapezoid with $\\overline{BC}\\parallel \\overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\\angle AXD = \\angle BYC = 90^\\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABC?$",
        "geo_code": "[asy] size(10cm); usepackage(\"mathptmx\"); import geometry; void perp(picture pic=currentpicture, pair O, pair M, pair B, real size=5, pen p=currentpen, filltype filltype = NoFill){ perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype); } pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6)); draw(A--B--C--D--cycle,p); draw(A--C,p); draw(B--Y,p); draw(D--X,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(X,q); dot(Y,q); label(\"2\",C--Y,S); label(\"1\",Y--X,S); label(\"3\",X--A,S); label(\"$A$\",A,2*E); label(\"$B$\",B,2*N); label(\"$C$\",C,2*W); label(\"$D$\",D,2*S); label(\"$Y$\",Y,2*sqrt(2)*NE); label(\"$X$\",X,2*N); perp(B,Y,C,8,p); perp(A,X,D,8,p); [/asy]",
        "answer": "$\\frac{6\\sqrt{35}}{5}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 166,
        "problem": "In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\\circ$. The enclosed area of the hexagon is $24\\sqrt{3}$. What is the perimeter of the hexagon?",
        "geo_code": "[asy] size(10cm); pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F; draw(C--D--E--F--A--B--cycle,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); label(\"$C$\",C,2*S); label(\"$D$\",D,2*S); label(\"$E$\",E,2*S); label(\"$F$\",F,2*dir(0)); label(\"$A$\",A,2*N); label(\"$B$\",B,2*W); [/asy]",
        "answer": "$24\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 167,
        "problem": "As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\\angle CDE = 110^\\circ$. Point $F$ lies on $\\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\\angle EFD$?",
        "geo_code": "[asy] size(6cm); pair A = (0,10); label(\"$A$\", A, N); pair B = (0,0); label(\"$B$\", B, S); pair C = (10,0); label(\"$C$\", C, S); pair D = (10,10); label(\"$D$\", D, SW); pair EE = (15,11.8); label(\"$E$\", EE, N); pair F = (3,10); label(\"$F$\", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label(\"$110^\\circ$\", (15,9), SW); [/asy]",
        "answer": "$10$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 168,
        "problem": "In square $ABCD$, points $P$ and $Q$ lie on $\\overline{AD}$ and $\\overline{AB}$, respectively. Segments $\\overline{BP}$ and $\\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square?",
        "geo_code": "[asy] size(170); defaultpen(linewidth(0.6)+fontsize(10)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot(\"$A$\",A,S); dot(\"$B$\",B,S); dot(\"$C$\",C,N); dot(\"$D$\",D,N); dot(\"$Q$\",Q,S); dot(\"$P$\",P,W); dot(\"$R$\",R,1.3*S); label(\"$7$\",(P+R)/2,NE); label(\"$6$\",(R+B)/2,NE); [/asy]",
        "answer": "$117$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 169,
        "problem": "Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\\overline{BD} \\perp \\overline{BC}$. The line $\\ell$ through $D$ parallel to line $BC$ intersects sides $\\overline{AB}$ and $\\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\\ell$ such that $F$ is between $E$ and $G$, $\\triangle AFG$ is isosceles, and the ratio of the area of $\\triangle AFG$ to the area of $\\triangle BED$ is $8:9$. Find $FC$.",
        "geo_code": "[asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label(\"$A$\",A,dir(90)); label(\"$B$\",B,dir(225)); label(\"$C$\",C,dir(-45)); label(\"$D$\",D,dir(180)); label(\"$E$\",E,dir(-45)); label(\"$F$\",F,dir(225)); label(\"$G$\",G,dir(0)); label(\"$\\ell$\",midpoint(E--F),dir(90)); [/asy]",
        "answer": "$504$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 170,
        "problem": "Let $ABCD$ be an isoceles trapezoid having parallel bases $\\overline{AB}$ and $\\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\\frac{CD}{AB}?$",
        "geo_code": "[asy] unitsize(100); pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); draw(A--B--C--D--cycle, black); draw(A--P, black); draw(B--P, black); draw(C--P, black); draw(D--P, black); label(\"$A$\",A,(-1,0)); label(\"$B$\",B,(1,0)); label(\"$C$\",C,(1,-0)); label(\"$D$\",D,(-1,0)); label(\"$2$\",E,(0,0)); label(\"$3$\",F,(0,0)); label(\"$4$\",G,(0,0)); label(\"$5$\",H,(0,0)); dot(A^^B^^C^^D^^P); [/asy]",
        "answer": "$2-\\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 171,
        "problem": "The figure is constructed from $11$ line segments, each of which has length $2$. Find the area of pentagon $ABCDE$.",
        "geo_code": "[asy] pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label(\"A\",A,N); label(\"B\",B,W); label(\"C\",C,S); label(\"D\",D,S); label(\"E\",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy]",
        "answer": "$\\sqrt{12}+\\sqrt{11}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 172,
        "problem": "Equilateral $\\triangle ABC$ has side length $3$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?",
        "geo_code": "[asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); pair B = (0,0); pair C = (1,0); pair A = rotate(60,B)*C; pair E = rotate(270,A)*B; pair D = rotate(270,E)*A; pair F = rotate(90,A)*C; pair G = rotate(90,F)*A; pair I = rotate(270,B)*C; pair H = rotate(270,I)*B; draw(A--B--C--cycle); draw(A--E--D--B); draw(A--F--G--C); draw(B--I--H--C); draw(E--F); draw(D--I); draw(I--H); draw(H--G); label(\"$A$\",A,N); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,W); label(\"$E$\",E,W); label(\"$F$\",F,E); label(\"$G$\",G,E); label(\"$H$\",H,SE); label(\"$I$\",I,SW); [/asy]",
        "answer": "$27+9\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 173,
        "problem": "Three unit squares and two line segments connecting two pairs of vertices are shown. Each square has side length 5. What is the area of $\\triangle ABC$?",
        "geo_code": "[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); draw(A--(2,0)); draw((0,-1)--(2,-1)); draw((0,-2)--(1,-2)); draw(A--(0,-2)); draw(B--(1,-2)); draw((2,0)--(2,-1)); draw(A--(2,-1)); draw(B--(0,-2)); pair[] ps={A,B,C}; dot(ps); label(\"$A$\",A,N); label(\"$B$\",B,N); label(\"$C$\",C,W); [/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 174,
        "problem": "Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\\overline{BC}$, $Y$ on $\\overline{DE}$, and $Z$ on $\\overline{EF}$. Suppose that $AB=40$, and $EF=41(\\sqrt{3}-1)$. What is the perimeter of the square?",
        "geo_code": "[asy] size(200); defaultpen(linewidth(1)); pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60); pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A; draw(A--B--C--D--E--F--cycle); draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype(\"2 2\")); dot(\"$A$\",A,W,linewidth(4)); dot(\"$B$\",B,dir(0),linewidth(4)); dot(\"$C$\",C,dir(0),linewidth(4)); dot(\"$D$\",D,dir(20),linewidth(4)); dot(\"$E$\",E,dir(100),linewidth(4)); dot(\"$F$\",F,W,linewidth(4)); dot(\"$X$\",X,dir(0),linewidth(4)); dot(\"$Y$\",Y,N,linewidth(4)); dot(\"$Z$\",Z,W,linewidth(4)); [/asy]",
        "answer": "$116\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 175,
        "problem": "Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the value of the ratio of the rise of the liquid level in the wide cone to the rise of the liquid level in the narrow cone?",
        "geo_code": "[asy] size(350); defaultpen(linewidth(0.8)); real h1 = 10, r = 3.1, s=0.75; pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q; path e = ellipse((0,h1),r,0.9), ep = ellipse((0,h1*s),r*s,0.9); draw(ellipse(origin,r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill(origin--Pp--Qp--cycle,gray(0.8)); draw((-r,h1)--(0,0)--(r,h1)^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype(\"4 4\")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(0,Qp.y),Arrows(size=8)); draw(origin--(0,12),linetype(\"4 4\")); draw(origin--(r*(s-0.1),0)); label(\"$3$\",(-0.9,h1*s),N,fontsize(10)); real h2 = 7.5, r = 6, s=0.6, d = 14; pair P = (d+r-0.05,h2-0.15), Q = (d-r+0.05,h2-0.15), Pp = s * P + (1-s)*(d,0), Qp = s * Q + (1-s)*(d,0); path e = ellipse((d,h2),r,1), ep = ellipse((d,h2*s+0.09),r*s,1); draw(ellipse((d,0),r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill((d,0)--Pp--Qp--cycle,gray(0.8)); draw(P--(d,0)--Q^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype(\"4 4\")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(d,Qp.y),Arrows(size=8)); draw((d,0)--(d,10),linetype(\"4 4\")); draw((d,0)--(d+r*(s-0.1),0)); label(\"$6$\",(d-r/4,h2*s-0.06),N,fontsize(10)); [/asy]",
        "answer": "$\\frac{1}{4}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 176,
        "problem": "A square piece of paper has side length $6$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\\overline{AD}$ at point $C'$, and edge $\\overline{BC}$ intersects edge $\\overline{AB}$ at point $E$. Suppose that $C'D = \\frac{1}{3}$. What is the perimeter of triangle $\\bigtriangleup AEC' ?$",
        "geo_code": "[asy] pair A=(0,1); pair CC=(0.666666666666,1); pair D=(1,1); pair F=(1,0.440062); pair C=(1,0); pair B=(0,0); pair G=(0,0.22005); pair H=(-0.13,0.41); pair E=(0,0.5); dot(A^^CC^^D^^C^^B^^E); draw(E--A--D--F); draw(G--B--C--F, dashed); fill(E--CC--F--G--H--E--CC--cycle, gray); draw(E--CC--F--G--H--E--CC); label(\"A\",A,NW); label(\"B\",B,SW); label(\"C\",C,SE); label(\"D\",D,NE); label(\"E\",E,NW); label(\"C'\",CC,N); label(\"F\",F,NE); [/asy]",
        "answer": "$12$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 177,
        "problem": "In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?",
        "geo_code": "[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy]",
        "answer": "$7$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 178,
        "problem": "Rectangles $R_1$ and $R_2,$ and squares $S_1,\\,S_2,\\,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the perimeter of $S_2$ in units?",
        "geo_code": "[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw((3,0)--(3,1)--(0,1)); draw((3,1)--(3,2)--(5,2)); draw((3,2)--(2,2)--(2,1)--(2,3)); label(\"$R_1$\",(3/2,1/2)); label(\"$S_3$\",(4,1)); label(\"$S_2$\",(5/2,3/2)); label(\"$S_1$\",(1,2)); label(\"$R_2$\",(7/2,5/2)); [/asy]",
        "answer": "$2604$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 179,
        "problem": "Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\\times$ $27$ rectangle with side $\\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\\overline{AH}$, and partition the octagon into 7 triangles by drawing segments $\\overline{JB}$, $\\overline{JC}$, $\\overline{JD}$, $\\overline{JE}$, $\\overline{JF}$, and $\\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.",
        "geo_code": "[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label(\"$A$\", A, W*r); label(\"$B$\", B, S*r); label(\"$C$\", C, S*r); label(\"$D$\", D, E*r); label(\"$E$\", EE, E*r); label(\"$F$\", F, N*r); label(\"$G$\", G, N*r); label(\"$H$\", H, W*r); label(\"$J$\", J, W*r); [/asy]",
        "answer": "$184$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 180,
        "problem": "A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper? Assume $w$=2 and $h$=3.",
        "geo_code": "[asy] size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label(\"$A$\",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype(\"2.5 2.5\")+linewidth(.5)); draw((3,0)--(-3,0),linetype(\"2.5 2.5\")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label(\"$A$\",(6.5,0),NW); dot((6.5,0)); [/asy]",
        "answer": "$50$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 181,
        "problem": "The twelve-sided figure shown has been drawn on $3 \\text{ cm}\\times 3 \\text{ cm}$ graph paper. What is the area of the figure in $\\text{cm}^2$?",
        "geo_code": "[asy] unitsize(8mm); for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray); draw((0,i+1)--(7,i+1),gray); } draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--(4,1)--(3,2)--(2,2)--cycle,black+2bp); [/asy]",
        "answer": "$117$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 182,
        "problem": "In the figure shown, $\\overline{US}$ and $\\overline{UT}$ are line segments each of length 1, and $m\\angle TUS = 60^\\circ$. Arcs $\\overarc{TR}$ and $\\overarc{SR}$ are each one-sixth of a circle with radius 1. What is the area of the region shown?",
        "geo_code": "[asy]draw((1,1.732)--(2,3.464)--(3,1.732)); draw(arc((0,0),(2,0),(1,1.732))); draw(arc((4,0),(3,1.732),(2,0))); label(\"$U$\", (2,3.464), N); label(\"$S$\", (1,1.732), W); label(\"$T$\", (3,1.732), E); label(\"$R$\", (2,0), S);[/asy]",
        "answer": "$\\sqrt{3}-\\frac{\\pi}{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 183,
        "problem": "In the non-convex quadrilateral $ABCD$ shown below, $\\angle BCD$ is a right angle, $AB=24$, $BC=8$, $CD=6$, and $AD=26$. What is the area of quadrilateral $ABCD$?",
        "geo_code": "[asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0)); label(\"$B$\", (0, 0), SW); label(\"$A$\", (12, 0), ESE); label(\"$C$\", (2.4, 3.6), SE); label(\"$D$\", (0, 5), N);[/asy]",
        "answer": "$96$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 184,
        "problem": "In the figure below, choose point $D$ on $\\overline{BC}$ so that $\\triangle ACD$ and $\\triangle ABD$ have equal perimeters. What is the area of $\\triangle ACD$?",
        "geo_code": "[asy]draw((0,0)--(4,0)--(0,3)--(0,0)); label(\"$A$\", (0,0), SW); label(\"$B$\", (4,0), ESE); label(\"$C$\", (0, 3), N); label(\"$3$\", (0, 1.5), W); label(\"$4$\", (2, 0), S); label(\"$5$\", (2, 1.5), NE);[/asy]",
        "answer": "$\\frac{18}{5}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 185,
        "problem": "In the given figure, $BD = DC$ and $\\angle BCD = 70^\\circ$. By how many times does the measure of $\\angle ADB$ exceed that of $\\angle BCD$?",
        "geo_code": "[asy] size(300); defaultpen(linewidth(0.8)); pair A=(-1,0),C=(1,0),B=dir(40),D=origin; draw(A--B--C--A); draw(D--B); dot(\"$A$\", A, SW); dot(\"$B$\", B, NE); dot(\"$C$\", C, SE); dot(\"$D$\", D, S); label(\"$70^\\circ$\",C,2*dir(180-35));[/asy]",
        "answer": "$2$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 186,
        "problem": "A straight one-mile stretch of highway, which has a width of 40 feet, has been shut down. Robert rides his bike on a path composed of semicircles as shown. If he rides at 2.5 miles per hour, how many hours will it take to cover the one-mile stretch?  Note: 1 mile = 5280 feet",
        "geo_code": "[asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,-1)--(5.5,-1));[/asy]",
        "answer": "$\\frac{\\pi}{5}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 187,
        "problem": "Equilateral $\\triangle ABC$ has side length \\$1$, and squares $ABDE$, $BCHI$, and $CAFG$ are constructed externally on its sides. What is the value of the area of hexagon $DEFGHI$ minus the area of the three squares?",
        "geo_code": "[asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); pair B = (0,0); pair C = (1,0); pair A = rotate(60,B)*C; pair E = rotate(270,A)*B; pair D = rotate(270,E)*A; pair F = rotate(90,A)*C; pair G = rotate(90,F)*A; pair I = rotate(270,B)*C; pair H = rotate(270,I)*B; draw(A--B--C--cycle); draw(A--E--D--B); draw(A--F--G--C); draw(B--I--H--C); draw(E--F); draw(D--I); draw(I--H); draw(H--G); label(\"$A$\",A,N); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,W); label(\"$E$\",E,W); label(\"$F$\",F,E); label(\"$G$\",G,E); label(\"$H$\",H,SE); label(\"$I$\",I,SW); [/asy]",
        "answer": "$\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 188,
        "problem": "In a stack of four cubes arranged as shown, what is twice the length of the portion of $\\overline{XY}$ that is contained in the cube with edge length \\$3$?",
        "geo_code": "[asy] dotfactor = 3; size(10cm); dot((0, 10)); label(\"$X$\", (0,10),W,fontsize(8pt)); dot((6,2)); label(\"$Y$\", (6,2),E,fontsize(8pt)); draw((0, 0)--(0, 10)--(1, 10)--(1, 9)--(2, 9)--(2, 7)--(3, 7)--(3,4)--(4, 4)--(4, 0)--cycle); draw((0,9)--(1, 9)--(1.5, 9.5)--(1.5, 10.5)--(0.5, 10.5)--(0, 10)); draw((1, 10)--(1.5,10.5)); draw((1.5, 10)--(3,10)--(3,8)--(2,7)--(0,7)); draw((2,9)--(3,10)); draw((3,8.5)--(4.5,8.5)--(4.5,5.5)--(3,4)--(0,4)); draw((3,7)--(4.5,8.5)); draw((4.5,6)--(6,6)--(6,2)--(4,0)); draw((4,4)--(6,6)); label(\"$1$\", (1,9.5), W,fontsize(8pt)); label(\"$2$\", (2,8), W,fontsize(8pt)); label(\"$3$\", (3,5.5), W,fontsize(8pt)); label(\"$4$\", (4,2), W,fontsize(8pt)); [/asy]",
        "answer": "$\\frac{6\\sqrt{33}}{5}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 189,
        "problem": "Six regular hexagons are arranged around a central regular hexagon with side length \\$1$, as illustrated. What is twice the area of $\\triangle ABC$?",
        "geo_code": "[asy] draw((0,0)--(-5,8.66025404)--(0, 17.3205081)--(10, 17.3205081)--(15,8.66025404)--(10, 0)--(0, 0)); draw((30,0)--(25,8.66025404)--(30, 17.3205081)--(40, 17.3205081)--(45, 8.66025404)--(40, 0)--(30, 0)); draw((30,0)--(25,-8.66025404)--(30, -17.3205081)--(40, -17.3205081)--(45, -8.66025404)--(40, 0)--(30, 0)); draw((0,0)--(-5, -8.66025404)--(0, -17.3205081)--(10, -17.3205081)--(15, -8.66025404)--(10, 0)--(0, 0)); draw((15,8.66025404)--(10, 17.3205081)--(15, 25.9807621)--(25, 25.9807621)--(30, 17.3205081)--(25, 8.66025404)--(15, 8.66025404)); draw((15,-8.66025404)--(10, -17.3205081)--(15, -25.9807621)--(25, -25.9807621)--(30, -17.3205081)--(25, -8.66025404)--(15, -8.66025404)); label(\"A\", (0,0), W); label(\"B\", (30, 17.3205081), NE); label(\"C\", (30, -17.3205081), SE); draw((0,0)--(30, 17.3205081)--(30, -17.3205081)--(0, 0)); //(Diagram Creds-DivideBy0) [/asy]",
        "answer": "$6\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 190,
        "problem": "Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the area of the square window?",
        "geo_code": "[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill((6,0)--(8,0)--(8,26)--(6,26)--cycle,gray); fill((12,0)--(14,0)--(14,26)--(12,26)--cycle,gray); fill((18,0)--(20,0)--(20,26)--(18,26)--cycle,gray); fill((24,0)--(26,0)--(26,26)--(24,26)--cycle,gray); fill((0,0)--(26,0)--(26,2)--(0,2)--cycle,gray); fill((0,12)--(26,12)--(26,14)--(0,14)--cycle,gray); fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); [/asy]",
        "answer": "$676$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 191,
        "problem": "The radii of the circles in the following figure are all 1cm, and they are tangent to each other. Calculate the area of the bounding rectangle.",
        "geo_code": "[asy] real r = 3;\npair A = (r, -r);       \npair B = (3*r, -r);\npair C = (6, -8.2);\nreal width = 4*r;\nreal top = 0;\nreal bottom = C.y - r;\ndraw(box((0,top),(width,bottom)), linewidth(0.8bp));\ndraw(circle(A, r), blue);\ndraw(circle(B, r), blue);\ndraw(circle(C, r), blue);\n\ndraw(A--(A+(0,r)), red+linewidth(1));\nlabel(\"$1,\\mathrm{cm}$\", A+(0,r/2), W, red);\ndraw(B--(B+(0,r)), red+linewidth(1));\nlabel(\"$1,\\mathrm{cm}$\", B+(0,r/2), E, red);\ndraw(C--(C+(0,-r)), red+linewidth(1));\nlabel(\"$1,\\mathrm{cm}$\", C+(0,-r/2), E, red);\ndot(midpoint(A--B));\ndot(midpoint(A--C));\ndot(midpoint(B--C));\ndot(A); label(\"$A$\", A, W);\ndot(B); label(\"$B$\", B, E);\ndot(C); label(\"$C$\", C, S);[/asy]",
        "answer": "$8+4\\sqrt{3}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 192,
        "problem": "How many pairs of parallel edges does a cube have, excluding the pairs $\\overline{AB}$ and $\\overline{GH}$, and $\\overline{EH}$ and $\\overline{FG}$?",
        "geo_code": "[asy] import three; currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */ draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1)); draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); label(\"$D$\",(0,0,0),S); label(\"$A$\",(0,0,1),N); label(\"$H$\",(0,1,0),S); label(\"$E$\",(0,1,1),N); label(\"$C$\",(1,0,0),S); label(\"$B$\",(1,0,1),N); label(\"$G$\",(1,1,0),S); label(\"$F$\",(1,1,1),N); [/asy]",
        "answer": "$16$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 193,
        "problem": "A triangle with vertices at $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a \\$6\\times5$ grid. How many times as large is the area of the grid compared to the area of the triangle?",
        "geo_code": "[asy] draw((1,0)--(1,5),linewidth(.5)); draw((2,0)--(2,5),linewidth(.5)); draw((3,0)--(3,5),linewidth(.5)); draw((4,0)--(4,5),linewidth(.5)); draw((5,0)--(5,5),linewidth(.5)); draw((6,0)--(6,5),linewidth(.5)); draw((0,1)--(6,1),linewidth(.5)); draw((0,2)--(6,2),linewidth(.5)); draw((0,3)--(6,3),linewidth(.5)); draw((0,4)--(6,4),linewidth(.5)); draw((0,5)--(6,5),linewidth(.5)); draw((0,0)--(0,6),EndArrow); draw((0,0)--(7,0),EndArrow); draw((1,3)--(4,4)--(5,1)--cycle); label(\"$y$\",(0,6),W); label(\"$x$\",(7,0),S); label(\"$A$\",(1,3),dir(210)); label(\"$B$\",(5,1),SE); label(\"$C$\",(4,4),dir(100)); [/asy]",
        "answer": "$6$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 194,
        "problem": "In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas \\$18$ and \\$32$ respectively, $\\triangle JBK$ is equilateral, and $FE = BC$. How many times as large is the area of $ABJI$ compared to the area of $\\triangle KBC$?",
        "geo_code": "[asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label(\"$I$\",(-4,8+6*sqrt(2)),dir(100)); label(\"$J$\",(4,8+6*sqrt(2)),dir(80)); label(\"$A$\",(-4,6*sqrt(2)),dir(280)); label(\"$B$\",(4,6*sqrt(2)),dir(250)); label(\"$C$\",(8,0),W); label(\"$D$\",(4,-6*sqrt(2)),NW); label(\"$E$\",(-4,-6*sqrt(2)),NE); label(\"$F$\",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label(\"$K$\",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label(\"$H$\",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label(\"$G$\",(-8-6*sqrt(2),-4),W); label(\"$32$\",(-10,-8),N); label(\"$18$\",(0,6*sqrt(2)+2),N); [/asy]",
        "answer": "$\\frac{3}{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 195,
        "problem": "The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\\tfrac{3}{2}$ and center $(0,\\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the twice the area of the shark's fin falcata?",
        "geo_code": "[asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP(\"x\",(3.5,0),S)--(0,0)--MP(\"\\frac{3}{2}\",(0,3/2),W)--MP(\"y\",(0,3.5),W)); path P=(0,0)--MP(\"3\",(3,0),S)..(3*dir(45))..MP(\"3\",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy]",
        "answer": "$\\frac{9\\pi}{4}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 196,
        "problem": "In the upper half-plane, consider a configuration of circles, each tangent to the $x$-axis, built in successive layers as follows. The initial layer $L_0$ consists of two mutually externally tangent circles with radii \\$70^2$ and \\$73^2$. For each layer $k \\ge 1$, take all circles from the union $\\bigcup_{j=0}^{k-1}L_j$ and order them by the $x$-coordinates of their points of tangency with the $x$-axis. For every pair of adjacent circles in this sequence, construct a new circle that is externally tangent to both circles in the pair and also tangent to the $x$-axis. Define layer $L_k$ as the set of \\$2^{k-1}$ new circles added in this way. Let $S = \\bigcup_{j=0}^6 L_j$, and let $r(C)$ denote the radius of the circle $C$. Compute\n\\[\n\\sum_{C \\in S} \\frac{1}{\\sqrt{r(C)}}.\n\\]",
        "geo_code": "[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]",
        "answer": "$\\frac{143}{14}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 197,
        "problem": "In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\\overline{AD}$. Points $F$ and $G$ lie on $\\overline{CE}$, and $H$ and $J$ lie on $\\overline{AB}$ and $\\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\\overline{GH}$, and $M$ and $N$ lie on $\\overline{AD}$ and $\\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. How many times as large is the area of $FGHJ$ as that of $KLMN$?.",
        "geo_code": "[asy] pair A,B,C,D,E,F,G,H,J,K,L,M,N; B=(0,0); real m=7*sqrt(55)/5; J=(m,0); C=(7*m/2,0); A=(0,7*m/2); D=(7*m/2,7*m/2); E=(A+D)/2; H=(0,2m); N=(0,2m+3*sqrt(55)/2); G=foot(H,E,C); F=foot(J,E,C); draw(A--B--C--D--cycle); draw(C--E); draw(G--H--J--F); pair X=foot(N,E,C); M=extension(N,X,A,D); K=foot(N,H,G); L=foot(M,H,G); draw(K--N--M--L); label(\"$A$\",A,NW); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$E$\",E,dir(90)); label(\"$F$\",F,NE); label(\"$G$\",G,NE); label(\"$H$\",H,W); label(\"$J$\",J,S); label(\"$K$\",K,SE); label(\"$L$\",L,SE); label(\"$M$\",M,dir(90)); label(\"$N$\",N,dir(180)); [/asy]",
        "answer": "$\\frac{49}{9}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 198,
        "problem": "In rectangle $ABCD$, point $M$ is the midpoint of $\\overline{AD}$. The area of $\\triangle AMC$ is \\$12$, and $\\frac{AD}{AB} = \\frac{3}{2}$. Find the length of side $AD$.",
        "geo_code": "[asy]size(4cm);draw((0,2)--(0,0)--(3,0)--(3,4)--(0,4)--(0,2)--(3,4)--(0,0));label(\"$A$\", (0,0), SW); label(\"$B$\", (3, 0), SE); label(\"$C$\", (3,4), NE); label(\"$D$\", (0, 4), NW); label(\"$M$\", (0, 2), W); [/asy]",
        "answer": "$8$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 199,
        "problem": "Rectangle $DEFA$ below is a $3 \\times 4$ rectangle with $DC=CB=BA=1$. What is the area of rectangle $DEFA$ minus the area of the shaded \"bat wings\" region?",
        "geo_code": "[asy] draw((0,0)--(3,0)--(3,4)--(0,4)--(0,0)--(2,4)--(3,0)); draw((3,0)--(1,4)--(0,0)); fill((0,0)--(1,4)--(1.5,3)--cycle, black); fill((3,0)--(2,4)--(1.5,3)--cycle, black); label(\"$A$\",(3.05,4.2)); label(\"$B$\",(2,4.2)); label(\"$C$\",(1,4.2)); label(\"$D$\",(0,4.2)); label(\"$E$\", (0,-0.2)); label(\"$F$\", (3,-0.2)); label(\"$1$\", (0.5, 4), N); label(\"$1$\", (1.5, 4), N); label(\"$1$\", (2.5, 4), N); label(\"$4$\", (3.2, 2), E); [/asy]",
        "answer": "$9$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 200,
        "problem": "A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the perimeter, in feet, of the inner rectangle",
        "geo_code": "[asy] size(6cm); defaultpen(fontsize(9pt)); path rectangle(pair X, pair Y){ return X--(X.x,Y.y)--Y--(Y.x,X.y)--cycle; } filldraw(rectangle((0,0),(7,5)),gray(0.5)); filldraw(rectangle((1,1),(6,4)),gray(0.75)); filldraw(rectangle((2,2),(5,3)),white); label(\"$1$\",(0.5,2.5)); draw((0.3,2.5)--(0,2.5),EndArrow(TeXHead)); draw((0.7,2.5)--(1,2.5),EndArrow(TeXHead)); label(\"$1$\",(1.5,2.5)); draw((1.3,2.5)--(1,2.5),EndArrow(TeXHead)); draw((1.7,2.5)--(2,2.5),EndArrow(TeXHead)); label(\"$1$\",(4.5,2.5)); draw((4.5,2.7)--(4.5,3),EndArrow(TeXHead)); draw((4.5,2.3)--(4.5,2),EndArrow(TeXHead)); label(\"$1$\",(4.1,1.5)); draw((4.1,1.7)--(4.1,2),EndArrow(TeXHead)); draw((4.1,1.3)--(4.1,1),EndArrow(TeXHead)); label(\"$1$\",(3.7,0.5)); draw((3.7,0.7)--(3.7,1),EndArrow(TeXHead)); draw((3.7,0.3)--(3.7,0),EndArrow(TeXHead)); [/asy]",
        "answer": "$6$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 201,
        "problem": "Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the diameter in inches of the scrap cookie?",
        "geo_code": "[asy] draw(circle((0,0),3)); draw(circle((0,0),1)); draw(circle((1,sqrt(3)),1)); draw(circle((-1,sqrt(3)),1)); draw(circle((-1,-sqrt(3)),1)); draw(circle((1,-sqrt(3)),1)); draw(circle((2,0),1)); draw(circle((-2,0),1)); [/asy]",
        "answer": "$2\\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 202,
        "problem": "Let $ABCD$ be a rectangle with $AB = 5$ and $BC = 4$. On side $\\overline{AB}$, let $E$ be the point such that $EB = 1$. On side $\\overline{BC}$, let $G$ be the point such that $CG = 1$. On side $\\overline{CD}$, let $F$ be the point such that $DF = 2$. The lines $\\overline{AG}$ and $\\overline{AC}$ meet $\\overline{EF}$ at points $Q$ and $P$, respectively. Compute the value of $\\dfrac{PQ}{EF}$.",
        "geo_code": "[asy]pair A1=(2,0),A2=(4,4); pair B1=(0,4),B2=(5,1); pair C1=(5,0),C2=(0,4); draw(A1--A2); draw(B1--B2); draw(C1--C2); draw((0,0)--B1--(5,4)--C1--cycle); dot((20/7,12/7)); dot((3.07692307692,2.15384615384)); label(\"$Q$\",(3.07692307692,2.15384615384),N); label(\"$P$\",(20/7,12/7),W); label(\"$A$\",(0,4), NW); label(\"$B$\",(5,4), NE); label(\"$C$\",(5,0),SE); label(\"$D$\",(0,0),SW); label(\"$F$\",(2,0),S); label(\"$G$\",(5,1),E); label(\"$E$\",(4,4),N);[/asy]",
        "answer": "$\\frac{10}{91}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 203,
        "problem": "In $\\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\\overline{BC}$, and $\\overline{AD}$ bisects $\\angle BAC$. Point $E$ lies on $\\overline{AC}$, and $\\overline{BE}$ bisects $\\angle ABC$. The bisectors intersect at $F$. Compute the value of $\\dfrac{AF}{FD}$.",
        "geo_code": "[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,NE); label(\"$E$\",E,NW); label(\"$F$\",F,1.5*N); [/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 204,
        "problem": "In $\\triangle ABC$, let $AB=7$, $BC=8$, and $CA=9$. The altitude from $A$ to $BC$ is $\\overline{AH}$. Let points $D$ and $E$ be located on $\\overline{AC}$ and $\\overline{AB}$, respectively, such that $\\overline{BD}$ and $\\overline{CE}$ are the angle bisectors of $\\angle ABC$ and $\\angle BCA$, respectively. These angle bisectors intersect $\\overline{AH}$ at points $Q$ and $P$. Find the length of segment $PQ$.",
        "geo_code": "[asy] import graph; size(9cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.381056062031275, xmax = 15.020004395092375, ymin = -4.051697595316909, ymax = 10.663513514111651; /* image dimensions */ draw((0.,0.)--(4.714285714285714,7.666518779999279)--(7.,0.)--cycle); /* draw figures */ draw((0.,0.)--(4.714285714285714,7.666518779999279)); draw((4.714285714285714,7.666518779999279)--(7.,0.)); draw((7.,0.)--(0.,0.)); label(\"7\",(3.2916797119724284,-0.07831656949355523),SE*labelscalefactor); label(\"9\",(2.0037562070503783,4.196493361737088),SE*labelscalefactor); label(\"8\",(6.114150371695219,3.785453945272603),SE*labelscalefactor); draw((0.,0.)--(6.428571428571427,1.9166296949998194)); draw((7.,0.)--(2.2,3.5777087639996634)); draw((4.714285714285714,7.666518779999279)--(3.7058823529411766,0.)); /* dots and labels */ dot((0.,0.),dotstyle); label(\"$A$\", (-0.2432592696221352,-0.5715638692509372), NE * labelscalefactor); dot((7.,0.),dotstyle); label(\"$B$\", (7.0458397156813835,-0.48935598595804014), NE * labelscalefactor); dot((3.7058823529411766,0.),dotstyle); label(\"$E$\", (3.8123296394941084,0.16830708038513573), NE * labelscalefactor); dot((4.714285714285714,7.666518779999279),dotstyle); label(\"$C$\", (4.579603216894479,7.895848109917452), NE * labelscalefactor); dot((2.2,3.5777087639996634),linewidth(3.pt) + dotstyle); label(\"$D$\", (2.1407693458718726,3.127790878929427), NE * labelscalefactor); dot((6.428571428571427,1.9166296949998194),linewidth(3.pt) + dotstyle); label(\"$H$\", (6.004539860638023,1.9494778850645704), NE * labelscalefactor); dot((5.,1.49071198499986),linewidth(3.pt) + dotstyle); label(\"$Q$\", (4.935837377830365,1.7302568629501784), NE * labelscalefactor); dot((3.857142857142857,1.1499778169998918),linewidth(3.pt) + dotstyle); label(\"$P$\", (3.538303361851119,1.2370095631927964), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]",
        "answer": "$\\frac{8}{15}\\sqrt{5}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 205,
        "problem": "In the non-convex quadrilateral $ABCD$ shown below, $\\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. How many times as large is the area of quadrilateral $ABCD$ as the area of $\\triangle BCD$?",
        "geo_code": "[asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0)); label(\"$B$\", (0, 0), SW); label(\"$A$\", (12, 0), ESE); label(\"$C$\", (2.4, 3.6), SE); label(\"$D$\", (0, 5), N);[/asy]",
        "answer": "$4$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 206,
        "problem": "Tamara\u2019s garden contains six rectangular flower beds, each measuring \\$6$ feet by \\$2$ feet, arranged in three rows with two beds per row. There are \\$1$-foot-wide paths both between each bed and around the entire arrangement. What is the combined area, in square feet, of all the walkways?",
        "geo_code": "[asy] draw((0,0)--(0,10)--(15,10)--(15,0)--cycle); fill((0,0)--(0,10)--(15,10)--(15,0)--cycle, lightgray); draw((1,1)--(1,3)--(7,3)--(7,1)--cycle); fill((1,1)--(1,3)--(7,3)--(7,1)--cycle, white); draw((1,4)--(1,6)--(7,6)--(7,4)--cycle); fill((1,4)--(1,6)--(7,6)--(7,4)--cycle, white); draw((1,7)--(1,9)--(7,9)--(7,7)--cycle); fill((1,7)--(1,9)--(7,9)--(7,7)--cycle, white); draw((8,1)--(8,3)--(14,3)--(14,1)--cycle); fill((8,1)--(8,3)--(14,3)--(14,1)--cycle, white); draw((8,4)--(8,6)--(14,6)--(14,4)--cycle); fill((8,4)--(8,6)--(14,6)--(14,4)--cycle, white); draw((8,7)--(8,9)--(14,9)--(14,7)--cycle); fill((8,7)--(8,9)--(14,9)--(14,7)--cycle, white); defaultpen(fontsize(8, lineskip=1)); label(\"2\", (1.2, 2)); label(\"6\", (4, 1.2)); defaultpen(linewidth(.2)); draw((0,8)--(1,8), arrow=Arrows); draw((7,8)--(8,8), arrow=Arrows); draw((14,8)--(15,8), arrow=Arrows); draw((11,0)--(11,1), arrow=Arrows); draw((11,3)--(11,4), arrow=Arrows); draw((11,6)--(11,7), arrow=Arrows); label(\"1\", (.5,7.8)); label(\"1\", (7.5,7.8)); label(\"1\", (14.5,7.8)); label(\"1\", (10.8,.5)); label(\"1\", (10.8,3.5)); label(\"1\", (10.8,6.5)); [/asy]",
        "answer": "$78$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 207,
        "problem": "Point $E$ is the midpoint of side $\\overline{CD}$ in square $ABCD,$ and $\\overline{BE}$ meets diagonal $\\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the side length of $ABCD?$",
        "geo_code": "[asy] size(5cm); draw((0,0)--(6,0)--(6,6)--(0,6)--cycle); draw((0,6)--(6,0)); draw((3,0)--(6,6)); label(\"$A$\",(0,6),NW); label(\"$B$\",(6,6),NE); label(\"$C$\",(6,0),SE); label(\"$D$\",(0,0),SW); label(\"$E$\",(3,0),S); label(\"$F$\",(4,2),E); [/asy]",
        "answer": "$6\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 208,
        "problem": "Let $ABCDEFGH$ be a cube where $C$ and $E$ are a pair of opposite vertices. Points $J$ and $I$ are the midpoints of $\\overline{FB}$ and $\\overline{HD}$, respectively. Consider the cross-section of the cube determined by points $E$, $J$, $C$, and $I$. If $R$ is the ratio of the area of quadrilateral $EJCI$ to the area of a face of the cube, what is the value of $R^2$?",
        "geo_code": "[asy] size(6cm); pair A,B,C,D,EE,F,G,H,I,J; C = (0,0); B = (-1,1); D = (2,0.5); A = B+D; G = (0,2); F = B+G; H = G+D; EE = G+B+D; I = (D+H)/2; J = (B+F)/2; filldraw(C--I--EE--J--cycle,lightgray,black); draw(C--D--H--EE--F--B--cycle); draw(G--F--G--C--G--H); draw(A--B,dashed); draw(A--EE,dashed); draw(A--D,dashed); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(I); dot(J); label(\"$A$\",A,E); label(\"$B$\",B,W); label(\"$C$\",C,S); label(\"$D$\",D,E); label(\"$E$\",EE,N); label(\"$F$\",F,W); label(\"$G$\",G,N); label(\"$H$\",H,E); label(\"$I$\",I,E); label(\"$J$\",J,W); [/asy]",
        "answer": "$\\frac{3}{2}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 209,
        "problem": "In the figure below, $N$ congruent semicircles lie along the diameter of a larger semicircle, with their diameters exactly covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of all the small semicircles and $B$ be the area of the region inside the large semicircle but outside the small semicircles. If the ratio $A:B$ is \\$1:18$ and if the area of each small semicircle is \\$2$, what is the value of $A$?",
        "geo_code": "[asy] size(4cm);draw((0,0)--(18,0)); draw(arc((9,0),9,0,180)); filldraw(arc((1,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((3,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((5,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((7,0),1,0,180)--cycle,gray(0.8)); label(\"...\",(9,0.5)); filldraw(arc((11,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((13,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((15,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((17,0),1,0,180)--cycle,gray(0.8)); [/asy]",
        "answer": "$38$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 210,
        "problem": "Consider a rectangular box in which $AB = 3$, $BC = 1$, and $CG = 2$. Let $M$ denote the midpoint of edge $\\overline{FG}$. Find the volume of the pyramid whose base is the rectangle $BCHE$ and whose apex is the point $M$.",
        "geo_code": "[asy] size(250); defaultpen(fontsize(10pt)); pair A =origin; pair B = (4.75,0); pair E1=(0,3); pair F = (4.75,3); pair G = (5.95,4.2); pair C = (5.95,1.2); pair D = (1.2,1.2); pair H= (1.2,4.2); pair M = ((4.75+5.95)/2,3.6); draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H); draw(B--C); draw(F--G); draw(A--D--H--C--D,dashed); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,E); label(\"$D$\",D,W); label(\"$E$\",E1,W); label(\"$F$\",F,SW); label(\"$G$\",G,NE); label(\"$H$\",H,NW); label(\"$M$\",M,N); dot(A); dot(B); dot(E1); dot(F); dot(G); dot(C); dot(D); dot(H); dot(M); label(\"3\",A/2+B/2,S); label(\"2\",C/2+G/2,E); label(\"1\",C/2+B/2,SE); [/asy]",
        "answer": "$2$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "volume"
    },
    {
        "index": 211,
        "problem": "Farmer Pythagoras owns a plot of land shaped like a right triangle, with legs measuring \\$3$ and \\$4$ units, respectively. At the vertex where these sides form the right angle, he reserves a small square region $S$, unplanted, making it appear from above like a right angle marker. The remainder of the field is planted. If the minimum distance from the square $S$ to the hypotenuse of the triangle is \\$2$ units, what fraction of the triangular field has been planted?",
        "geo_code": "[asy] size(160); pair A, B, C, D, F; A = origin; B = (4,0); C = (0,3); D = (2/7,2/7); F = foot(D,B,C); fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray); draw(A--B--C--cycle); draw((2/7,0)--D--(0,2/7)); label(\"$4$\", midpoint(A--B), N); label(\"$3$\", midpoint(A--C), E); label(\"$2$\", midpoint(D--F), SE); label(\"$S$\", midpoint(A--D)); draw(D--F, dashed); [/asy]",
        "answer": "$\\frac{145}{147}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 212,
        "problem": "Start with a \\$23 \\times 27$ rectangle and remove four congruent right triangles with side lengths \\$6$, \\$8$, and \\$10$ from its corners, creating an octagon $ABCDEFGH$ such that $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$, with $\\overline{AH}$ positioned along one of the rectangle\u2019s shorter sides. Let $J$ be the midpoint of $\\overline{AH}$. Connect $J$ to vertices $B$, $C$, $D$, $E$, $F$, and $G$ to partition the octagon into \\$7$ triangles. Determine the area of the convex polygon whose vertices are the centroids of these \\$7$ triangles.",
        "geo_code": "[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label(\"$A$\", A, W*r); label(\"$B$\", B, S*r); label(\"$C$\", C, S*r); label(\"$D$\", D, E*r); label(\"$E$\", EE, E*r); label(\"$F$\", F, N*r); label(\"$G$\", G, N*r); label(\"$H$\", H, W*r); label(\"$J$\", J, W*r); [/asy]",
        "answer": "$184$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 213,
        "problem": "Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, What is the perimeter, in feet, of rectangle $ABCD$?",
        "geo_code": "[asy] draw((0,0)--(3,0)); draw((0,0)--(0,2)); draw((0,2)--(3,2)); draw((3,2)--(3,0)); dot((0,0)); dot((0,2)); dot((3,0)); dot((3,2)); draw((2,0)--(2,2)); draw((0,1)--(2,1)); label(\"A\",(0,0),S); label(\"B\",(3,0),S); label(\"C\",(3,2),N); label(\"D\",(0,2),N); [/asy]",
        "answer": "$50$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 214,
        "problem": "A square contains \\$81$ equally spaced grid points, including those on its boundary. Let $P$ be the grid point at the exact center of the square. If point $Q$ is randomly selected from the remaining \\$80$ grid points, what is the probability that the segment $PQ$ forms a line of symmetry of the square?",
        "geo_code": "[asy] draw((0,0)--(0,8)); draw((0,8)--(8,8)); draw((8,8)--(8,0)); draw((8,0)--(0,0)); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((0,6)); dot((0,7)); dot((0,8)); dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((1,6)); dot((1,7)); dot((1,8)); dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((2,6)); dot((2,7)); dot((2,8)); dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((3,6)); dot((3,7)); dot((3,8)); dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); dot((4,6)); dot((4,7)); dot((4,8)); dot((5,0)); dot((5,1)); dot((5,2)); dot((5,3)); dot((5,4)); dot((5,5)); dot((5,6)); dot((5,7)); dot((5,8)); dot((6,0)); dot((6,1)); dot((6,2)); dot((6,3)); dot((6,4)); dot((6,5)); dot((6,6)); dot((6,7)); dot((6,8)); dot((7,0)); dot((7,1)); dot((7,2)); dot((7,3)); dot((7,4)); dot((7,5)); dot((7,6)); dot((7,7)); dot((7,8)); dot((8,0)); dot((8,1)); dot((8,2)); dot((8,3)); dot((8,4)); dot((8,5)); dot((8,6)); dot((8,7)); dot((8,8)); label(\"P\",(4,4),NE); [/asy]",
        "answer": "$\\frac{2}{5}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 215,
        "problem": "In triangle $\\triangle ABC$, point $D$ divides side $\\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\\overline{BD}$ and let $F$ be the point of intersection of line $\\overline{BC}$ and line $\\overline{AE}$. How many times as large is the area of $\\triangle ABC$ compared to the area of $\\triangle EBF$",
        "geo_code": "[asy] unitsize(2cm); pair A,B,C,DD,EE,FF; B = (0,0); C = (3,0); A = (1.2,1.7); DD = (2/3)*A+(1/3)*C; EE = (B+DD)/2; FF = intersectionpoint(B--C,A--A+2*(EE-A)); draw(A--B--C--cycle); draw(A--FF); draw(B--DD);dot(A); label(\"$A$\",A,N); dot(B); label(\"$B$\", B,SW);dot(C); label(\"$C$\",C,SE); dot(DD); label(\"$D$\",DD,NE); dot(EE); label(\"$E$\",EE,NW); dot(FF); label(\"$F$\",FF,S); [/asy]",
        "answer": "$12$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 216,
        "problem": "The figure below depicts line $\\ell$ and a regular, infinite, repeating arrangement made up of squares and line segments. Excluding the identity transformation, how many of the following types of rigid motions will map the figure onto itself?",
        "geo_code": "[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0)); [/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 217,
        "problem": "Let $ABC$ be an isosceles right triangle with legs of length \\$1$, and let $C$ be the right angle. On the hypotenuse $\\overline{AC}$, construct triangle $ACD$ externally such that $ACD$ is a right triangle with the right angle at $C$ and the perimeters of triangles $ABC$ and $ACD$ are equal. What is the value of $\\sin(2\\angle BAD)$?",
        "geo_code": "[asy] import graph; size(8.016233639805293cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.001920114613276, xmax = 4.014313525192017, ymin = -2.552570341575814, ymax = 5.6249093771911145; /* image dimensions */ draw((-1.6742337260757447,-1.)--(-1.6742337260757445,-0.6742337260757447)--(-2.,-0.6742337260757447)--(-2.,-1.)--cycle, linewidth(2.)); draw((-1.7696484586262846,2.7696484586262846)--(-1.5392969172525692,3.)--(-1.7696484586262846,3.2303515413737154)--(-2.,3.)--cycle, linewidth(2.)); /* draw figures */ draw((-2.,3.)--(-2.,-1.), linewidth(2.)); draw((-2.,-1.)--(2.,-1.), linewidth(2.)); draw((2.,-1.)--(-2.,3.), linewidth(2.)); draw((-0.6404058554606791,4.3595941445393205)--(-2.,3.), linewidth(2.)); draw((-0.6404058554606791,4.3595941445393205)--(2.,-1.), linewidth(2.)); label(\"$D$\",(-0.9382446143428628,4.887784444795223),SE*labelscalefactor,fontsize(14)); label(\"$A$\",(1.9411496528285788,-1.0783204767840298),SE*labelscalefactor,fontsize(14)); label(\"$B$\",(-2.5046350956841272,-0.9861798602345433),SE*labelscalefactor,fontsize(14)); label(\"$C$\",(-2.5737405580962416,3.5747806589650395),SE*labelscalefactor,fontsize(14)); label(\"$1$\",(-2.665881174645728,1.2712652452278765),SE*labelscalefactor,fontsize(14)); label(\"$1$\",(-0.3393306067712029,-1.3547423264324894),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((-2.,3.),linewidth(4.pt) + dotstyle); dot((-2.,-1.),linewidth(4.pt) + dotstyle); dot((2.,-1.),linewidth(4.pt) + dotstyle); dot((-0.6404058554606791,4.3595941445393205),linewidth(4.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]",
        "answer": "$\\frac{7}{9}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 218,
        "problem": "Rectangle $ABCD$ is inscribed in a semicircle with diameter $\\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ Find $AB$",
        "geo_code": "[asy] draw(arc((0,0),34,180,0)); draw((-34,0)--(34,0)); fill((-16,0)--(-16,30)--(16,30)--(16,0)--cycle, 1.5*grey); draw((-16,0)--(-16,30)--(16,30)--(16,0)--cycle); dot(\"$A$\",(16,0), 1.25*S); dot(\"$B$\",(16,30), 1.25*N); dot(\"$C$\",(-16,30), 1.25*N); dot(\"$D$\",(-16,0), 1.25*S); dot(\"$E$\",(34,0), 1.25*S); dot(\"$F$\",(-34,0), 1.25*S); label(\"$16$\",(0,0),N); label(\"$9$\",(25,0),N); label(\"$9$\",(-25,0),N); [/asy]",
        "answer": "$15$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 219,
        "problem": "A large square is entirely covered with $n^2$ gray square tiles, each with side length $s$ inches. Each tile is separated from adjacent tiles and the boundary of the large square by a uniform border of width $d$ inches. The diagram below illustrates the arrangement for $n=3$. If $n=24$, so there are \\$576$ gray tiles, and these tiles collectively occupy \\$64\\%$ of the total area of the entire large square, what is the value of the ratio $\\frac{d}{s}$ for this configuration?",
        "geo_code": "[asy] draw((0,0)--(13,0)--(13,13)--(0,13)--cycle); filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle, mediumgray); filldraw((1,5)--(4,5)--(4,8)--(1,8)--cycle, mediumgray); filldraw((1,9)--(4,9)--(4,12)--(1,12)--cycle, mediumgray); filldraw((5,1)--(8,1)--(8,4)--(5,4)--cycle, mediumgray); filldraw((5,5)--(8,5)--(8,8)--(5,8)--cycle, mediumgray); filldraw((5,9)--(8,9)--(8,12)--(5,12)--cycle, mediumgray); filldraw((9,1)--(12,1)--(12,4)--(9,4)--cycle, mediumgray); filldraw((9,5)--(12,5)--(12,8)--(9,8)--cycle, mediumgray); filldraw((12,12)--(12,9)--(9,9)--(9,12)--cycle, mediumgray); [/asy]",
        "answer": "$\\frac{6}{25}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 220,
        "problem": "Let $\\triangle AMC$ be an isosceles triangle with $AM = AC$. The medians $\\overline{MV}$ and $\\overline{CU}$ are equal in length and are perpendicular to each other, where $MV=CU=12$. Find the area of triangle $AMC$.",
        "geo_code": "[asy] draw((-4,0)--(4,0)--(0,12)--cycle); draw((-2,6)--(4,0)); draw((2,6)--(-4,0)); label(\"M\", (-4,0), W); label(\"C\", (4,0), E); label(\"A\", (0, 12), N); label(\"V\", (2, 6), NE); label(\"U\", (-2, 6), NW); label(\"P\", (0, 3.6), S); [/asy]",
        "answer": "$96$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 221,
        "problem": "A square sheet of paper with vertices labeled $A$, $B$, $C$, and $D$ in order and with side length \\$1$ is folded so that vertex $C$ lands on edge $\\overline{AD}$ at the point $C'$. The fold causes edge $\\overline{BC}$ to cross edge $\\overline{AB}$ at point $E$. If the distance from $C'$ to $D$ is $\\frac{1}{3}$, what is the perimeter of triangle $AEC'$?",
        "geo_code": "[asy] pair A=(0,1); pair CC=(0.666666666666,1); pair D=(1,1); pair F=(1,0.440062); pair C=(1,0); pair B=(0,0); pair G=(0,0.22005); pair H=(-0.13,0.41); pair E=(0,0.5); dot(A^^CC^^D^^C^^B^^E); draw(E--A--D--F); draw(G--B--C--F, dashed); fill(E--CC--F--G--H--E--CC--cycle, gray); draw(E--CC--F--G--H--E--CC); label(\"A\",A,NW); label(\"B\",B,SW); label(\"C\",C,SE); label(\"D\",D,NE); label(\"E\",E,NW); label(\"C'\",CC,N); label(\"F\",F,NE); [/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 222,
        "problem": "Let $ABCD$ be an isosceles trapezoid with $AB$ and $CD$ as its parallel sides, where $AB > CD$. Suppose a point is chosen inside the trapezoid, and segments are drawn from this point to each of the four vertices, subdividing $ABCD$ into four triangles. Starting with the triangle whose base is $\\overline{CD}$ and proceeding clockwise, the areas of the triangles are \\$2$, \\$3$, \\$4$, and \\$5$, respectively. Find the value of the ratio $\\dfrac{AB}{CD}$.",
        "geo_code": "[asy] unitsize(100); pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); draw(A--B--C--D--cycle, black); draw(A--P, black); draw(B--P, black); draw(C--P, black); draw(D--P, black); label(\"$A$\",A,(-1,0)); label(\"$B$\",B,(1,0)); label(\"$C$\",C,(1,-0)); label(\"$D$\",D,(-1,0)); label(\"$2$\",E,(0,0)); label(\"$3$\",F,(0,0)); label(\"$4$\",G,(0,0)); label(\"$5$\",H,(0,0)); dot(A^^B^^C^^D^^P); [/asy]",
        "answer": "$2+\\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 223,
        "problem": "Let $ABC$ be an equilateral triangle with side length \\$840$. Point $D$ is positioned on the same side of $BC$ as $A$ so that $\\overline{BD}$ is perpendicular to $\\overline{BC}$. Through $D$, draw a line $\\ell$ parallel to $BC$; let $\\ell$ intersect $AB$ at $E$ and $AC$ at $F$. Let $G$ be a point on $\\ell$ such that $F$ lies strictly between $E$ and $G$, and triangle $AFG$ is isosceles. If the ratio of the area of $\\triangle AFG$ to the area of $\\triangle BED$ is \\$8:9$, find the length $AF$.",
        "geo_code": "[asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label(\"$A$\",A,dir(90)); label(\"$B$\",B,dir(225)); label(\"$C$\",C,dir(-45)); label(\"$D$\",D,dir(180)); label(\"$E$\",E,dir(-45)); label(\"$F$\",F,dir(225)); label(\"$G$\",G,dir(0)); label(\"$\\ell$\",midpoint(E--F),dir(90)); [/asy]",
        "answer": "$336$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 224,
        "problem": "In square $ABCD$, points $P$ and $Q$ lie on $\\overline{AD}$ and $\\overline{AB}$, respectively. Segments $\\overline{BP}$ and $\\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the side length of the square?",
        "geo_code": "[asy] size(170); defaultpen(linewidth(0.6)+fontsize(10)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot(\"$A$\",A,S); dot(\"$B$\",B,S); dot(\"$C$\",C,N); dot(\"$D$\",D,N); dot(\"$Q$\",Q,S); dot(\"$P$\",P,W); dot(\"$R$\",R,1.3*S); label(\"$7$\",(P+R)/2,NE); label(\"$6$\",(R+B)/2,NE); [/asy]",
        "answer": "$\\sqrt{117}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 225,
        "problem": "As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\\angle CDE = 110^\\circ$. Point $F$ lies on $\\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What multiple of $\\angle CDE$ is the measure of $\\angle AFE$?",
        "geo_code": "[asy] size(6cm); pair A = (0,10); label(\"$A$\", A, N); pair B = (0,0); label(\"$B$\", B, S); pair C = (10,0); label(\"$C$\", C, S); pair D = (10,10); label(\"$D$\", D, SW); pair EE = (15,11.8); label(\"$E$\", EE, N); pair F = (3,10); label(\"$F$\", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label(\"$110^\\circ$\", (15,9), SW); [/asy]",
        "answer": "$\\frac{17}{11}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 226,
        "problem": "In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\\circ$. The enclosed area of the hexagon is $6\\sqrt{3}$. What is the perimeter of the hexagon?",
        "geo_code": "[asy] size(10cm); pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F; draw(C--D--E--F--A--B--cycle,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); label(\"$C$\",C,2*S); label(\"$D$\",D,2*S); label(\"$E$\",E,2*S); label(\"$F$\",F,2*dir(0)); label(\"$A$\",A,2*N); label(\"$B$\",B,2*W); [/asy]",
        "answer": "$12\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 227,
        "problem": "Let $ABCD$ be an isosceles trapezoid with $AB = CD$ and $\\overline{BC} \\parallel \\overline{AD}$. Points $X$ and $Y$ are located on diagonal $\\overline{AC}$ with $X$ lying between $A$ and $Y$. It is given that $\\angle AXD = \\angle BYC = 90^\\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. Determine the area of the trapezoid $ABCD$.",
        "geo_code": "[asy] size(10cm); usepackage(\"mathptmx\"); import geometry; void perp(picture pic=currentpicture, pair O, pair M, pair B, real size=5, pen p=currentpen, filltype filltype = NoFill){ perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype); } pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6)); draw(A--B--C--D--cycle,p); draw(A--C,p); draw(B--Y,p); draw(D--X,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(X,q); dot(Y,q); label(\"2\",C--Y,S); label(\"1\",Y--X,S); label(\"3\",X--A,S); label(\"$A$\",A,2*E); label(\"$B$\",B,2*N); label(\"$C$\",C,2*W); label(\"$D$\",D,2*S); label(\"$Y$\",Y,2*sqrt(2)*NE); label(\"$X$\",X,2*N); perp(B,Y,C,8,p); perp(A,X,D,8,p); [/asy]",
        "answer": "$3\\sqrt{35}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 228,
        "problem": "The diagram below presents a polygon $ABCDEFGH$ made from rectangles and right triangles. If this shape is cut out and folded along the dashed lines, it forms a triangular prism. Given that $AH = EF = 8$ and $GH = 14$, find the volume of the resulting prism.",
        "geo_code": "[asy] usepackage(\"mathptmx\"); size(275); defaultpen(linewidth(0.8)); real r = 2, s = 2.5, theta = 14; pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta); pair N = (B+G)/2, J = N + s/2 * dir(180+theta); pair E = F + r * dir(- 45 - theta/2), D = I+E-F; pair H = J + r * dir(135 + theta/2), A = B+H-J; draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B)); draw(J--B--G^^C--F--I,linetype (\"4 4\")); dot(\"$A$\",A,N); dot(\"$B$\",B,1.2*N); dot(\"$C$\",C,N); dot(\"$D$\",D,dir(0)); dot(\"$E$\",E,S); dot(\"$F$\",F,1.5*dir(-100)); dot(\"$G$\",G,S); dot(\"$H$\",H,W); dot(\"$I$\",I,NE); dot(\"$J$\",J,1.5*S); [/asy]",
        "answer": "$192$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "volume"
    },
    {
        "index": 229,
        "problem": "In rhombus $ABCD$, point $P$ lies on segment $\\overline{AD}$ so that $\\overline{BP}$ $\\perp$ $\\overline{AD}$, $AP = 3$, and $PD = 2$. Find $BP$?",
        "geo_code": "[asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label(\"$A$\",A,SW); label(\"$B$\", B, NW); label(\"$C$\",C,NE); label(\"$D$\",D,SE); label(\"$P$\",P,S); [/asy]",
        "answer": "$4$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 230,
        "problem": "A regular heptagon (regular \\$7$-gon) is inscribed in a circle of radius \\$1$. Consider all \\$21$ distinct edges and diagonals of the heptagon. What is the sum of the fourth powers of their lengths?",
        "geo_code": "[asy] import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } for(int i = 0; i < 7; ++i) { dot(dir(i * 360/7),5+black); } [/asy]",
        "answer": "$147$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 231,
        "problem": "Consider a rectangle whose sides are parallel to the coordinate axes and whose opposite vertices are at $(15, 3)$ and $(16, 5)$. One line passes through points $A(0, 0)$ and $B(3, 1)$, and a second line passes through points $C(0, 10)$ and $D(2, 9)$. How many points lying on the rectangle also lie on at least one of these two lines?",
        "geo_code": "[asy] usepackage(\"mathptmx\"); size(9cm); draw((0,-.5)--(0,11),EndArrow(size=.15cm)); draw((1,0)--(1,11),mediumgray); draw((2,0)--(2,11),mediumgray); draw((3,0)--(3,11),mediumgray); draw((4,0)--(4,11),mediumgray); draw((5,0)--(5,11),mediumgray); draw((6,0)--(6,11),mediumgray); draw((7,0)--(7,11),mediumgray); draw((8,0)--(8,11),mediumgray); draw((9,0)--(9,11),mediumgray); draw((10,0)--(10,11),mediumgray); draw((11,0)--(11,11),mediumgray); draw((12,0)--(12,11),mediumgray); draw((13,0)--(13,11),mediumgray); draw((14,0)--(14,11),mediumgray); draw((15,0)--(15,11),mediumgray); draw((16,0)--(16,11),mediumgray); draw((-.5,0)--(17,0),EndArrow(size=.15cm)); draw((0,1)--(17,1),mediumgray); draw((0,2)--(17,2),mediumgray); draw((0,3)--(17,3),mediumgray); draw((0,4)--(17,4),mediumgray); draw((0,5)--(17,5),mediumgray); draw((0,6)--(17,6),mediumgray); draw((0,7)--(17,7),mediumgray); draw((0,8)--(17,8),mediumgray); draw((0,9)--(17,9),mediumgray); draw((0,10)--(17,10),mediumgray); draw((-.13,1)--(.13,1)); draw((-.13,2)--(.13,2)); draw((-.13,3)--(.13,3)); draw((-.13,4)--(.13,4)); draw((-.13,5)--(.13,5)); draw((-.13,6)--(.13,6)); draw((-.13,7)--(.13,7)); draw((-.13,8)--(.13,8)); draw((-.13,9)--(.13,9)); draw((-.13,10)--(.13,10)); draw((1,-.13)--(1,.13)); draw((2,-.13)--(2,.13)); draw((3,-.13)--(3,.13)); draw((4,-.13)--(4,.13)); draw((5,-.13)--(5,.13)); draw((6,-.13)--(6,.13)); draw((7,-.13)--(7,.13)); draw((8,-.13)--(8,.13)); draw((9,-.13)--(9,.13)); draw((10,-.13)--(10,.13)); draw((11,-.13)--(11,.13)); draw((12,-.13)--(12,.13)); draw((13,-.13)--(13,.13)); draw((14,-.13)--(14,.13)); draw((15,-.13)--(15,.13)); draw((16,-.13)--(16,.13)); label(scale(.7)*\"$1$\", (1,-.13), S); label(scale(.7)*\"$2$\", (2,-.13), S); label(scale(.7)*\"$3$\", (3,-.13), S); label(scale(.7)*\"$4$\", (4,-.13), S); label(scale(.7)*\"$5$\", (5,-.13), S); label(scale(.7)*\"$6$\", (6,-.13), S); label(scale(.7)*\"$7$\", (7,-.13), S); label(scale(.7)*\"$8$\", (8,-.13), S); label(scale(.7)*\"$9$\", (9,-.13), S); label(scale(.7)*\"$10$\", (10,-.13), S); label(scale(.7)*\"$11$\", (11,-.13), S); label(scale(.7)*\"$12$\", (12,-.13), S); label(scale(.7)*\"$13$\", (13,-.13), S); label(scale(.7)*\"$14$\", (14,-.13), S); label(scale(.7)*\"$15$\", (15,-.13), S); label(scale(.7)*\"$16$\", (16,-.13), S); label(scale(.7)*\"$1$\", (-.13,1), W); label(scale(.7)*\"$2$\", (-.13,2), W); label(scale(.7)*\"$3$\", (-.13,3), W); label(scale(.7)*\"$4$\", (-.13,4), W); label(scale(.7)*\"$5$\", (-.13,5), W); label(scale(.7)*\"$6$\", (-.13,6), W); label(scale(.7)*\"$7$\", (-.13,7), W); label(scale(.7)*\"$8$\", (-.13,8), W); label(scale(.7)*\"$9$\", (-.13,9), W); label(scale(.7)*\"$10$\", (-.13,10), W); dot((0,0),linewidth(4)); label(scale(.75)*\"$A$\", (0,0), NE); dot((3,1),linewidth(4)); label(scale(.75)*\"$B$\", (3,1), NE); dot((0,10),linewidth(4)); label(scale(.75)*\"$C$\", (0,10), NE); dot((2,9),linewidth(4)); label(scale(.75)*\"$D$\", (2,9), NE); draw((15,3)--(16,3)--(16,5)--(15,5)--cycle,linewidth(1.125)); dot((15,3),linewidth(4)); dot((16,3),linewidth(4)); dot((16,5),linewidth(4)); dot((15,5),linewidth(4)); [/asy]",
        "answer": "$1$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 232,
        "problem": "A bicycle race course has \\$7$ water stations that are placed at equal intervals between the start and finish. Additionally, \\$2$ repair stations are positioned at equal intervals along the same route. If the \\$3$rd water station is \\$2$ miles beyond the \\$1$st repair station, what is the total length of the race, in miles?",
        "geo_code": "[asy] usepackage(\"mathptmx\"); size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray*0.4 + lightgray*0.6); draw((11,11)--(171,11),linetype(\"2 2\")+white+linewidth(1.2)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.85)*rotate(45)*\"Water 1\", (23,-13.5)); label(scale(.85)*rotate(45)*\"Water 2\", (43,-13.5)); label(scale(.85)*rotate(45)*\"Water 7\", (143,-13.5)); filldraw(circle((103,-13.5),.2)); filldraw(circle((98,-13.5),.2)); filldraw(circle((93,-13.5),.2)); filldraw(circle((88,-13.5),.2)); filldraw(circle((83,-13.5),.2)); label(scale(.85)*rotate(90)*\"Start\", (5.5,11)); label(scale(.85)*rotate(270)*\"Finish\", (176.5,11)); [/asy]",
        "answer": "$48$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 233,
        "problem": "A regular octahedron consists of eight equilateral triangular faces, with four faces coming together at each vertex. Jun constructs a regular octahedron by folding a paper pattern whose faces are numbered. After assembly, which numbered face will be immediately to the right of vertex $Q$?",
        "geo_code": "[asy] import graph; // The Solid // To save processing time, do not use three (dimensions) // Project (roughly) to two size(15cm); pair Fr, Lf, Rt, Tp, Bt, Bk; Lf=(0,0); Rt=(12,1); Fr=(7,-1); Bk=(5,2); Tp=(6,6.7); Bt=(6,-5.2); draw(Lf--Fr--Rt); draw(Lf--Tp--Rt); draw(Lf--Bt--Rt); draw(Tp--Fr--Bt); draw(Lf--Bk--Rt,dashed); draw(Tp--Bk--Bt,dashed); label(rotate(-8.13010235)*slant(0.1)*\"$Q$\", (4.2,1.6)); label(rotate(21.8014095)*slant(-0.2)*\"$?$\", (8.5,2.05)); pair g = (-8,0); // Define Gap transform real a = 8; draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow // Time for the NET pair DA,DB,DC,CD,O; DA = (4*sqrt(3),0); DB = (2*sqrt(3),6); DC = (DA+DB)/3; CD = conj(DC); O=(0,0); transform trf=shift(3g+(0,3)); path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB); draw(trf*NET); label(\"$7$\",trf*DC); label(\"$Q$\",trf*DC+DA-DB); label(\"$5$\",trf*DC-DB); label(\"$3$\",trf*DC-DA-DB); label(\"$6$\",trf*CD); label(\"$4$\",trf*CD-DA); label(\"$2$\",trf*CD-DA-DB); label(\"$1$\",trf*CD-2DA); [/asy]",
        "answer": "$1$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 234,
        "problem": "A set of concentric circles is drawn, starting with a circle of radius \\(1\\), and each next circle has its radius increased by \\(1\\) from the previous one. There are an even number of circles, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius \\(2\\) but outside the circle of radius \\(1\\), then the region inside the circle of radius \\(4\\) but outside the circle of radius \\(3\\), and so on. What is the least even number of circles needed so that the total shaded area is at least \\(2023\\pi\\)?",
        "geo_code": "[asy] filldraw(circle((0,0),8),gray); filldraw(circle((-1,0),7),white); filldraw(circle((-2,0),6),gray); filldraw(circle((-3,0),5),white); filldraw(circle((-4,0),4),gray); filldraw(circle((-5,0),3),white); filldraw(circle((-6,0),2),gray); filldraw(circle((-7,0),1),white); [/asy]",
        "answer": "$64$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 235,
        "problem": "A square of area \\(2\\) is inscribed, with its sides not parallel, inside a larger square of area \\(3\\) such that their centers coincide. This arrangement creates four congruent right triangles at the corners of the larger square. In one of these triangles, what is the ratio of the shorter leg to the longer leg?",
        "geo_code": "[asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy]",
        "answer": "$2-\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 236,
        "problem": "Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the diameter of $C_4$?",
        "geo_code": "[asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label(\"$C_4$\", D); label(\"$C_1$\", (-1.375, 0)); label(\"$C_2$\", (1.375,0)); label(\"$\\frac{1}{2}$\", (0, -.125)); label(\"$C_3$\", (-0.4, -0.4)); label(\"$1$\", (-.85, 0.70)); label(\"$1$\", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy]",
        "answer": "$\\frac{3}{14}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 237,
        "problem": "Four squares with side lengths \\(4\\), \\(7\\), \\(9\\), and \\(10\\) are placed so that their left sides and bottom sides are all aligned, and their sizes increase from smallest to largest. The colors of the squares alternate: the smallest square is white, the next is gray, then white again, and the largest is gray. What is the total area of the gray region that remains visible (not covered by any other square), in square units?",
        "geo_code": "[asy] size(150); filldraw((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(9,0)--(9,9)--(0,9)--cycle,white,linewidth(1)); filldraw((0,0)--(7,0)--(7,7)--(0,7)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(4,0)--(4,4)--(0,4)--cycle,white,linewidth(1)); draw((11,0)--(11,4),linewidth(1)); draw((11,6)--(11,10),linewidth(1)); label(\"$10$\",(11,5),fontsize(14pt)); draw((10.75,0)--(11.25,0),linewidth(1)); draw((10.75,10)--(11.25,10),linewidth(1)); draw((0,11)--(3,11),linewidth(1)); draw((5,11)--(9,11),linewidth(1)); draw((0,11.25)--(0,10.75),linewidth(1)); draw((9,11.25)--(9,10.75),linewidth(1)); label(\"$9$\",(4,11),fontsize(14pt)); draw((-1,0)--(-1,1),linewidth(1)); draw((-1,3)--(-1,7),linewidth(1)); draw((-1.25,0)--(-0.75,0),linewidth(1)); draw((-1.25,7)--(-0.75,7),linewidth(1)); label(\"$7$\",(-1,2),fontsize(14pt)); draw((0,-1)--(1,-1),linewidth(1)); draw((3,-1)--(4,-1),linewidth(1)); draw((0,-1.25)--(0,-.75),linewidth(1)); draw((4,-1.25)--(4,-.75),linewidth(1)); label(\"$4$\",(2,-1),fontsize(14pt)); [/asy]",
        "answer": "$52$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 238,
        "problem": "The coordinates of $\\triangle ABC$ are $A(5,7)$, $B(11,7)$, and $C(3,y)$, with $y>7$. The area of $\\triangle ABC$ is 12. Find the length of $BC$?",
        "geo_code": "[asy] draw((3,11)--(11,7)--(5,7)--(3,11)); dot((5,7)); label(\"$A(5,7)$\",(5,7),S); dot((11,7)); label(\"$B(11,7)$\",(11,7),S); dot((3,11)); label(\"$C(3,y)$\",(3,11),NW); [/asy]",
        "answer": "$4\\sqrt{5}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 239,
        "problem": "Three concentric circles with center \\(O\\) have radii \\(1\\), \\(2\\), and \\(3\\). Points \\(B\\) and \\(C\\) both lie on the outermost circle. The entire annulus between the circles of radius \\(1\\) and \\(2\\) is shaded, and an additional sector (defined by central angle \\(BOC\\)) of the annulus between the circles of radius \\(2\\) and \\(3\\) is also shaded. If the total shaded area is equal to the unshaded area, what is the degree measure of \\(\\angle BOC\\)?",
        "geo_code": "[asy] size(150); import graph; draw(circle((0,0),3)); real radius = 3; real angleStart = -54; // starting angle of the sector real angleEnd = 54; // ending angle of the sector label(\"$O$\",(0,0),W); pair O = (0, 0); filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, gray); filldraw(circle((0,0),2),gray); filldraw(circle((0,0),1),white); draw((1.763,2.427)--(0,0)--(1.763,-2.427)); label(\"$B$\",(1.763,2.427),NE); label(\"$C$\",(1.763,-2.427),SE); [/asy]",
        "answer": "$108$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 240,
        "problem": "Jean creates a stained glass artwork shaped like two adjacent mountains, as shown in the diagram below. The first peak has a height of $8$ feet and the second peak has a height of $12$ feet. Each mountain peak forms a right angle ($90^\\circ$), and both of the straight sides of the peaks rise at an angle of $45^\\circ$ from the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$",
        "geo_code": "[asy] unitsize(.3cm); filldraw((0,0)--(8,9)--(11,6)--(18,12)--(30,0)--cycle,gray(0.7),linewidth(1)); draw((-1,0)--(-1,9),linewidth(.75)); draw((-1.4,0)--(-.6,0),linewidth(.75)); draw((-1.4,9)--(-.6,9),linewidth(.75)); label(\"$8$\",(-1,4),W); label(\"$12$\",(31,6),E); draw((-1,9)--(8,9),dashed); draw((31,0)--(31,12),linewidth(.75)); draw((30.6,0)--(31.4,0),linewidth(.75)); draw((30.6,12)--(31.4,12),linewidth(.75)); draw((31,12)--(18,12),dashed); label(\"$45^{\\circ}$\",(.75,0),NE,fontsize(10pt)); label(\"$45^{\\circ}$\",(29.25,0),NW,fontsize(10pt)); draw((8,9)--(7.5,8.5)--(8,8)--(8.5,8.5)--cycle); draw((18,12)--(17.5,11.5)--(18,11)--(18.5,11.5)--cycle); draw((11,6)--(11,0),dashed); label(\"$h$\",(11,2.5),E);  [/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 241,
        "problem": "Let \\(\\triangle ABC\\) be a triangle with area \\(240\\). Points \\(X\\), \\(Y\\), and \\(Z\\) are located on sides \\(AB\\), \\(BC\\), and \\(CA\\) respectively, such that \\(\\frac{AX}{BX} = 3\\), \\(\\frac{BY}{CY} = 4\\), and \\(\\frac{CZ}{AZ} = 5\\). What is the area of triangle \\(XYZ\\)?",
        "geo_code": "[asy]\nsize(175);\ndefaultpen(linewidth(0.8));\npair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;\ndraw(A--B--C--cycle^^X--Y--Z--cycle);\nlabel(\"$A$\",A,N);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,E);\nlabel(\"$X$\",X,W);\nlabel(\"$Y$\",Y,S);\nlabel(\"$Z$\",Z,NE);[/asy]",
        "answer": "$122$",
        "category": "Local Relation Composition",
        "source": "olympiads",
        "problem_type": "area"
    },
    {
        "index": 242,
        "problem": "In the figure below, \\(\\angle ABC\\) is a right angle. Point \\(D\\) lies on segment \\(\\overline{BC}\\), and segment \\(\\overline{AD}\\) is the angle bisector of \\(\\angle CAB\\). Points \\(E\\) and \\(F\\) are located on segments \\(\\overline{AB}\\) and \\(\\overline{AC}\\), respectively, such that \\(AE = 3\\), \\(AF = 10\\), \\(EB = 9\\), and \\(FC = 27\\). Let \\(G\\) denote the point where \\(\\overline{DF}\\) intersects \\(\\overline{CE}\\). Determine the integer nearest to the area of quadrilateral \\(DCFG\\).",
        "geo_code": "[asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$D$\", D, dir(point--D)); label(\"$E$\", E, dir(point--E)); label(\"$F$\", F, dir(point--F)); label(\"$G$\", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label(\"10\", A--F, dir(90)*dir(A--F)); label(\"27\", F--C, dir(90)*dir(F--C)); label(\"3\", (0,10), W); label(\"9\", (0,4), W); [/asy]",
        "answer": "$148$",
        "category": "Local Relation Composition",
        "source": "AIME-83-24",
        "problem_type": "area"
    },
    {
        "index": 243,
        "problem": "Consider a fixed regular  $n$ -gon of unit side. When a second regular  $n$ -gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line  $\\kappa$  as in the figure.\n\n\n\nLet  $10$  be the area of a regular  $n$ -gon of unit side, and let  $5$  be the area of a regular  $n$ -gon of unit circumradius. Calculate the area enclosed by  $\\kappa$.",
        "geo_code": "[asy]\nint n=9;\ndraw(polygon(n));\nfor (int i = 0; i<n;++i) {\n draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4));\n draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2));\n}\n[/asy]",
        "answer": "$50$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 244,
        "problem": "Let  $n\\ge 2$  be an integer. Thibaud the Tiger lays  $n$   $2\\times 2$  overlapping squares out on a table, such that the centers of the squares are equally spaced along the line  $y=x$  from  $(0,0)$  to  $(1,1)$  (including the two endpoints). For example, for  $n=4$  the resulting figure is shown below, and it covers a total area of  $\\frac{23}{3}$ .\n\nFind the minimum  $n$  such that the figure covers an area of at least  $\\sqrt{63}$ .",
        "geo_code": "[asy]\nfill((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--cycle, lightgrey);\nfill((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--cycle, lightgrey);\nfill((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--cycle, lightgrey);\nfill((1,1)--(3,1)--(3,3)--(1,3)--cycle, lightgrey);\ndraw((0.33333333333333,2)--(2,2)--(2,0.333333333333), dashed+grey+linewidth(0.4));\ndraw((0.66666666666666,2.3333333333333)--(2.3333333333333,2.3333333333333)--(2.3333333333333,0.66666666666), dashed+grey+linewidth(0.4));\ndraw((1,2.666666666666)--(2.666666666666,2.666666666666)--(2.666666666666,1), dashed+grey+linewidth(0.4));\ndraw((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--(0,0),linewidth(0.4));\ndraw((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--(0.333333333333,.333333333333),linewidth(0.4));\ndraw((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--(0.6666666666666,0.6666666666666),linewidth(0.4));\ndraw((1,1)--(3,1)--(3,3)--(1,3)--cycle,linewidth(0.4));\n[/asy]",
        "answer": "$17$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 245,
        "problem": "In the figure, AQPB and ASRC are squares, and AQS is an equilateral triangle. If QS\u00a0\u00a0= 12 and  BC\u00a0\u00a0= x, what is the value of x?",
        "geo_code": "[asy]\nunitsize(16);\npair A,B,C,P,Q,R,T;\nA=(3.4641016151377544, 2);\nB=(0, 0);\nC=(6.928203230275509, 0);\nP=(-1.9999999999999991, 3.464101615137755);\nQ=(1.4641016151377544, 5.464101615137754);\nR=(8.928203230275509, 3.4641016151377544);\nT=(5.464101615137754, 5.464101615137754);\ndot(A);dot(B);dot(C);dot(P);\ndot(Q);dot(R);dot(T);\nlabel(\" $A$ \", (3.4641016151377544, 2),E);\nlabel(\" $B$ \", (0, 0),S);\nlabel(\" $C$ \", (6.928203230275509, 0),S);\nlabel(\" $P$ \", (-1.9999999999999991, 3.464101615137755), W);\nlabel(\" $Q$ \", (1.4641016151377544, 5.464101615137754),N);\nlabel(\" $R$ \", (8.928203230275509, 3.4641016151377544),E);\nlabel(\" $S$ \", (5.464101615137754, 5.464101615137754),N);\ndraw(B--C--A--B);\ndraw(B--P--Q--A--B);\ndraw(A--C--R--T--A);\ndraw(Q--T--A--Q);\nlabel(\" $x$ \", (3.4641016151377544, 0), S);\nlabel(\" $4$ \", (Q+T)/2, N);[/asy]",
        "answer": "$12\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 246,
        "problem": "In an acute triangle  $ABC$  let  $K,L,$  and  $M$  be the midpoints of sides  $AB,BC,$  and  $CA,$  respectively. From each of  $K,L,$  and  $M$  drop two perpendiculars to the other two sides of the triangle; e.g., drop perpendiculars from  $K$  to sides  $BC$  and  $CA,$  etc. The resulting  $6$  perpendiculars intersect at points  $Q,S,$  and  $T$  as in the figure to form a hexagon  $KQLSMT$  inside triangle  $ABC.$  The area of triangle $ABC$ is $n$ times of the area of hexagon  $KQLSMT$. Find the value of $n$",
        "geo_code": "[asy]import graph; size(10cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = 11.888712276357234, xmax = 17.841346447833423, ymin = 10.61620970860601, ymax = 15.470685507068502;\npen zzttqq = rgb(0.6,0.2,0.); pen qqwuqq = rgb(0.,0.39215686274509803,0.); \npair A = (12.488234161849352,12.833838721895551), B = (16.50823416184936,15.093838721895553), C = (16.28823416184936,11.353838721895551), K = (14.498234161849355,13.963838721895552), L = (16.39823416184936,13.223838721895552), M = (14.388234161849356,12.093838721895551), D = (13.615830174638527,13.467760858438725), F = (15.75135711740064,11.562938202365055), G = (15.625830174638523,14.597760858438724), H = (16.435061748056253,13.849907687412797), T = (14.02296781802369,12.74356027153236), Q = (16.032967818023693,13.873560271532357), O = (16.325061748056253,11.979907687412794); \n\ndraw(A--B--C--cycle, zzttqq); \ndraw((13.426050287639166,13.361068683160477)--(13.532742462917415,13.171288796161116)--(13.722522349916774,13.277980971439364)--D--cycle, qqwuqq); \ndraw((14.054227993863618,12.223925334689998)--(14.133240861538676,12.426796211152979)--(13.930369985075695,12.505809078828037)--(13.851357117400637,12.302938202365056)--cycle, qqwuqq); \ndraw((16.337846386707046,12.19724654447628)--(16.12050752964356,12.210031183127075)--(16.107722890992765,11.992692326063588)--O--cycle, qqwuqq); \ndraw((15.830369985075697,11.765809078828037)--(15.627499108612716,11.844821946503092)--(15.54848624093766,11.641951070040111)--F--cycle, qqwuqq); \ndraw((15.436050287639164,14.491068683160476)--(15.542742462917412,14.301288796161115)--(15.73252234991677,14.407980971439365)--G--cycle, qqwuqq); \ndraw((16.217722890992764,13.86269232606359)--(16.20493825234197,13.645353469000101)--(16.42227710940546,13.63256883034931)--H--cycle, qqwuqq); \nLabel laxis; laxis.p = fontsize(10); \nxaxis(xmin, xmax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true); \nyaxis(ymin, ymax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true);  \ndraw(A--B, zzttqq); \ndraw(B--C, zzttqq); \ndraw(C--A, zzttqq); \ndraw(M--D); \ndraw(K--(13.851357117400637,12.302938202365056)); \ndraw(F--L); \ndraw(L--G); \ndraw(K--H); \ndraw(M--O); \ndot(A,dotstyle); \nlabel(\" $A$ \", (12.52502834296331,12.93568440300881), NE * labelscalefactor); \ndot(B,dotstyle); \nlabel(\" $B$ \", (16.548187989892043,15.193580123223922), NE * labelscalefactor); \ndot(C,dotstyle); \nlabel(\" $C$ \", (16.332661580235147,11.457789022504372), NE * labelscalefactor); \ndot(K,linewidth(3.pt) + dotstyle); \nlabel(\" $K$ \", (14.536608166427676,14.02357961365791), NE * labelscalefactor); \ndot(L,linewidth(3.pt) + dotstyle); \nlabel(\" $L$ \", (16.43529320388129,13.28463192340569), NE * labelscalefactor); \ndot(M,linewidth(3.pt) + dotstyle); \nlabel(\" $M$ \", (14.433976542781535,12.155684063298134), NE * labelscalefactor); \ndot(D,linewidth(3.pt) + dotstyle); \ndot((13.851357117400637,12.302938202365056),linewidth(3.pt) + dotstyle); \ndot(F,linewidth(3.pt) + dotstyle); \ndot(G,linewidth(3.pt) + dotstyle); \ndot(H,linewidth(3.pt) + dotstyle); \ndot((15.922967818023695,12.003560271532354),linewidth(3.pt) + dotstyle); \nlabel(\" $S$ \", (15.96318773510904,12.063315602016607), NE * labelscalefactor); \ndot(T,linewidth(3.pt) + dotstyle); \nlabel(\" $T$ \", (14.064502697655428,12.802263292268826), NE * labelscalefactor); \ndot(Q,linewidth(3.pt) + dotstyle); \nlabel(\" $Q$ \", (16.076082521119794,13.931211152376383), NE * labelscalefactor); \ndot(O,linewidth(3.pt) + dotstyle); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 247,
        "problem": "In the figure below, each side of the rhombus has length 8 centimeters.\n\nThe circle lies entirely within the rhombus.  The area of the circle is  $n$  square centimeters, where  $n$  is a positive integer.  Compute the number of possible values of  $n$ .",
        "geo_code": "[asy]\n  import graph;\n  unitsize(2.0cm);\n  real w = sqrt(3);\n  draw((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- cycle);\n  filldraw(Circle((-0.5, 0), 0.8 / sqrt(pi)), gray);\n  label(\" $60^\\circ$ \", (w - 0.1, 0), W);\n[/asy]",
        "answer": "$37$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 248,
        "problem": "The grid below contains six rows with six points in each row. Points that are adjacent either horizontally or vertically with equal distance 13. Find the area of the irregularly shaped ten sided figure shown.",
        "geo_code": "[asy]\nimport graph; size(5cm); \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);\npen dotstyle = black;\ndraw((-2,5)--(-3,4), linewidth(1.6)); \ndraw((-3,4)--(-2,1), linewidth(1.6)); \ndraw((-2,1)--(1,0), linewidth(1.6)); \ndraw((1,0)--(2,1), linewidth(1.6)); \ndraw((2,1)--(1,3), linewidth(1.6)); \ndraw((1,3)--(1,4), linewidth(1.6)); \ndraw((1,4)--(2,5), linewidth(1.6)); \ndraw((2,5)--(0,5), linewidth(1.6)); \ndraw((-2,5)--(-1,4), linewidth(1.6)); \ndraw((-1,4)--(0,5), linewidth(1.6)); \ndot((-3,5),linewidth(6pt) + dotstyle); \ndot((-2,5),linewidth(6pt) + dotstyle); \ndot((-1,5),linewidth(6pt) + dotstyle); \ndot((0,5),linewidth(6pt) + dotstyle); \ndot((1,5),linewidth(6pt) + dotstyle); \ndot((2,5),linewidth(6pt) + dotstyle); \ndot((2,4),linewidth(6pt) + dotstyle); \ndot((2,3),linewidth(6pt) + dotstyle); \ndot((2,2),linewidth(6pt) + dotstyle); \ndot((2,1),linewidth(6pt) + dotstyle); \ndot((2,0),linewidth(6pt) + dotstyle); \ndot((-3,4),linewidth(6pt) + dotstyle); \ndot((-3,3),linewidth(6pt) + dotstyle); \ndot((-3,2),linewidth(6pt) + dotstyle); \ndot((-3,1),linewidth(6pt) + dotstyle); \ndot((-3,0),linewidth(6pt) + dotstyle); \ndot((-2,0),linewidth(6pt) + dotstyle); \ndot((-2,1),linewidth(6pt) + dotstyle); \ndot((-2,2),linewidth(6pt) + dotstyle); \ndot((-2,3),linewidth(6pt) + dotstyle); \ndot((-2,4),linewidth(6pt) + dotstyle); \ndot((-1,4),linewidth(6pt) + dotstyle); \ndot((0,4),linewidth(6pt) + dotstyle); \ndot((1,4),linewidth(6pt) + dotstyle); \ndot((1,3),linewidth(6pt) + dotstyle); \ndot((0,3),linewidth(6pt) + dotstyle); \ndot((-1,3),linewidth(6pt) + dotstyle); \ndot((-1,2),linewidth(6pt) + dotstyle); \ndot((-1,1),linewidth(6pt) + dotstyle); \ndot((-1,0),linewidth(6pt) + dotstyle); \ndot((0,0),linewidth(6pt) + dotstyle); \ndot((1,0),linewidth(6pt) + dotstyle); \ndot((1,1),linewidth(6pt) + dotstyle); \ndot((1,2),linewidth(6pt) + dotstyle); \ndot((0,2),linewidth(6pt) + dotstyle); \ndot((0,1),linewidth(6pt) + dotstyle); [/asy]",
        "answer": "$214.5$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 249,
        "problem": "In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If  $AG=2$ ,  $GF=13$ ,  $FC=1$ , and  $HJ=7$, the length of $DE$ is ?",
        "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=sqrt(22);\npair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J);\ndot(A^^B^^C^^D^^E^^F^^G^^H^^J);\ndraw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7));\nlabel(\" $A$ \", A, N);\nlabel(\" $B$ \", B, dir(210));\nlabel(\" $C$ \", C, dir(330));\nlabel(\" $D$ \", D, SW);\nlabel(\" $E$ \", E, SE);\nlabel(\" $F$ \", F, dir(170));\nlabel(\" $G$ \", G, dir(250));\nlabel(\" $H$ \", H, SE);\nlabel(\" $J$ \", J, dir(0));\nlabel(\"2\", A--G, dir(30));\nlabel(\"13\", F--G, dir(180+30));\nlabel(\"1\", F--C, dir(30));\nlabel(\"7\", H--J, dir(-30));[/asy]",
        "answer": "$2\\sqrt{22}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 250,
        "problem": "An equilateral triangle has sides of length  $6$ cm. At each vertex, a circle with radius  $3$ cm is drawn, as shown in the figure below. The total area of the unshaded regions of the three circles is  $a\\cdot \\pi \\text{cm}^2$ . Determine  $a$ .\n\n[center][/center]",
        "geo_code": "[asy]\nsize(2.5cm);\ndraw(circle((0,2sqrt(3)/3),1));\ndraw(circle((1,-sqrt(3)/3),1));\ndraw(circle((-1,-sqrt(3)/3),1));\ndraw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle);\nfill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray);\ndraw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle);\nfill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray);\ndraw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle);\nfill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray);\n[/asy]",
        "answer": "$22.5$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 251,
        "problem": "Suppose two circles  $\\Omega_1$  and  $\\Omega_2$  with centers  $O_1$  and  $O_2$  have radii  $6$  and  $8$ , respectively. Suppose that points  $A$  and  $B$  lie on circles  $\\Omega_1$  and  $\\Omega_2$ , respectively, such that segments  $AB$  and  $O_1O_2$  intersect and that  $AB$  is tangent to  $\\Omega_1$  and  $\\Omega_2$ . If  $O_1O_2=50$ , find the lenght of line $AB$",
        "geo_code": "[asy]\nimport graph; size(12cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = -12.81977592804657, xmax = 32.13023014338037, ymin = -14.185056097058798, ymax = 12.56855801985179;  \ndraw(circle((-3.4277328104418046,-1.4524996726688195), 3), linewidth(1.2)); \ndraw(circle((21.572267189558197,-1.4524996726688195), 4), linewidth(1.2)); \ndraw((-2.5877328104418034,1.4275003273311748)--(20.452267189558192,-5.2924996726687885), linewidth(1.2)); \n\ndot((-3.4277328104418046,-1.4524996726688195),linewidth(3pt) + dotstyle); \nlabel(\" $O_1$ \", (-4.252707018231291,-1.545940604327141), N * labelscalefactor); \ndot((21.572267189558197,-1.4524996726688195),linewidth(3pt) + dotstyle); \nlabel(\" $O_2$ \", (21.704189347819636,-1.250863978037686), NE * labelscalefactor); \ndot((-2.5877328104418034,1.4275003273311748),linewidth(3pt) + dotstyle); \nlabel(\" $A$ \", (-2.3937351324858342,1.6999022848568643), NE * labelscalefactor); \ndot((20.452267189558192,-5.2924996726687885),linewidth(3pt) + dotstyle); \nlabel(\" $B$ \", (20.671421155806545,-4.9885012443707835), NE * labelscalefactor); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]",
        "answer": "$48$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 252,
        "problem": "Suppose circles  $\\mathit{W}_1$  and  $\\mathit{W}2$ , with centres  $\\mathit{O}_1$  and  $\\mathit{O}_2$  respectively, intersect at points  $\\mathit{M}$  and  $\\mathit{N}$ . Let the tangent on  $\\mathit{W}_2$  at point  $\\mathit{N}$  intersect  $\\mathit{W}_1$  for the second time at  $\\mathit{B}_1$ . Similarly, let the tangent on  $\\mathit{W}_1$  at point  $\\mathit{N}$  intersect  $\\mathit{W}_2$  for the second time at  $\\mathit{B}_2$ . Let  $\\mathit{A}_1$  be a point on  $\\mathit{W}_1$  which is on arc  $\\mathit{B}_1\\mathit{N}$  not containing  $\\mathit{M}$  and suppose line  $\\mathit{A}_1\\mathit{N}$  intersects  $\\mathit{W}_2$  at point  $\\mathit{A}_2$ . Denote the incentres of triangles  $\\mathit{B}_1\\mathit{A}_1\\mathit{N}$  and  $\\mathit{B}_2\\mathit{A}_2\\mathit{N}$  by  $\\mathit{I}_1$  and  $\\mathit{I}_2$ , respectively.If the degree of $\\angle\\mathit{I}_1\\mathit{MI}_2$ is 37, find the degree of angle $\\angle\\mathit{O}_1\\mathit{MO}_2 + \\angle\\mathit{I}_1\\mathit{MI}_2$.",
        "geo_code": "[asy]import graph; size(10.1cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = -0.9748626324969808, xmax = 13.38440254515721, ymin = 0.5680051903627492, ymax = 10.99430986899034; \npair O_2 = (7.682929606970993,6.084708172218866), O_1 = (2.180000000000002,6.760000000000007), M = (4.560858774883258,8.585242858926296), B_2 = (10.07334553576748,9.291873850408265), A_2 = (11.49301008867042,4.866805580476367), B_1 = (2.113311869970955,9.759258690628950), A_1 = (0.2203184186713625,4.488514120712773); \ndraw(circle(O_2, 4.000000000000000)); \ndraw(circle(O_1, 3.000000000000000)); \ndraw((4.048892687647541,4.413249028538064)--B_2); \ndraw(B_2--A_2); \ndraw(A_2--(4.048892687647541,4.413249028538064)); \ndraw((4.048892687647541,4.413249028538064)--B_1); \ndraw(B_1--A_1); \ndraw(A_1--(4.048892687647541,4.413249028538064)); \ndot(O_2,dotstyle); \nlabel(\" $O_2$ \", (7.788512439159622,6.243082420501817), NE * labelscalefactor); \ndot(O_1,dotstyle); \nlabel(\" $O_1$ \", (2.298205165350667,6.929370829727937), NE * labelscalefactor); \ndot(M,dotstyle); \nlabel(\" $M$ \", (4.383466101076183,8.935444641311980), NE * labelscalefactor); \ndot((4.048892687647541,4.413249028538064),dotstyle); \nlabel(\" $N$ \", (3.855551940133015,3.761885864068922), NE * labelscalefactor); \ndot(B_2,dotstyle); \nlabel(\" $B_2$ \", (10.19052187145104,9.463358802255147), NE * labelscalefactor); \ndot(A_2,dotstyle); \nlabel(\" $A_2$ \", (11.80066006232771,4.659339937672310), NE * labelscalefactor); \ndot(B_1,dotstyle); \nlabel(\" $B_1$ \", (1.981456668784765,10.09685579538695), NE * labelscalefactor); \ndot(A_1,dotstyle); \nlabel(\" $A_1$ \", (0.08096568938935705,3.973051528446190), NE * labelscalefactor); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]",
        "answer": "$74$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 253,
        "problem": "The diagram below shows nine points on a circle where  $AB=BC=CD=DE=EF=FG=GH$ .  Given that  $\\angle GHJ=117^\\circ$  and  $\\overline{BH}$  is perpendicular to  $\\overline{EJ}$ , there are relatively prime positive integers  $m$  and  $n$  so that the degree measure of  $\\angle AJB$  is  $\\frac{m}{n}$ .  Find  $m\times n$ .",
        "geo_code": "[asy]\nsize(175);\ndefaultpen(linewidth(0.6));\ndraw(unitcircle,linewidth(0.9));\nstring labels[] = {\"A\",\"B\",\"C\",\"D\",\"E\",\"F\",\"G\"};\nint start=110,increment=20;\npair J=dir(210),x[],H=dir(start-7*increment);\nfor(int i=0;i<=6;i=i+1)\n{\nx[i]=dir(start-increment*i);\ndraw(J--x[i]--H);\ndot(x[i]);\nlabel(\" $\"+labels[i]+\"$ \",x[i],dir(origin--x[i]));\n}\ndraw(J--H);\ndot(H^^J);\nlabel(\" $H$ \",H,dir(origin--H));\nlabel(\" $J$ \",J,dir(origin--J));\n[/asy]",
        "answer": "$54$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 254,
        "problem": "In the following diagram,  $O$  is the center of the circle. If three angles  $\\alpha, \\beta$  and  $\\gamma$  be equal, find the degree of angle $\\gamma.$",
        "geo_code": "[asy]\nunitsize(40);\nimport graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2,0.4); pen qqwuqq = rgb(0,0.39,0);\ndraw(circle((0,0),2.33),ttttff+linewidth(2.8pt)); draw((-1.95,-1.27)--(0.64,2.24),ffttww+linewidth(2pt)); draw((0.64,2.24)--(1.67,-1.63),ffttww+linewidth(2pt)); draw((-1.95,-1.27)--(1.06,0.67),ffttww+linewidth(2pt)); draw((1.67,-1.63)--(-0.6,0.56),ffttww+linewidth(2pt)); draw((-0.6,0.56)--(1.06,0.67),ffttww+linewidth(2pt)); pair parametricplot0_cus(real t){\nreturn (0.6*cos(t)+0.64,0.6*sin(t)+2.24);\n}\ndraw(graph(parametricplot0_cus,-2.2073069497794027,-1.3111498158746024)--(0.64,2.24)--cycle,qqwuqq); pair parametricplot1_cus(real t){\nreturn (0.6*cos(t)+-0.6,0.6*sin(t)+0.56);\n}\ndraw(graph(parametricplot1_cus,0.06654165390165974,0.9342857038103908)--(-0.6,0.56)--cycle,qqwuqq); pair parametricplot2_cus(real t){\nreturn (0.6*cos(t)+-0.6,0.6*sin(t)+0.56);\n}\ndraw(graph(parametricplot2_cus,-0.766242589858673,0.06654165390165967)--(-0.6,0.56)--cycle,qqwuqq);\ndot((0,0),ds); label(\" $O$ \", (-0.2,-0.38), NE*lsf); dot((0.64,2.24),ds); label(\" $A$ \", (0.72,2.36), NE*lsf); dot((-1.95,-1.27),ds); label(\" $B$ \", (-2.2,-1.58), NE*lsf); dot((1.67,-1.63),ds); label(\" $C$ \", (1.78,-1.96), NE*lsf); dot((1.06,0.67),ds); label(\" $E$ \", (1.14,0.78), NE*lsf); dot((-0.6,0.56),ds); label(\" $D$ \", (-0.92,0.7), NE*lsf); label(\" $\\alpha$ \", (0.48,1.38),NE*lsf); label(\" $\\beta$ \", (-0.02,0.94),NE*lsf); label(\" $\\gamma$ \", (0.04,0.22),NE*lsf); clip((-8.84,-9.24)--(-8.84,8)--(11.64,8)--(11.64,-9.24)--cycle);\n[/asy]",
        "answer": "$50$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 255,
        "problem": "Square  $ABCD$  is divided into four rectangles by  $EF$  and  $GH$ .  $EF$  is parallel to  $AB$  and  $GH$  parallel to  $BC$ .   $\\angle BAF = 18^\\circ$ .  $EF$  and  $GH$  meet at point  $P$ . The area of rectangle  $PFCH$  is twice that of rectangle  $AGPE$ . Given that the value of  $\\angle FAH$  in degrees is  $3y$ , find the nearest integer to  $y$ .",
        "geo_code": "[asy]\nsize(100); defaultpen(linewidth(0.7)+fontsize(10)); \npair D2(pair P) {\ndot(P,linewidth(3)); return P;\n}\npair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];\ndraw(A--B--C--D--cycle);\ndraw(F--A--H); draw(E--F); draw(G--H); \nlabel(\" $A$ \",D2(A),NW);\nlabel(\" $B$ \",D2(B),SW);\nlabel(\" $C$ \",D2(C),SE);\nlabel(\" $D$ \",D2(D),NE);\nlabel(\" $E$ \",D2(E),plain.N);\nlabel(\" $F$ \",D2(F),S);\nlabel(\" $G$ \",D2(G),W);\nlabel(\" $H$ \",D2(H),plain.E);\nlabel(\" $P$ \",D2(P),SE);\n[/asy]",
        "answer": "$15$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 256,
        "problem": "In the following diagram (not to scale),  $A$ ,  $B$ ,  $C$ ,  $D$  are four consecutive vertices of an 18-sided regular polygon with center  $O$ . Let  $P$  be the midpoint of  $AC$  and  $Q$  be the midpoint of  $DO$ . Find the degree of $\\angle OPQ$.",
        "geo_code": "[asy]\npathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype(\"4 4\"); real n = 10, start = 360/n*6-15;\npair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D(\"O\",O,NE)--D(\"A\",A,W)--D(\"B\",B,SW)--D(\"C\",C,S)--D(\"D\",D,SE)--O--D(\"P\",P,1.6*dir(95))--D(\"Q\",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd);\n[/asy]",
        "answer": "$30$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 257,
        "problem": "In the diagram below  $ \\angle CAB, \\angle CBD$ , and  $\\angle CDE$  are all right angles with side lengths  $AC = 3$ ,  $BC = 5$ ,  $BD = 12$ , and  $DE = 84$ . The distance from point  $E$  to the line  $AB$  can be expressed as the ratio of two relatively prime positive integers,  $m$  and  $n$ . Find  $m\times n$ .",
        "geo_code": "[asy]\nsize(300);\ndefaultpen(linewidth(0.8));\ndraw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8));\nlabel(\" $A$ \",origin,SW);\nlabel(\" $B$ \",(0,4),dir(160));\nlabel(\" $C$ \",(3,0),S);\nlabel(\" $D$ \",(6,8),dir(80));\nlabel(\" $E$ \",(30,-4),E);[/asy]",
        "answer": "$346320$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 258,
        "problem": "As in the following diagram, square  $ABCD$  and square  $CEFG$  are placed side by side. If  $CE = 18$ ,  $AB > 19$ , compute the minimal area of  $\\triangle AEG$ .",
        "geo_code": "[asy]\nsize(120); defaultpen(linewidth(0.7)+fontsize(10)); \npair D2(real x, real y) {\n pair P = (x,y);\n dot(P,linewidth(3)); return P;\n}\nint big = 30, small = 14;\nfilldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5));\ndraw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare);\nlabel(\" $A$ \",D2(0,big),NW);\nlabel(\" $B$ \",D2(0,0),SW);\nlabel(\" $C$ \",D2(big,0),SW);\nlabel(\" $D$ \",D2(big,big),N);\nlabel(\" $E$ \",D2(big+small,0),SE);\nlabel(\" $F$ \",D2(big+small,small),NE);\nlabel(\" $G$ \",D2(big,small),NE);\n[/asy]",
        "answer": "$162$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 259,
        "problem": "In the diagram, the circle has radius  $\\sqrt{\\frac{31}{3}}$  and and centre  $O.$  Points  $A, B$  and  $C$  are on the circle. If  $\\angle BOC=120^\\circ$  and  $AC = AB + 1,$  determine the length of  $AB.$",
        "geo_code": "[asy]\nimport graph; size(120); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen xdxdff = rgb(0.49,0.49,1); pen fftttt = rgb(1,0.2,0.2);\ndraw(circle((2.34,2.4),2.01),qqttff); draw((2.34,2.4)--(1.09,0.82),fftttt); draw((2.34,2.4)--(4.1,1.41),fftttt); draw((1.09,0.82)--(1.4,4.18),fftttt); draw((4.1,1.41)--(1.4,4.18),fftttt);\ndot((2.34,2.4),ds); label(\" $O$ \", (2.1,2.66),NE*lsf); dot((1.09,0.82),ds); label(\" $B$ \", (0.86,0.46),NE*lsf); dot((4.1,1.41),ds); label(\" $C$ \", (4.2,1.08),NE*lsf); dot((1.4,4.18),ds); label(\" $A$ \", (1.22,4.48),NE*lsf); clip((-4.34,-10.94)--(-4.34,6.3)--(16.14,6.3)--(16.14,-10.94)--cycle);\n[/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 260,
        "problem": "The diagram below shows the regular hexagon  $BCEGHJ$  surrounded by the rectangle  $ADFI$ . Let  $\\theta$ be the measure of the acute angle between the side  $\\overline{EG}$  of the hexagon and the diagonal of the rectangle  $\\overline{AF}$ . There are relatively prime positive integers  $m$  and  $n$  so that  $\\sin^2\\theta = \\frac{m}{n}$ . Find  $n - m$ .",
        "geo_code": "[asy]\nimport graph; size(3.2cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);\ndraw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle); \ndraw((-1,3)--(-1,2)); \ndraw((-1,2)--(-0.13,1.5)); \ndraw((-0.13,1.5)--(0.73,2)); \ndraw((0.73,2)--(0.73,3)); \ndraw((0.73,3)--(-0.13,3.5)); \ndraw((-0.13,3.5)--(-1,3)); \ndraw((-1,3.5)--(0.73,3.5)); \ndraw((0.73,3.5)--(0.73,1.5)); \ndraw((-1,1.5)--(0.73,1.5)); \ndraw((-1,3.5)--(-1,1.5)); \nlabel(\" $ A $ \",(-1.4,3.9),SE*labelscalefactor); \nlabel(\" $ B $ \",(-1.4,3.28),SE*labelscalefactor); \nlabel(\" $ C $ \",(-1.4,2.29),SE*labelscalefactor); \nlabel(\" $ D $ \",(-1.4,1.45),SE*labelscalefactor); \nlabel(\" $ E $ \",(-0.3,1.4),SE*labelscalefactor); \nlabel(\" $ F $ \",(0.8,1.45),SE*labelscalefactor); \nlabel(\" $ G $ \",(0.8,2.24),SE*labelscalefactor); \nlabel(\" $ H $ \",(0.8,3.26),SE*labelscalefactor); \nlabel(\" $ I $ \",(0.8,3.9),SE*labelscalefactor); \nlabel(\" $ J $ \",(-0.25,3.9),SE*labelscalefactor); [/asy]",
        "answer": "$1$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 261,
        "problem": "The diagram below shows an isosceles triangle with base  $42$  and height  $56$ .  Inscribed in the triangle is a square.  Find the area of the shaded regione.",
        "geo_code": "[asy]\nsize(170);\ndefaultpen(linewidth(0.8));\ndraw((0,0)--(1,1));\npair A=(5,0),B=(-5,0),C=(0,14), invis[]={(1,2),(-1,2)};\npair intsquare[]={extension(origin,invis[0],A,C),extension(origin,invis[1],B,C)};\npath triangle=A--B--C--cycle,square=(intsquare[0]--intsquare[1]--(intsquare[1].x,0)--(intsquare[0].x,0)--cycle);\nfill(triangle,gray);\nunfill(square);\ndraw(triangle^^square);\n[/asy]",
        "answer": "$600$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 262,
        "problem": "A square is inscribed in a circle of radius  $8$ . A quarter circle is inscribed in the square, as shown in the diagram below. Given the area of the region inside the circle but outside the quarter circle is  $n\\pi$  for some positive integer  $n$ , what is  $n$ ?",
        "geo_code": "[asy]\n        size(5 cm);\n        draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);\n        draw(circle((1,1),1.41));\n        draw(arc((0,0),2,0,90));[/asy]",
        "answer": "$32$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 263,
        "problem": "In the following diagram, let  $ABCD$  be a square and let  $M,N,P$  and  $Q$  be the midpoints of its sides. How many times of the area of square $ABCD$ compared to the area of quadrilateral $A'B'C'D'$",
        "geo_code": "[asy]\nimport graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1);\ndraw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt));\ndot((0,4),ds); label(\" $A$ \", (0.07,4.12), NE*lsf); dot((0,0),ds); label(\" $D$ \", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label(\" $C$ \", (4.14,-0.39), NE*lsf); dot((4,4),ds); label(\" $B$ \", (4.08,4.12), NE*lsf); dot((2,4),ds); label(\" $M$ \", (2.08,4.12), NE*lsf); dot((4,2),ds); label(\" $N$ \", (4.2,1.98), NE*lsf); dot((2,0),ds); label(\" $P$ \", (1.99,-0.49), NE*lsf); dot((0,2),ds); label(\" $Q$ \", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label(\" $A'$ \", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label(\" $B'$ \", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label(\" $C'$ \", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label(\" $D'$ \", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle);\n[/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 264,
        "problem": "In the diagram line segments  $AB$  and  $CD$  are of length 1 while angles  $ABC$  and  $CBD$  are  $90^\\circ$  and  $30^\\circ$  respectively.  Find  $AC$ .",
        "geo_code": "[asy]\nimport geometry;\nimport graph;\n\nunitsize(1.5 cm);\n\npair A, B, C, D;\n\nB = (0,0);\nD = (3,0);\nA = 2*dir(120);\nC = extension(B,dir(30),A,D);\n\ndraw(A--B--D--cycle);\ndraw(B--C);\ndraw(arc(B,0.5,0,30));\n\nlabel(\" $A$ \", A, NW);\nlabel(\" $B$ \", B, SW);\nlabel(\" $C$ \", C, NE);\nlabel(\" $D$ \", D, SE);\nlabel(\" $30^\\circ$ \", (0.8,0.2));\nlabel(\" $90^\\circ$ \", (0.1,0.5));\n\nperpendicular(B,NE,C-B);\n[/asy]",
        "answer": "$\\sqrt[3]{2}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 265,
        "problem": "In the following diagram, a chord $AN$ intersects its diameter $MN$ at $B$. Given that  $MB : BN = 2 : 3$  and  $MN = 10$ . If  $AN = x$ , find  $x^2$ .",
        "geo_code": "[asy]\nsize(120); defaultpen(linewidth(0.7)+fontsize(10)); \npair D2(pair P) {\ndot(P,linewidth(3)); return P;\n}\nreal r = sqrt(80)/5;\npair M=(-1,0), N=(1,0), A=intersectionpoints(arc((M+N)/2, 1, 0, 180),circle(N,r))[0], C=intersectionpoints(circle(A,1),circle(N,1))[0], B=intersectionpoints(circle(C,1),M--N)[0];\ndraw(arc((M+N)/2, 1, 0, 180)--cycle); draw(A--N); draw(arc(C,1,180,180+2*aSin(r/2)));\nlabel(\" $A$ \",D2(A),NW);\nlabel(\" $B$ \",D2(B),SW);\nlabel(\" $M$ \",D2(M),S);\nlabel(\" $N$ \",D2(N),SE);\n[/asy]",
        "answer": "$80$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 266,
        "problem": "$DEB$  is a chord of a circle such that  $DE=6$  and  $EB=5$ . Let  $O$  be the centre of the circle. Join  $OE$  and extend  $OE$  to cut the circle at  $C$. Given  $EC=2$ , find the radius of the circle.",
        "geo_code": "[asy]\nsize(6cm);\n\npair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle);\n\ndraw(unitcircle);\ndraw(O--C);\ndraw(B--D);\n\ndot(O);\ndot(B);\ndot(C);\ndot(D);\ndot(E);\n\nlabel(\" $B$ \", B, B);\nlabel(\" $C$ \", C, C);\nlabel(\" $D$ \", D, D);\nlabel(\" $E$ \", E, dir(280));\nlabel(\" $O$ \", O, dir(270));\n[/asy]",
        "answer": "$14$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 267,
        "problem": "The figure below shows a  $9\\times7$  arrangement of  $3\\times3$  squares. As shown in the diagram, find the area of the shaded region in the grid.",
        "geo_code": "[asy]\nsize(5cm);\ndefaultpen(linewidth(.6));\nfill((0,1)--(1,1)--(1,0)--cycle^^(0,3)--(1,3)--(1,2)--cycle^^(1,2)--(2,2)--(2,1)--cycle^^(2,1)--(3,1)--(3,0)--cycle,rgb(.76,.76,.76));\nfill((0,5)--(1,5)--(1,4)--cycle^^(1,4)--(2,4)--(2,3)--cycle^^(2,3)--(3,3)--(3,2)--cycle^^(3,2)--(4,2)--(4,1)--cycle^^(4,1)--(5,1)--(5,0)--cycle,rgb(.76,.76,.76));\nfill((0,7)--(1,7)--(1,6)--cycle^^(1,6)--(2,6)--(2,5)--cycle^^(2,5)--(3,5)--(3,4)--cycle^^(3,4)--(4,4)--(4,3)--cycle^^(4,3)--(5,3)--(5,2)--cycle^^(5,2)--(6,2)--(6,1)--cycle^^(6,1)--(7,1)--(7,0)--cycle,rgb(.76,.76,.76));\nfill((2,7)--(3,7)--(3,6)--cycle^^(3,6)--(4,6)--(4,5)--cycle^^(4,5)--(5,5)--(5,4)--cycle^^(5,4)--(6,4)--(6,3)--cycle^^(6,3)--(7,3)--(7,2)--cycle^^(7,2)--(8,2)--(8,1)--cycle^^(8,1)--(9,1)--(9,0)--cycle,rgb(.76,.76,.76));\nfill((4,7)--(5,7)--(5,6)--cycle^^(5,6)--(6,6)--(6,5)--cycle^^(6,5)--(7,5)--(7,4)--cycle^^(7,4)--(8,4)--(8,3)--cycle^^(8,3)--(9,3)--(9,2)--cycle,rgb(.76,.76,.76));\nfill((6,7)--(7,7)--(7,6)--cycle^^(7,6)--(8,6)--(8,5)--cycle^^(8,5)--(9,5)--(9,4)--cycle,rgb(.76,.76,.76));\nfill((8,7)--(9,7)--(9,6)--cycle,rgb(.76,.76,.76));\ndraw((0,0)--(0,7)^^(1,0)--(1,7)^^(2,0)--(2,7)^^(3,0)--(3,7)^^(4,0)--(4,7)^^(5,0)--(5,7)^^(6,0)--(6,7)^^(7,0)--(7,7)^^(8,0)--(8,7)^^(9,0)--(9,7));\ndraw((0,0)--(9,0)^^(0,1)--(9,1)^^(0,2)--(9,2)^^(0,3)--(9,3)^^(0,4)--(9,4)^^(0,5)--(9,5)^^(0,6)--(9,6)^^(0,7)--(9,7));\ndraw((0,1)--(1,0)^^(0,3)--(3,0)^^(0,5)--(5,0)^^(0,7)--(7,0)^^(2,7)--(9,0)^^(4,7)--(9,2)^^(6,7)--(9,4)^^(8,7)--(9,6));\n[/asy]",
        "answer": "$144$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 268,
        "problem": "A semicircle with diameter length  $16$  contains a circle radius  $3$  tangent both to the inside of the semicircle and its diameter as shown. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be written as  $\\frac{n + k\\sqrt{2}}{m}$  where  $m$ ,  $n$ , and  $k$  are positive integers and  $m$  and  $n$  have no factors in common. Find  $m + n - k$ .",
        "geo_code": "[asy]\nsize(200);\npair O=(0,0);\nreal R=10, r=4.7;\ndraw(arc(O,R,0,180)--cycle);\npair P=(sqrt((R-r)^2-r^2),r),Q;\ndraw(circle(P,r));\nreal a=0,b=r,c;\nfor(int k=0;k<20;++k)\n{\nc=(a+b)/2;\nQ=(-sqrt((R-c)^2-c^2),c);\nif(abs(P-Q)>c+r) a=c; else b=c;\n}\ndraw(circle(Q,c));[/asy]",
        "answer": "$169$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 269,
        "problem": "Three lines are drawn parallel to each of the three sides of  $\\triangle ABC$  so that the three lines intersect in the interior of  $ABC$ . The resulting three smaller triangles have areas  $1$ ,  $4$ , and  $16$ . Find the area of  $\\triangle ABC$ .",
        "geo_code": "[asy]\ndefaultpen(linewidth(0.7)); size(120);\npair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); }\npair B = (0,0), C = (1,0), A = (0.3, 0.8), D = relpt(relpt(A,B,3,3),relpt(A,C,3,3),1,2);\ndraw(A--B--C--cycle);\nlabel(\" $A$ \",A,N); label(\" $B$ \",B,S); label(\" $C$ \",C,S);\nfilldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7));\nfilldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7));\nfilldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy]",
        "answer": "$49$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 270,
        "problem": "In the figure on the right,  $O$  is the center of the circle,  $OK$  and  $OA$  are perpendicular to one another,  $M$  is the midpoint of  $OK$ ,  $BN$   is parallel to  $OK$ , and  $\\angle AMN=\\angle NMO$ . Determine the measure of  $\\angle ABN$  in degrees.",
        "geo_code": "[asy]\nsize(120);defaultpen(linewidth(0.7)+fontsize(10));\npair O=origin, A=dir(90), K=dir(180), M=1/2*dir(180), N=0.4*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90);\ndraw(K--O--A--M--N--B--A^^circle((0,0),1));\nlabel(\" $A$ \", A, dir(O--A));\nlabel(\" $K$ \", K, dir(O--K));\nlabel(\" $B$ \", B, dir(O--B));\nlabel(\" $N$ \", N, E);\nlabel(\" $M$ \", M, S);\nlabel(\" $O$ \", O, SE);[/asy]",
        "answer": "$36$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 271,
        "problem": "$\\triangle ABC$  has area  $360$ . Points  $X, Y, Z$  lie on sides  $AB$ ,  $BC$ , and  $CA$ , respectively. Given that  $\\frac{AX}{BX} = 3$ ,  $\\frac{BY}{CY} = 4$ , and  $\\frac{CZ}{AZ} = 5$ , find the area of  $\\triangle XYZ$ .",
        "geo_code": "[asy]\nsize(175);\ndefaultpen(linewidth(0.8));\npair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;\ndraw(A--B--C--cycle^^X--Y--Z--cycle);\nlabel(\" $A$ \",A,N);\nlabel(\" $B$ \",B,S);\nlabel(\" $C$ \",C,E);\nlabel(\" $X$ \",X,W);\nlabel(\" $Y$ \",Y,S);\nlabel(\" $Z$ \",Z,NE);[/asy]",
        "answer": "$183$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 272,
        "problem": "In acute triangle  $ABC$ ,  $AB > AC$ . Let  $M$  be the midpoint of side  $BC$ . The exterior angle bisector of  $\\widehat{BAC}$  meet ray  $BC$  at  $P$ . Point  $K$  and  $F$  lie on line  $PA$  such that  $MF \\perp BC$  and  $MK \\perp PA$ . What is $\\frac{BC^2}{PF\\cdot AK}$ ?",
        "geo_code": "[asy] pair incenter(pair A=(0,0), pair B=(0,0), pair C=(0,0)){pair P,Q;P=rotate((angle(C-A)-angle(B-A))*90/pi,A)*B;Q=rotate((angle(A-B)-angle(C-B))*90/pi,B)*C;return extension(A,P,B,Q);}pair foot(pair P,pair A, pair B){real s;s=dot(P-A,unit(B-A));return (scale(s)*unit(B-A)+A);}defaultpen(fontsize(10)); size(7cm);pair A = (5.6,5), B = (0,0), C = (5,0), M = midpoint(B--C), I = incenter(A,B,C), P = extension(A, A+dir(I--A)*dir(-90), B,C), K = foot(M,A,P), F = extension(M, (M.x, M.x+1), A,P);draw(K--M--F--P--B--A--C);pair point = I;pair[] p={A,B,C,M,P,F,K};string s = \"A,B,C,M,P,F,K\";int size = p.length;real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;}string[] k= split(s,\",\");for(int i = 0;i<p.length;++i) {label(\"$\"+k[i]+\"$\",p[i],mult[i]*dir(point--p[i])*dir(d[i]));}[/asy]",
        "answer": "$4$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 273,
        "problem": "Let  $A_1A_2A_3A_4A_5$  be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. How many times of the area of the 10-sided polygon compared to the shaded area?",
        "geo_code": "[asy]\nsize(8cm);\ndefaultpen(fontsize(10pt)); \npair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705); \nfilldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8));\ndraw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle);\n\npair O = (A_1+A_2+A_3+A_4+A_5)/5;\nlabel(\" $A_1$ \",A_1, 2dir(A_1-O));\nlabel(\" $A_2$ \",A_2, 2dir(A_2-O));\nlabel(\" $A_3$ \",A_3, 2dir(A_3-O));\nlabel(\" $A_4$ \",A_4, 2dir(A_4-O));\nlabel(\" $A_5$ \",A_5, 2dir(A_5-O));\nlabel(\" $B_1$ \",B_1, 2dir(B_1-O));\nlabel(\" $B_2$ \",B_2, 2dir(B_2-O));\nlabel(\" $B_3$ \",B_3, 2dir(B_3-O));\nlabel(\" $B_4$ \",B_4, 2dir(B_4-O));\nlabel(\" $B_5$ \",B_5, 2dir(B_5-O));\n[/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 274,
        "problem": "Point  $P$  lies outside a circle, and two rays are drawn from  $P$  that intersect the circle as shown. One ray intersects the circle at points  $A$  and  $B$  while the other ray intersects the circle at  $M$  and  $N$ .  $AN$  and  $MB$  intersect at  $X$ . Given that  $\\angle AXB$  measures  $127^{\\circ}$  and the minor arc  $AM$  measures  $14^{\\circ}$ , compute the measure of the angle at  $P$ .",
        "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10pt));\npair P=(60,10),C=(-20,10),K=(-20,-10);\npath CC=circle((0,0),20), PC=P--C, PK=P--K;\npair A=intersectionpoints(CC,PC)[0],\nB=intersectionpoints(CC,PC)[1],\nM=intersectionpoints(CC,PK)[0],\nN=intersectionpoints(CC,PK)[1],\nX=intersectionpoint(A--N,B--M);\ndraw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M);\nlabel(\" $A$ \",A,plain.NE);label(\" $B$ \",B,plain.NW);label(\" $M$ \",M,SE);\nlabel(\" $P$ \",P,E);label(\" $N$ \",N,dir(250));label(\" $X$ \",X,plain.N);[/asy]",
        "answer": "$39$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 275,
        "problem": "Equilateral triangle  $ T$  is inscribed in circle  $ A$ , which has radius  $ 10$ . Circle  $ B$  with radius  $ 3$  is internally tangent to circle  $ A$  at one vertex of  $ T$ . Circles  $ C$  and  $ D$ , both with radius  $ 2$ , are internally tangent to circle  $ A$  at the other two vertices of  $ T$ . Circles  $ B$ ,  $ C$ , and  $ D$  are all externally tangent to circle  $ E$ , which has radius  $ \\frac {m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m - n$ .",
        "geo_code": "[asy]unitsize(2.2mm);\ndefaultpen(linewidth(.8pt)+fontsize(8pt));\ndotfactor=4;\n\npair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90);\npair Ep=(0,4-27/5);\npair[] dotted={A,B,C,D,Ep};\n\ndraw(Circle(A,10));\ndraw(Circle(B,3));\ndraw(Circle(C,2));\ndraw(Circle(D,2));\ndraw(Circle(Ep,27/5));\n\ndot(dotted);\nlabel(\" $E$ \",Ep,E);\nlabel(\" $A$ \",A,W);\nlabel(\" $B$ \",B,W);\nlabel(\" $C$ \",C,W);\nlabel(\" $D$ \",D,E);[/asy]",
        "answer": "$22$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 276,
        "problem": "Let  $ABCD$  be a rectangle with  $AB=10$  and  $BC=66$ . Let  $\\omega_1$  be the circle with diameter  $\\overline{AB}$  and  $\\omega_2$  be the circle with diameter  $\\overline{CD}$ . Suppose  $\\ell$  is a common internal tangent to  $\\omega_1$  and  $\\omega_2$  and that  $\\ell$  intersects  $AD$  and  $BC$  at  $E$  and  $F$  respectively. What is  $EF$ ?",
        "geo_code": "[asy]\n        size(10cm);\n        draw((0,0)--(26,0)--(26,10)--(0,10)--cycle);\n        draw((1,0)--(25,10));\n        draw(circle((0,5),5));\n        draw(circle((26,5),5));\n        dot((1,0));\n        dot((25,10));\n        label(\" $E$ \",(1,0),SE);\n        label(\" $F$ \",(25,10),NW);\n        label(\" $A$ \", (0,0), SW);\n        label(\" $B$ \", (0,10), NW);\n        label(\" $C$ \", (26,10), NE);\n        label(\" $D$ \", (26,0), SE);\n        dot((0,0));\n        dot((0,10));\n        dot((26,0));\n        dot((26,10));\n[/asy]",
        "answer": "$66$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 277,
        "problem": "Point  $P$  is in the interior of  $\\triangle ABC$ . The side lengths of  $ABC$  are  $AB = 7$ ,  $BC = 8$ ,  $CA = 9$ . The three foots of perpendicular lines from  $P$  to sides  $BC$ ,  $CA$ ,  $AB$  are  $D$ ,  $E$ ,  $F$  respectively. Suppose the minimal value of  $\\frac{BC}{PD} + \\frac{CA}{PE} + \\frac{AB}{PF}$  can be written as  $\\frac{a}{b}\\sqrt{c}$ , where  $\\gcd(a,b) = 1$  and  $c$  is square free, calculate  $a+b+c$ .",
        "geo_code": "[asy]\nsize(120); pathpen = linewidth(0.7); pointfontpen = fontsize(10); // pointpen = black;\npair B=(0,0), C=(8,0), A=IP(CR(B,7),CR(C,9)), P = (2,1.6), D=foot(P,B,C), E=foot(P,A,C), F=foot(P,A,B);\nD(A--B--C--cycle); D(P--D); D(P--E); D(P--F);\nD(MP(\"A\",A,N)); D(MP(\"B\",B)); D(MP(\"C\",C)); D(MP(\"D\",D)); D(MP(\"E\",E,NE)); D(MP(\"F\",F,NW)); D(MP(\"P\",P,SE));\n[/asy]",
        "answer": "$34$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 278,
        "problem": "A rectangular piece of paper  $ABCD$  has sides of lengths  $AB = 3$ ,  $BC = 6$ . The rectangle is folded in half such that  $AD$  coincides with  $BC$  and  $EF$  is the folding line. Then fold the paper along a line  $BM$  such that the corner  $A$  falls on line  $EF$ . How large, in degrees, is  $\\angle AMB$ ?",
        "geo_code": "[asy]\nsize(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype(\"4 4\"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);\npair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A;\nD(B--M--D(\"N\",N,NE)--B--D(\"C\",C,SE)--D(\"D\",D,NE)--M); D(D(\"M\",M,plain.N)--D(\"A\",A,NW)--D(\"B\",B,SW),dd); D(D(\"E\",E,W)--D(\"F\",F,plain.E),dd); \n[/asy]",
        "answer": "$60$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "angle"
    },
    {
        "index": 279,
        "problem": "Two concentric circles have radii  $1$  and  $4$ .  Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown.  The three lightly shaded circles are internally tangent to the circle with radius  $4$  while the three darkly shaded circles are externally tangent to the circle with radius  $1$ .  The radius of the six congruent circles can be written  $\\frac{k+\\sqrt m}{n}$ , where  $k,m,$  and  $n$  are integers with  $k$  and  $n$  relatively prime.  Find  $k\times m\times n$ .",
        "geo_code": "[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\nreal r = (sqrt(133)-9)/2;\ndraw(circle(origin,1)^^circle(origin,4));\nfor(int i=0;i<=2;i=i+1)\n{\nfilldraw(circle(dir(90 + i*120)*(4-r),r),gray);\n}\nfor(int j=0;j<=2;j=j+1)\n{\nfilldraw(circle(dir(30+j*120)*(1+r),r),darkgray);\n}\n[/asy]",
        "answer": "$-2394$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 280,
        "problem": "Inside the  $7\\times 10$  rectangle below, one point is chosen a distance  $\\sqrt2$  from the left side and a distance  $\\sqrt7$  from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region.",
        "geo_code": "[asy]\nimport graph;\nsize(4cm);\npair A = (0,0);\npair B = (9,0);\npair C = (9,7);\npair D = (0,7);\npair P = (1.5,3);\ndraw(A--B--C--D--cycle,linewidth(1.5));\nfilldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5));\nfilldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5));\n[/asy]",
        "answer": "$35$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 281,
        "problem": "The pattern in the figure below continues inward infinitely. The base of the biggest triangle is 1. All triangles are equilateral. The area of the shaded region can be written as $\\frac{\\sqrt{m}{n}}$ where $m$ and $n$ are relatively prime integers. Find the value of $m+n$.",
        "geo_code": "[asy]\ndefaultpen(linewidth(0.8));\npen blu = rgb(0,112,191);\nreal r=sqrt(3);\nfill((8,0)--(0,8r)--(-8,0)--cycle, blu);\nfill(origin--(4,4r)--(-4,4r)--cycle, white);\nfill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);\nfill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]",
        "answer": "$8$",
        "category": "Global Abstract Integration",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 282,
        "problem": "The target below is made up of concentric circles with diameters  $2$ ,  $4$ ,  $8$ ,  $12$ , and  $16$ . The area of the dark region is  $n\\pi$ . Find  $n$ .",
        "geo_code": "[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\nint i;\nfor(i=5;i>=1;i=i-1)\n{\nif (floor(i/2)==i/2)\n{\nfilldraw(circle(origin,4*i),white);\n}\nelse\n{\nfilldraw(circle(origin,4*i),red);\n}\n}\n[/asy]",
        "answer": "$41$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 283,
        "problem": "Find 600 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.",
        "geo_code": "[asy]\ndefaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);\ndraw(unitcircle,dg);\nfor(int i = 0; i < 12; ++i) {\n draw(dir(30*i+theta)--dir(30*(i+1)+theta), db);\n dot(dir(30*i+theta),Fill(rgb(0.8,0,0)));\n} dot(dir(theta),Fill(dr)); dot((0,0),Fill(dr));\n[/asy]",
        "answer": "$1800$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 284,
        "problem": "Circles  $\\omega$  and  $\\gamma$  are drawn such that  $\\omega$  is internally tangent to  $\\gamma$ , the distance between their centers are  $12$ , and the area inside of  $\\gamma$  but outside of  $\\omega$  is  $360\\pi$ . What is the sum of the radii of the circles?",
        "geo_code": "[asy]\nsize(3cm);\nreal lw=0.4, dr=0.3;\nreal r1=14, r2=9;\npair A=(0,0), B=(r1-r2,0);\ndraw(A--B,dashed);\ndraw(circle(A,r1),linewidth(lw)); draw(circle(B,r2),linewidth(lw));\nfilldraw(circle(A,dr)); filldraw(circle(B,dr));\nlabel(\" $5$ \",(A+B)/2,dir(-90));\nlabel(\" $\\gamma$ \",A+r1*dir(135),dir(135)); label(\" $\\omega$ \",B+r2*dir(135),dir(135));\n[/asy]",
        "answer": "$23$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 285,
        "problem": "The cross below is made up of five congruent squares. The perimeter of the cross is  $120$ . Find its area.",
        "geo_code": "[asy]\nimport graph;\nsize(3cm);\npair A = (0,0);\npair temp = (1,0);\npair B = rotate(45,A)*temp;\npair C = rotate(90,B)*A;\npair D = rotate(270,C)*B;\npair E = rotate(270,D)*C;\npair F = rotate(90,E)*D;\npair G = rotate(270,F)*E;\npair H = rotate(270,G)*F;\npair I = rotate(90,H)*G;\npair J = rotate(270,I)*H;\npair K = rotate(270,J)*I;\npair L = rotate(90,K)*J;\ndraw(A--B--C--D--E--F--G--H--I--J--K--L--cycle);\n[/asy]",
        "answer": "$500$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "area"
    },
    {
        "index": 286,
        "problem": "As shown in figure, circle  $\\Omega$ has $\\overline{AB}=65$ and $CD=16$. Moreover, let  $AD, BC, AC,$  and  $BD$  all have integer lengths. Two other circles, $\\omega_1$  and  $\\omega_2$ , have  $\\overline{AC}$  and  $\\overline{BD}$  as their diameters, respectively. Let circle  $\\omega_1$  intersect  $AB$  at a point  $E \\neq A$  and let circle  $\\omega_2$  intersect  $AB$  at a point  $F \\neq B$ . Then  $EF=\\frac{m}{n}$ , for relatively prime integers  $m$  and  $n$ . Find  $m+n$ .",
        "geo_code": "[asy]\nsize(7cm);\npair A=(0,0), B=(64,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0);\ndraw(A--B--C--D--cycle);\ndraw(A--C);\ndraw(B--D);\ndot(\" $A$ \", A, SW);\ndot(\" $B$ \", B, SE);\ndot(\" $C$ \", C, NE);\ndot(\" $D$ \", D, NW);\ndot(\" $E$ \", E, S);\ndot(\" $F$ \", F, S);\ndraw(circle((A + C)/2, abs(A - C)/2));\ndraw(circle((B + D)/2, abs(B - D)/2));\ndraw(circle((A + B)/2, abs(A - B)/2));\nlabel(\" $\\mathcal P$ \", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45));\nlabel(\" $\\mathcal Q$ \", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210));\nlabel(\" $\\mathcal R$ \", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70));\n[/asy]",
        "answer": "$961$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "count"
    },
    {
        "index": 287,
        "problem": "In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is $\\frac{2}{5}$ that of triangle AOB. How many time of a compared to b?",
        "geo_code": "[asy]\n\nsize(120);\npair A = (0,3);\npair B = (0,0);\npair C = (3,0);\npair D = (0,1.5);\npair E = (0.35,0);\npair F = (1.1,1.9);\npair J = (0.17,0);\npair Y = (0.17,0.75);\n\npair Z = (1.6,0.4);\ndraw(A--B);\ndraw(B--C);\ndraw(C--A);\ndraw(D--F--Z--E--D);\ndraw(\" $O$ \", B, dir(180));\ndraw(\" $B$ \", A, dir(45));\ndraw(\" $A$ \", C, dir(45));\ndraw(\" $Q$ \", E, dir(45));\ndraw(\" $P$ \", D, dir(45));\ndraw(\" $R$ \", Z, dir(45));\ndraw(\" $S$ \", F, dir(45));\ndraw(\" $a$ \", Y, dir(210));\ndraw(\" $b$ \", J, dir(100));\n[/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 288,
        "problem": "Squares $ABCD$ and $EFGH$ are congruent, $G$ is the center of square $ABCD$ and $AB=5$. The area of the region in the plane covered by these squares is",
        "geo_code": "[asy] draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((5,5)--(12,-2)--(5,-9)--(-2,-2)--cycle); label(\"A\", (0,0), W); label(\"B\", (10,0), E); label(\"C\", (10,10), NE); label(\"D\", (0,10), NW); label(\"G\", (5,5), N); label(\"F\", (12,-2), E); label(\"E\", (5,-9), S); label(\"H\", (-2,-2), W); dot((-2,-2)); dot((5,-9)); dot((12,-2)); dot((0,0)); dot((10,0)); dot((10,10)); dot((0,10)); dot((5,5)); [/asy]",
        "answer": "$\\frac{175}{4}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 289,
        "problem": "Triangle $ABC$ is inscribed in a circle, and $\\angle B = \\angle C = 4\\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $3n$ sides inscribed in this circle, then $n=$",
        "geo_code": "[asy] draw(Circle((0,0), 10)); draw((0,10)--(6,-8)--(-6,-8)--cycle); label(\"A\", (0,10), N); label(\"B\", (-6,-8), SW); label(\"C\", (6,-8), SE); dot((0,10)); dot((6,-8)); dot((-6,-8)); [/asy]",
        "answer": "$3$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 290,
        "problem": "A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $n=5$, what is the value of $m$?",
        "geo_code": "[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle; draw(p); draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p); draw(shift((0,-2-sqrt(2)))*p); draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy]",
        "answer": "$10$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 291,
        "problem": "Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $2$ and passes through points $P$ and $R$. The area of the rectangle is",
        "geo_code": "[asy] pair A,B,C,D,P,Q,R; A = (0,4); B = (8,4); C = (8,0); D = (0,0); P = (2,2); Q = (4,2); R = (6,2); dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(R); draw(A--B--C--D--cycle); draw(circle(P,2)); draw(circle(Q,2)); draw(circle(R,2)); label(\"$A$\",A,NW); label(\"$B$\",B,NE); label(\"$C$\",C,SE); label(\"$D$\",D,SW); label(\"$P$\",P,W); label(\"$Q$\",Q,W); label(\"$R$\",R,W); [/asy]",
        "answer": "$8$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 292,
        "problem": "The horizontal and vertical distances between adjacent points equal 3 unit. What is the area of triangle $ABC$?",
        "geo_code": "[asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 4; ++b) { dot((a,b)); } } draw((0,0)--(3,2)--(4,3)--cycle); label(\"$A$\",(0,0),SW); label(\"$B$\",(3,2),SE); label(\"$C$\",(4,3),NE); [/asy]",
        "answer": "$\\frac{9}{2}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 293,
        "problem": "The measure of angle $ABC$ is $60^\\circ$, $\\overline{AD}$ bisects angle $BAC$, and $\\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is",
        "geo_code": "[asy] pair A,B,C,D; A = (0,0); B = (9,10); C = (10,0); D = (6.66,3); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--cycle); draw(A--D--C); label(\"$A$\",A,SW); label(\"$B$\",B,N); label(\"$C$\",C,SE); label(\"$D$\",D,N); label(\"$60^\\circ $\",(9.4,8.8),SW); [/asy]",
        "answer": "$120$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 294,
        "problem": "Assume the square have side length of 3. Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The area of the inscribed square is",
        "geo_code": "[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2)); draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2)); draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8)); draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2)); draw((2,0)--(3,2)--(1,3)--(0,1)--cycle); [/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 295,
        "problem": "In the figure,  $E$ is the midpoint of $\\overline{AD}$, $ABCD$ is a $1 \\times 1$ square, and $F$ is on $\\overline{BE}$. The area of quadrilateral $CDEF$ is",
        "geo_code": "[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = (0,2); pair B = origin; pair C = (2,0); pair D = (2,2); pair E = midpoint(A--D); pair F = foot(C,B,E); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label(\"$A$\",A,N);label(\"$B$\",B,S);label(\"$C$\",C,S);label(\"$D$\",D,N);label(\"$E$\",E,N);label(\"$F$\",F,NW); draw(A--B--C--D--cycle); draw(B--E); draw(C--F); draw(rightanglemark(B,F,C,4));[/asy]",
        "answer": "$\\frac{11}{20}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 296,
        "problem": "$BD$ and $CE$ are medians of triangle $ABC$, $CE=6$, and $BD=4$. The area of triangle $ABC$ is",
        "geo_code": "[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (1.25,1); pair C = (2,0); pair D = midpoint(A--C); pair E = midpoint(A--B); pair G = intersectionpoint(E--C,B--D); dot(A);dot(B);dot(C);dot(D);dot(E);dot(G); label(\"$A$\",A,S);label(\"$B$\",B,N);label(\"$C$\",C,S);label(\"$D$\",D,S);label(\"$E$\",E,NW);label(\"$G$\",G,NE); draw(A--B--C--cycle); draw(B--D); draw(E--C); draw(rightanglemark(C,G,D,3));[/asy]",
        "answer": "$16$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 297,
        "problem": "A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle $ABC$ as shown. To the nearest hundredth, what is the radius of the circle?",
        "geo_code": "[asy] defaultpen(linewidth(.8pt)); dotfactor=3; pair A = origin; pair B = (1,0); pair C = (0,sqrt(3)); pair O = (2.33,2.33); dot(A);dot(B);dot(C);dot(O); label(\"$A$\",A,SW);label(\"$B$\",B,SE);label(\"$C$\",C,W);label(\"$O$\",O,NW); label(\"$1$\",midpoint(A--B),S);label(\"$60^\\circ$\",B,2W + N); draw((3,0)--A--(0,3)); draw(B--C); draw(Arc(O,2.33,163,288.5));[/asy]",
        "answer": "$\\frac{3}{2}+\\frac{\\sqrt{3}}{2}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 298,
        "problem": "In the figure, $B$, $C$, and $D$ are squares with sides of length $2$; polygons $A$, $E$, and $F$ are isosceles right triangles; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is",
        "geo_code": "[asy] size(6cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); draw((-1,1)--(2,1)); draw((-1,0)--(1,0)); draw((-1,1)--(-1,0)); draw((0,-1)--(0,3)); draw((1,2)--(1,0)); draw((-1,1)--(1,1)); draw((0,2)--(1,2)); draw((0,3)--(1,2)); draw((0,-1)--(2,1)); draw((0,-1)--((0,-1) + sqrt(2)*dir(-15))); draw(((0,-1) + sqrt(2)*dir(-15))--(1,0)); label(\"$\\textbf{A}$\",foot((0,2),(0,3),(1,2)),SW); label(\"$\\textbf{B}$\",midpoint((0,1)--(1,2))); label(\"$\\textbf{C}$\",midpoint((-1,0)--(0,1))); label(\"$\\textbf{D}$\",midpoint((0,0)--(1,1))); label(\"$\\textbf{E}$\",midpoint((1,0)--(2,1)),NW); label(\"$\\textbf{F}$\",midpoint((0,-1)--(1,0)),NW); label(\"$\\textbf{G}$\",midpoint((0,-1)--(1,0)),2SE);[/asy]",
        "answer": "$frac{20}{3}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "volume"
    },
    {
        "index": 299,
        "problem": "A square with sides of length $2$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid.",
        "geo_code": "[asy] pointpen = black; pathpen = black; D(unitsquare); D((0,0)); D((1,0)); D((1,1)); D((0,1)); D(D((.5,.5))--D((1,.5))); D(D((.17,1))--(.5,.5)--D((.17,0))); MP(\"x\",(.58,1),N); [/asy]",
        "answer": "$\\frac{5}{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 300,
        "problem": "A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimter of each of the congruent rectangles is $18$. What is the area of the large square?",
        "geo_code": "[asy]pathpen = black+linewidth(0.7); D((0,0)--(7,0)--(7,7)--(0,7)--cycle); D((1,0)--(1,6)); D((0,6)--(6,6)); D((1,1)--(7,1)); D((6,7)--(6,1)); [/asy]",
        "answer": "$81$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 301,
        "problem": "As shown in the figure, the degree measure of angle $A$ is",
        "geo_code": "[asy] unitsize(12); draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle); draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW)); draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW)); draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW)); label(rotate(30)*\"$40^\\circ$\",(2,-8.9),ENE); label(\"$100^\\circ$\",(21/3,-2/3),SE); label(\"$110^\\circ$\",(900/83,-317/83),NNW); label(\"$A$\",(0,0),NW); [/asy]",
        "answer": "$30$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 302,
        "problem": "Segments $CM$ and $CN$ divide the square's area into three equal parts. Square $ABCD$ has sides of length 1. How long is segment $CM$?",
        "geo_code": "[asy] pair A,B,C,D,M,N; A = (0,0); B = (0,3); C = (3,3); D = (3,0); M = (0,1); N = (1,0); draw(A--B--C--D--cycle); draw(M--C--N); label(\"$A$\",A,SW); label(\"$M$\",M,W); label(\"$B$\",B,NW); label(\"$C$\",C,NE); label(\"$D$\",D,SE); label(\"$N$\",N,S); [/asy]",
        "answer": "$frac{\\sqrt{13}}{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 303,
        "problem": "A cube has edge length $2$. Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The ratio increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is",
        "geo_code": "[asy] draw((0,0)--(2,0)--(3,1)--(3,3)--(2,2)--(0,2)--cycle); draw((2,0)--(2,2)); draw((0,2)--(1,3)); draw((1,7/3)--(1,10/3)--(2,10/3)--(2,7/3)--cycle); draw((2,7/3)--(5/2,17/6)--(5/2,23/6)--(3/2,23/6)--(1,10/3)); draw((2,10/3)--(5/2,23/6)); draw((3,3)--(5/2,3));[/asy]",
        "answer": "$\\frac{1}{6}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 304,
        "problem": "Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below DO NOT change? (a) the length of the segment $MN$ (b) the perimeter of $\\triangle PAB$ (c) the area of $\\triangle PAB$ (d) the area of trapezoid $ABNM$",
        "geo_code": "[asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label(\"$A$\",(2,0),SW); label(\"$B$\",(8,0),SE); label(\"$M$\",(4,2),W); label(\"$N$\",(7,2),E); label(\"$P$\",(6,4),N); [/asy]",
        "answer": "$3$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 305,
        "problem": "If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\\overarc {AC}$ and $\\overarc{BC}$, and to $\\overline{AB}$. If the length of $\\overarc{BC}$ is $24$, then the radius of the circle is",
        "geo_code": "[asy] label(\"A\", (0,0), W); label(\"B\", (64,0), E); label(\"C\", (32, 32*sqrt(3)), N); draw(arc((0,0),64,0,60)); draw(arc((64,0),64,120,180)); draw((0,0)--(64,0)); draw(circle((32, 24), 24)); [/asy]",
        "answer": "$\\frac{27}{\\pi}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 306,
        "problem": "A square piece of paper, 8 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the large rectangles to the perimeter of the small rectangle?",
        "geo_code": "[asy] draw((0,8)--(0,0)--(4,0)--(4,8)--(0,8)--(3.5,8.5)--(3.5,8)); draw((2,-1)--(2,9),dashed); [/asy]",
        "answer": "$\\frac{6}{5}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 307,
        "problem": "The plane is tiled by congruent squares and congruent pentagons as indicated. The ratio of the plane that is enclosed by the pentagons is",
        "geo_code": "[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy]",
        "answer": "$\\frac{5}{9}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 308,
        "problem": "Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\\circ$ Art's cookies are trapezoids:  $\\circ$ Roger's cookies are rectangles:  $\\circ$ Paul's cookies are parallelograms:  $\\circ$ Trisha's cookies are triangles:  Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?",
        "geo_code": "[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label(\"5 in\", (2.5,0), S); label(\"3 in\", (5,1.5), E); label(\"3 in\", (3.5,3), N);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label(\"4 in\", (2,0), S); label(\"2 in\", (4,1), E);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label(\"3 in\", (1.5,0), S); label(\"2 in\", (2.5,1), W);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label(\"3 in\", (1.5,0), S); label(\"4 in\", (3,2), E);[/asy]",
        "answer": "$40$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 309,
        "problem": "Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\\circ$ Art's cookies are trapezoids:  $\\circ$ Roger's cookies are rectangles:  $\\circ$ Paul's cookies are parallelograms:  $\\circ$ Trisha's cookies are triangles:  How many cookies will be in one batch of Trisha's cookies?",
        "geo_code": "[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label(\"5 in\", (2.5,0), S); label(\"3 in\", (5,1.5), E); label(\"3 in\", (3.5,3), N);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label(\"4 in\", (2,0), S); label(\"2 in\", (4,1), E);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label(\"3 in\", (1.5,0), S); label(\"2 in\", (2.5,1), W);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label(\"3 in\", (1.5,0), S); label(\"4 in\", (3,2), E);[/asy]",
        "answer": "$24$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 310,
        "problem": "The area of trapezoid $ABCD$ is $164\\text{ cm}^2$. According to the figure, what is $BC$, in centimeters?",
        "geo_code": "[asy] size(4inch,2inch); draw((0,0)--(31,0)--(16,8)--(6,8)--cycle); draw((11,8)--(11,0), linetype(\"8 4\")); draw((11,1)--(12,1)--(12,0)); label(\"$A$\", (0,0), SW); label(\"$D$\", (31,0), SE); label(\"$B$\", (6,8), NW); label(\"$C$\", (16,8), NE); label(\"10\", (3,5), W); label(\"8\", (11,4), E); label(\"17\", (22.5,5), E);[/asy]",
        "answer": "$10$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 311,
        "problem": "Equilateral $\\triangle ABC$ has side length $1$, $M$ is the midpoint of $\\overline{AC}$, and $C$ is the midpoint of $\\overline{BD}$. What is the area of $\\triangle CDM$?",
        "geo_code": "[asy]defaultpen(linewidth(.8pt)+fontsize(8pt)); pair B = (0,0); pair A = 2*dir(60); pair C = (2,0); pair D = (4,0); pair M = midpoint(A--C); label(\"$A$\",A,NW);label(\"$B$\",B,SW);label(\"$C$\",C, SE);label(\"$M$\",M,NE);label(\"$D$\",D,SE); draw(A--B--C--cycle); draw(C--D--M--cycle);[/asy]",
        "answer": "$\\frac {\\sqrt {3}}{8}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 312,
        "problem": "The $8\\times18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $x$?",
        "geo_code": "[asy] unitsize(3mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((8,4)--(8,0)--(10,0)--(10,-4)); label(\"$A$\",(0,4),NW); label(\"$B$\",(18,4),NE); label(\"$C$\",(18,-4),SE); label(\"$D$\",(0,-4),SW); label(\"$x$\",(4,4),S); label(\"$x$\",(14,-4),N); label(\"$18$\",(9,4),N); label(\"$18$\",(9,-4),S); label(\"$8$\",(0,0),W); label(\"$8$\",(18,0),E); dot((0,4)); dot((18,4)); dot((18,-4)); dot((0,-4));[/asy]",
        "answer": "$6$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 313,
        "problem": "According to the figure, the equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $m+b$?",
        "geo_code": "[asy] size(150); defaultpen(linewidth(0.7)+fontsize(8)); draw(circle((2,4),4));draw(circle((14,9),9)); draw((0,-2)--(0,20));draw((-6,0)--(25,0)); draw((2,4)--(2,4)+4*expi(pi*4.5/11)); draw((14,9)--(14,9)+9*expi(pi*6/7)); label(\"4\",(2,4)+2*expi(pi*4.5/11),(-1,0)); label(\"9\",(14,9)+4.5*expi(pi*6/7),(1,1)); label(\"(2,4)\",(2,4),(0.5,-1.5));label(\"(14,9)\",(14,9),(1,-1)); draw((-4,120*-4/119+912/119)--(11,120*11/119+912/119)); dot((2,4)^^(14,9)); [/asy]",
        "answer": "$\\frac{1032}{119}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 314,
        "problem": "Hexagon $ABCDEF$ is divided into five rhombuses, as shown in the figure. Rhombuses $\\mathcal{P, Q, R,}$ and $\\mathcal{S}$ are congruent, and each has area $\\sqrt{2006}.$ Let $K$ be the area of rhombus $\\mathcal{T}$. Given that $K$ is a positive integer, find the number of possible values for $K$.",
        "geo_code": "[asy] size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label(\"$\\mathcal{T}$\",(2.1,-1.6)); label(\"$\\mathcal{P}$\",(0,-1),NE); label(\"$\\mathcal{Q}$\",(4.2,-1),NW); label(\"$\\mathcal{R}$\",(0,-2.2),SE); label(\"$\\mathcal{S}$\",(4.2,-2.2),SW); [/asy]",
        "answer": "$89$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 315,
        "problem": "Consider the $12$-sided polygon $ABCDEFGHIJKL$, as shown. Each of its sides has length $10$, and each two consecutive sides form a right angle. Suppose that $\\overline{AG}$ and $\\overline{CH}$ meet at $M$. What is the area of quadrilateral $ABCM$?",
        "geo_code": "[asy] unitsize(13mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2); pair M=intersectionpoints(A--G,H--C)[0]; draw(A--B--C--D--Ep--F--G--H--I--J--K--L--cycle); draw(A--G); draw(H--C); dot(M); label(\"$A$\",A,NW); label(\"$B$\",B,NE); label(\"$C$\",C,NE); label(\"$D$\",D,NE); label(\"$E$\",Ep,SE); label(\"$F$\",F,SE); label(\"$G$\",G,SE); label(\"$H$\",H,SW); label(\"$I$\",I,SW); label(\"$J$\",J,SW); label(\"$K$\",K,NW); label(\"$L$\",L,NW); label(\"$M$\",M,W); [/asy]",
        "answer": "$110$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 316,
        "problem": "In the figure, the outer equilateral triangle has side length $4$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids?",
        "geo_code": "[asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy]",
        "answer": "$\\frac{4\\sqrt{3}-1}{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 317,
        "problem": "Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$. According to the figure, what is the area of $\\triangle ABC$?",
        "geo_code": "[asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21)); pair D=foot(B,A,C); pair[] ps={B,C,A,D}; draw(A--B--C--cycle); draw(B--D); draw(rightanglemark(B,D,C)); dot(ps); label(\"$A$\",A,NW); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$3$\",midpoint(A--D),NE); label(\"$4$\",midpoint(D--C),NE); [/asy]",
        "answer": "$7\\sqrt3$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 318,
        "problem": "Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\\angle A = 2.5 \\angle B$. What is the degree measure of $\\angle D$?",
        "geo_code": "[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), Ep=dir(35), D=dir(-35), B=dir(145); pair A=intersectionpoints(Circle(B,1),C--(-1*Ep))[0]; pair[] ds={A,B,C,D,Ep}; dot(ds); draw(A--Ep--D--B--cycle); label(\"$A$\",A,SW); label(\"$B$\",B,NW); label(\"$C$\",C,N); label(\"$E$\",Ep,E); label(\"$D$\",D,E); [/asy]",
        "answer": "$52.5$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 319,
        "problem": "The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $y$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $y$?",
        "geo_code": "[asy] unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0)); [/asy]",
        "answer": "$100$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 320,
        "problem": "In quadrilateral $ABCD$, $AB = 6$, $BC = 17$, $CD = 6$, $DA = 7$, and $BD$ is an integer. What is $BD$?",
        "geo_code": "[asy] unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), B=(17,0); pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; pair[] dotted={A,B,C,D}; draw(D--A--B--C--D--B); dot(dotted); label(\"$D$\",D,NW); label(\"$C$\",C,W); label(\"$B$\",B,E); label(\"$A$\",A,NE); [/asy]",
        "answer": "$12$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 321,
        "problem": "The two circles pictured have the same center $C$. Chord $\\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $5$, and chord $\\overline{AD}$ has length $8$. What is the area between the two circles?",
        "geo_code": "[asy] unitsize(45); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,0.15)--(1.85,0.15)--(1.85,0)--(2,0)--cycle); draw(circle((2,1),2.24)); draw(circle((2,1),1)); draw((0,0)--(4,0)); draw((0,0)--(2,1)); draw((2,1)--(2,0)); draw((2,1)--(4,0)); dot((0,0),ds); label(\"$A$\", (-0.19,-0.23),NE*lsf); dot((2,0),ds); label(\"$B$\", (1.97,-0.31),NE*lsf); dot((2,1),ds); label(\"$C$\", (1.96,1.09),NE*lsf); dot((4,0),ds); label(\"$D$\", (4.07,-0.24),NE*lsf); clip((-3.1,-7.72)--(-3.1,4.77)--(11.74,4.77)--(11.74,-7.72)--cycle); [/asy]",
        "answer": "$16 \\pi$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 322,
        "problem": "Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What is the area of rectangle $AQRD$ that is shaded?",
        "geo_code": "[asy] filldraw((0,0)--(25,0)--(25,15)--(0,15)--cycle,white,black); label(\"D\",(0,0),S); label(\"R\",(25,0),S); label(\"Q\",(25,15),N); label(\"A\",(0,15),N); filldraw((10,0)--(15,0)--(15,15)--(10,15)--cycle,mediumgrey,black); label(\"S\",(10,0),S); label(\"C\",(15,0),S); label(\"B\",(15,15),N); label(\"P\",(10,15),N);[/asy]",
        "answer": "$225$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 323,
        "problem": "Quadrilateral $ABCD$ is a trapezoid. According to the figure, what is the perimeter of the trapezoid?",
        "geo_code": "[asy] pair A,B,C,D; A=(3,20); B=(35,20); C=(47,0); D=(0,0); draw(A--B--C--D--cycle); dot((0,0)); dot((3,20)); dot((35,20)); dot((47,0)); label(\"A\",A,N); label(\"B\",B,N); label(\"C\",C,S); label(\"D\",D,S); draw((19,20)--(19,0)); dot((19,20)); dot((19,0)); draw((19,3)--(22,3)--(22,0)); label(\"12\",(21,10),E); label(\"50\",(19,22),N); label(\"15\",(1,10),W); label(\"20\",(41,12),E);[/asy]",
        "answer": "$75$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 324,
        "problem": "A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?",
        "geo_code": "[asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,gray,black); filldraw(Circle((0,0),1), mediumgray,black); filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy]",
        "answer": "$\\frac{\\pi-1}{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 325,
        "problem": "In the given circle, $\\overline{AB}$ is parallel to $\\overline{ED}$, and the diameter $\\overline{EB}$ is parallel to $\\overline{DC}$. The angles $ABE$ and $AEB$ are in the ratio $5 : 4$. What is the degree measure of angle $DCB$?",
        "geo_code": "[asy] unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r=3; pair A=(-3cos(80),-3sin(80)); pair D=(3cos(80),3sin(80)), C=(-3cos(80),3sin(80)); pair O=(0,0), E=(-3,0), B=(3,0); path outer=Circle(O,r); draw(outer); draw(E--B); draw(E--A); draw(B--A); draw(E--D); draw(C--D); draw(B--C); pair[] ps={A,B,C,D,E,O}; dot(ps); label(\"$A$\",A,N); label(\"$B$\",B,NE); label(\"$C$\",C,S); label(\"$D$\",D,S); label(\"$E$\",E,NW); label(\"$$\",O,N); [/asy]",
        "answer": "$130$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 326,
        "problem": "In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , $X$ in centimeters?",
        "geo_code": "[asy] pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0); B=(7,0); C=(7,4); D=(8,4); E=(8,5); F=(10,5); G=(10,7); H=(7,7); I=(7,8); J=(5,8); K=(5,7); L=(4,7); M=(4,6); N=(0,6); O=(0,5); P=(2,5); Q=(2,3); R=(4,3); draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle); label(\"$X$\",(3.4,1.5)); label(\"6\",(7.6,1.5)); label(\"1\",(7.6,3.5)); label(\"1\",(8.4,4.6)); label(\"2\",(9.4,4.6)); label(\"2\",(10.4,6)); label(\"3\",(8.4,7.4)); label(\"1\",(7.5,7.8)); label(\"2\",(6,8.5)); label(\"1\",(4.7,7.8)); label(\"1\",(4.3,7.5)); label(\"1\",(3.5,6.5)); label(\"4\",(1.8,6.5)); label(\"1\",(-0.5,5.5)); label(\"2\",(0.8,4.5)); label(\"2\",(1.5,3.8)); label(\"2\",(2.8,2.6));[/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 327,
        "problem": "A square with area $16$ is inscribed in a square with area $8$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$?",
        "geo_code": "[asy] draw((0,2)--(2,2)--(2,0)--(0,0)--cycle); draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle); label(\"$a$\",(-0.1,0.15)); label(\"$b$\",(-0.1,1.15));[/asy]",
        "answer": "$4$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 328,
        "problem": "A circle of radius 2.5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?",
        "geo_code": "[asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy]",
        "answer": "$50$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 329,
        "problem": "Three circles with radius 4 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?",
        "geo_code": "[asy] filldraw((0,0)--(2,0)--(1,sqrt(3))--cycle,gray,gray); filldraw(circle((1,sqrt(3)),1),gray); filldraw(circle((0,0),1),gray); filldraw(circle((2,0),1),grey);[/asy]",
        "answer": "$40\\pi+16\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 330,
        "problem": "In the accompanying figure, the outer square $S$ has side length $20$. A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two closest vertices of $S'$. The result is a four-pointed starlike figure inscribed in $S$. The star figure is cut out and then folded to form a pyramid with base $S'$. Find the volume of this pyramid.",
        "geo_code": "[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]",
        "answer": "$\\frac{375}{4}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "volume"
    },
    {
        "index": 331,
        "problem": "Angle $ABC$ of $\\triangle ABC$ is a right angle. The sides of $\\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\\overline{AB}$ equals $8\\pi$, and the arc of the semicircle on $\\overline{AC}$ has length $8.5\\pi$. What is the diameter of the semicircle on $\\overline{BC}$?",
        "geo_code": "[asy] size(100);import graph; pair A,B,C; A=(0,8); B=(0,0); C=(18,0); draw((0,8)..(-4,4)..(0,0)--(0,8)); draw((0,0)..(9,-9)..(18,0)--(0,0)); real theta = aTan(8/18); draw(arc((9,4),10,-theta,180-theta)); draw((0,8)--(18,0)); dot(A); dot(B); dot(C); label(\"$A$\", A, NW); label(\"$B$\", B, SW); label(\"$C$\", C, SE); [/asy]",
        "answer": "$15$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 332,
        "problem": "Two squares with side  $6$  lie exactly on top of each other.\nOne square is rotated around a corner point through an angle of  $30$  degrees relative to the other square.\nDetermine the area of the common piece of the two squares.",
        "geo_code": "[asy]\nunitsize (2 cm);\n\npair A, B, C, D, Bp, Cp, Dp, P;\n\nA = (0,0);\nB = (-1,0);\nC = (-1,1);\nD = (0,1);\nBp = rotate(-30)*(B);\nCp = rotate(-30)*(C);\nDp = rotate(-30)*(D);\nP = extension(C, D, Bp, Cp);\n\nfill(A--Bp--P--D--cycle, gray(0.8));\ndraw(A--B--C--D--cycle);\ndraw(A--Bp--Cp--Dp--cycle);\n\nlabel(\" $30^\\circ$ \", (-0.5,0.1), fontsize(10));\n[/asy]",
        "answer": "$12\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 333,
        "problem": "The triangle ABC has sides AB = 137, AC = 241, and BC =200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of BD.",
        "geo_code": "[asy]\n\npair A = (2,6);\npair B = (0,0);\npair C = (10,0);\npair D = (3.5,0) ;\npair E = (3.1,2);\ndraw(A--B);\ndraw(B--C);\ndraw(C--A);\ndraw (A--D);\ndot ((3.1,1.7));\nlabel  (\"E\", E, dir(45));\nlabel (\"A\", A, dir(45));\nlabel (\"B\", B, dir(45));\nlabel (\"C\", C, dir(45));\nlabel (\"D\", D, dir(45));\ndraw(circle((1.8,1.3),1.3)); \ndraw(circle((4.9,1.7),1.75)); \n[/asy]",
        "answer": "$48$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 334,
        "problem": "In the figure, it is given that angle $C = 90^{\\circ}$, $\\overline{AD} = \\overline{DB}$, $DE \\perp AB$, $\\overline{AB} = 20$, and $\\overline{AC} = 12$. The area of triangle $BDE$ is:",
        "geo_code": "[asy] unitsize(7); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E; A=(0,0); B=(20,0); C=(36/5,48/5); D=(10,0); E=(10,75/10); draw(A--B--C--cycle); draw(D--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,S); label(\"$E$\",E,NE); draw(rightanglemark(B,D,E,30)); [/asy]",
        "answer": "$37\\frac{1}{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 335,
        "problem": "In $\\triangle ABC, AC = CD$ and $\\angle CAB - \\angle ABC = 40^\\circ$. Then $\\angle BAD$ is:",
        "geo_code": "[asy] defaultpen(linewidth(.8pt)); unitsize(2.5cm); pair A = origin; pair B = (2,0); pair C = (0.5,0.75); pair D = midpoint(C--B); draw(A--B--C--cycle); draw(A--D); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,NE);[/asy]",
        "answer": "$20$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 336,
        "problem": "Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $64$ square inches and the area of $\\triangle CEF$ is $50$ square inches. Then the number of inches in $AF$ is:",
        "geo_code": "[asy] size(6cm); pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7); draw(A--B--C--D--cycle); draw(F--C--E--B); label(\"$A$\", A, SW); label(\"$B$\", B, S); label(\"$C$\", C, N); label(\"$D$\", D, NW); label(\"$E$\", E, SE); label(\"$F$\", F, W); [/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 337,
        "problem": "$P$ is a point interior to rectangle $ABCD$ and such that $PA=6$ inches, $PD=8$ inches, and $PC=10$ inches. Then $PB$, in inches, equals:",
        "geo_code": "[asy] pair A, B, C, D, P; A = (0, 0); B = (6.5, 0); C = (6.5, 4.5); D = (0, 4.5); P = (2.5, 1.5); draw(A--B--C--D--cycle); draw(A--P); draw(C--P); draw(D--P); draw(B--P, dashed); label(\"$A$\", A, SW); label(\"$B$\", B, SE); label(\"$C$\", C, NE); label(\"$D$\", D, NW); label(\"$P$\", P, S); label(\"$6$\", midpoint(A--P), NW); label(\"$8$\", midpoint(D--P), NE); label(\"$10$\", midpoint(C--P), NW); [/asy]",
        "answer": "$6\\sqrt{2}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 338,
        "problem": "In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\\sqrt{3}$ and $8\\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\\tfrac{1}{2}$ % of itself, while the lengths of $AB$ and $CD$ remain unchanged. The decrease in the area of the square is:",
        "geo_code": "[asy] draw((0,0)--(10,20*sqrt(3)/2)--(20,0)--cycle,black+linewidth(.75)); draw((20,0)--(20,12)--(32,12)--(32,0)--cycle,black+linewidth(.75)); draw((32,0)--(37,10*sqrt(3)/2)--(42,0)--cycle,black+linewidth(.75)); MP(\"I\",(10,0),N);MP(\"II\",(26,0),N);MP(\"III\",(37,0),N); MP(\"A\",(0,0),S);MP(\"B\",(20,0),S);MP(\"C\",(32,0),S);MP(\"D\",(42,0),S); [/asy]",
        "answer": "$24$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 339,
        "problem": "A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at the point $5$ inches from the $A$ is:",
        "geo_code": "[asy] draw(arc((0,-1),2,30,150),dashed+linewidth(.75)); draw((-1.7,0)--(0,0)--(1.7,0),dot); draw((0,0)--(0,.98),dot); MP(\"A\",(-1.7,0),W);MP(\"B\",(1.7,0),E);MP(\"M\",(0,0),S);MP(\"C\",(0,1),N); [/asy]",
        "answer": "$7$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 340,
        "problem": "In $\\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoint of $BF$. The length of $EC$ divided by the length of $BE$ is",
        "geo_code": "[asy] size(2.5inch); pair A, B, C, E, F, G; A = (0,3); B = (-1,0); C = (4,0); E = (0,0); F = (1.14,2.14); G = intersectionpoint(B--F,A--E); draw(A--B--C--cycle); draw(A--E); draw(B--F); label(\"$A$\",A,N); label(\"$B$\",B,W); label(\"$C$\",C,dir(0)); label(\"$E$\",E,S); label(\"$F$\",F,NE); label(\"$G$\",G,SE); [/asy]",
        "answer": "$3$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 341,
        "problem": "Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE, AD$, and $BC$ at points $M, P$, and $Q$ respectively. The length of $MQ$ divided by the length of $PM$ is",
        "geo_code": "[asy] draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2)); label(\"D\",(0,1),NW);label(\"E\",(.4,1),N);label(\"C\",(1,1),NE); label(\"P\",(0,.6),W);label(\"M\",(.25,.55),E);label(\"Q\",(1,.2),E); label(\"A\",(0,0),SW);label(\"B\",(1,0),SE); [/asy]",
        "answer": "$\\frac{19}{5}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 342,
        "problem": "In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is $4$ square inch, then the area of $CMN$ in square inches is",
        "geo_code": "[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((.82,0)--(1,1)--(0,.76)--cycle); label(\"A\", (0,0), S); label(\"B\", (1,0), S); label(\"C\", (1,1), N); label(\"D\", (0,1), N); label(\"M\", (0,.76), W); label(\"N\", (.82,0), S);[/asy]",
        "answer": "$8\\sqrt{3}-12$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 343,
        "problem": "In the adjoining figure $TP$ and $T'Q$ are parallel tangents to a circle of radius $r$, with $T$ and $T'$ the points of tangency. $PT''Q$ is a third tangent with $T'''$ as a point of tangency. If $TP=8$ and $T'Q=18$ then $r$ is",
        "geo_code": "[asy] unitsize(45); pair O = (0,0); pair T = dir(90); pair T1 = dir(270); pair T2 = dir(25); pair P = (.61,1); pair Q = (1.61, -1); draw(unitcircle); dot(O); label(\"O\",O,W); label(\"T\",T,N); label(\"T'\",T1,S); label(\"T''\",T2,NE); label(\"P\",P,NE); label(\"Q\",Q,S); draw(O--T2); label(\"$r$\",midpoint(O--T2),NW); draw(T--P); label(\"8\",midpoint(T--P),N); draw(T1--Q); label(\"18\",midpoint(T1--Q),S); draw(P--Q);[/asy]",
        "answer": "$12$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 344,
        "problem": "In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $8$, then the area of the triangle $QPO$ is equal to",
        "geo_code": "[asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label(\"A\", (0,0), S); label(\"B\", (5,0), S); label(\"C\", (6,3), NE); label(\"D\", (1,3), NW); label(\"P\", (10,0), E); label(\"Q\", (-5,0), W); label(\"M\", (.5,1.5), NW); label(\"N\", (5.65, 1.5), NE); label(\"O\", (3.4,1.75));[/asy]",
        "answer": "$9$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 345,
        "problem": "In the adjoining figure triangle $ABC$ is such that $AB = 4$ and $AC = 8$. IF $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BM$?",
        "geo_code": "[asy] draw((-4,0)--(4,0)--(-1,4)--cycle); draw((-1, 4)--(0, 0.00001)); label(\"B\", (-4,0), S); label(\"C\", (4,0), S); label(\"A\", (-1, 4), N); label(\"M\", (0, 0.0001), S); [/asy]",
        "answer": "$\\sqrt{31}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 346,
        "problem": "In $\\triangle ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC, AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$ then $\\frac{GF}{EG}$ equals",
        "geo_code": "[asy] draw((0,0)--(12,0)--(14,7.75)--(0,0)); draw((0,0)--(13,3.875)); draw((5,0)--(8.75,4.84)); label(\"A\", (0,0), S); label(\"B\", (12,0), S); label(\"C\", (14,7.75), E); label(\"E\", (8.75,4.84), N); label(\"F\", (5,0), S); label(\"M\", (13,3.875), E); label(\"G\", (7,1)); [/asy]",
        "answer": "$\\frac{2}{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 347,
        "problem": "In the adjoining figure, circle $K$ has diameter $AB$; circle $L$ is tangent to circle $K$ and to $AB$ at the center of circle $K$; and circle $M$ tangent to circle $K$, to circle $L$ and $AB$. The area of circle $L$ divided by the area of circle $M$ is",
        "geo_code": "[asy] size(150); pair K=(0,0),B=(1,0),A=(-1,0),L=(0,0.5),M=(sqrt(2)/2,.25); draw(circle(K,1)^^A--B); draw(circle(L,0.5)^^circle(M,.25)); label(\"$A$\", A, W); label(\"$K$\", K, S); label(\"$B$\", B, E); label(\"$L$\", L); label(\"$M$\", M); [/asy]",
        "answer": "$4$",
        "category": "Primitive Recognition",
        "source": "aops_forum",
        "problem_type": "ratio"
    },
    {
        "index": 348,
        "problem": "Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius two, then the perimeter of the triangle is",
        "geo_code": "[asy] size(120); real t = 2/sqrt(3); real x = 1 + sqrt(3); pair A = t*dir(90), D = x*A; pair B = t*dir(210), E = x*B; pair C = t*dir(330), F = x*C; draw(D--E--F--cycle); draw(Circle(A, 1)); draw(Circle(B, 1)); draw(Circle(C, 1)); [/asy]",
        "answer": "$12+12\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "aops_forum",
        "problem_type": "length"
    },
    {
        "index": 349,
        "problem": "In $\\triangle ABC, AB = 10, AC = 8$ and $BC = 6$. Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$, respectively. The radius of circle P is",
        "geo_code": "[asy] size(100); real a=4, b=3; // import cse5; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP(\"A\",A)--MP(\"B\",B,N)--MP(\"C\",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP(\"R\",IP(CP((X+C)/2,C),B--C),NW)--MP(\"Q\",IP(CP((X+C)/2,C),A--C+(0.1,0)))); [/asy]",
        "answer": "$2.4$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 350,
        "problem": "If rectangle ABCD has area 120 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is",
        "geo_code": "[asy] draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle); draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle); label(\"$F$\",(0,0),E); label(\"$A$\",(-2,1),W); label(\"$B$\",(2,1),E); label(\"$C$\", (2,-1),E); label(\"$D$\",(-2,-1),WSW); label(\"$E$\",(-2,0),W); label(\"$G$\",(0,-1),S); [/asy]",
        "answer": "$30$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 351,
        "problem": "In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\\measuredangle DEB$ in degrees?",
        "geo_code": "[asy] real s=sqrt(3)/2; draw(box((0,0),(1,1))); draw((1+s,0.5)--(1,1)); draw((1+s,0.5)--(1,0)); draw((0,1)--(1+s,0.5)); label(\"$A$\",(1,1),N); label(\"$B$\",(1,0),S); label(\"$C$\",(0,0),W); label(\"$D$\",(0,1),W); label(\"$E$\",(1+s,0.5),E); [/asy]",
        "answer": "$45$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 352,
        "problem": "The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length two. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.",
        "geo_code": "[asy] size(150); import patterns; pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux; add(\"hatch\",hatch()); draw(rotate(100,D)*(A--B--C--D--cycle)); AA=rotate(100,D)*A; BB=rotate(100,D)*D; CC=rotate(100,D)*C; DD=rotate(100,D)*B; aux=midpoint(AA--BB); draw(BB--DD); P=midpoint(AA--aux); aux=midpoint(CC--DD); Q=midpoint(CC--aux); draw(AA--CC,dashed); dot(P); dot(Q); fill(DD--BB--CC--cycle,pattern(\"hatch\")); label(\"$A$\",AA,W); label(\"$B$\",BB,S); label(\"$C$\",CC,E); label(\"$D$\",DD,N); label(\"$P$\",P,S); label(\"$Q$\",Q,E); [/asy]",
        "answer": "$\\sqrt{2}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 353,
        "problem": "Circles with centers $A, B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals",
        "geo_code": "[asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP(\"$A$\",A)); D(MP(\"$B$\",B)); D(MP(\"$C$\",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP(\"$B'$\",BB[0],NW);MP(\"$C'$\",CC[1],NE); [/asy]",
        "answer": "$1+\\sqrt{3(r^2-1)}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 354,
        "problem": "In $\\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $2$ and $\\measuredangle BAC = 60^\\circ, \\measuredangle ABC = 100^\\circ, \\measuredangle ACB = 20^\\circ$ and $\\measuredangle DEC = 80^\\circ$, then the area of $\\triangle ABC$ plus twice the area of $\\triangle CDE$ equals",
        "geo_code": "[asy] size(200); import cse5; pathpen=black; anglefontpen=black; pointpen=black; anglepen=black; dotfactor=3; pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE; EE=(B+C)/2; D(MP(\"$A$\",A,W)--MP(\"$B$\",B,N)--MP(\"$C$\",C,E)--cycle); D(MP(\"$E$\",EE,N)--MP(\"$D$\",D,S)); D(D);D(EE); MA(\"80^\\circ\",8,D,EE,C,0.1); MA(\"20^\\circ\",8,EE,C,D,0.3,2,shift(1,3)*C); draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow()); MA(\"100^\\circ\",8,A,B,C,0.1,0); MA(\"60^\\circ\",8,C,A,B,0.1,0); [/asy]",
        "answer": "$\\frac{\\sqrt{3}}{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 355,
        "problem": "In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of $\\angle ADE$ is",
        "geo_code": "[asy] size(100);defaultpen(linewidth(0.7)+fontsize(10)); pair D=(0,0), C=D+dir(230), E=D+dir(310), F=E+dir(40), G=D+dir(40), A=D+dir(140), B=C+dir(140); draw(E--D--G--F--E--C--D--A--B--C); pair point=(0,0.5); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$D$\", D, dir(-15)); label(\"$E$\", E, dir(point--E)); label(\"$F$\", F, dir(point--F)); label(\"$G$\", G, dir(point--G));[/asy]",
        "answer": "$150$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 356,
        "problem": "If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\\overline{AQ}$, and $\\measuredangle QPC = 60^\\circ$, then the length of $PC$ divided by the length of $BQ$ is",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=(-1,0), B=(1,0), C=(0,1), D=(0,-1), Q=origin, P=(-0.5,0); draw(P--C--D^^A--B^^Circle(Q,1)); label(\"$A$\", A, W); label(\"$B$\", B, E); label(\"$C$\", C, N); label(\"$D$\", D, S); label(\"$P$\", P, S); label(\"$Q$\", Q, SE); label(\"$60^\\circ$\", P+0.0.5*dir(30), dir(30));[/asy]",
        "answer": "$\\frac{2\\sqrt{3}}{3}$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 357,
        "problem": "Sides $AB,BC,CD$ and $DA$ of convex polygon $ABCD$ have lengths 6, 8, 24, and 26, respectively, and $\\angle CBA$ is a right angle. The area of the quadrilateral is",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); real r=degrees((12,5)), s=degrees((3,4)); pair D=origin, A=(13,0), C=D+12*dir(r), B=A+3*dir(180-(90-r+s)); draw(A--B--C--D--cycle); markscalefactor=0.05; draw(rightanglemark(A,B,C)); pair point=incenter(A,C,D); label(\"$A$\", A, dir(A)); label(\"$B$\", B, dir(B)); label(\"$C$\", C, dir(C)); label(\"$D$\", D, dir(D)); label(\"$6$\", A--B, dir(A--B)*dir(-90)); label(\"$8$\", B--C, dir(B--C)*dir(-90)); label(\"$24$\", C--D, dir(C--D)*dir(-90)); label(\"$26$\", D--A, dir(D--A)*dir(-90));[/asy]",
        "answer": "$144$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 358,
        "problem": "In triangle $ABC$, $\\measuredangle CBA=72^\\circ$, $E$ is the midpoint of side $AC$, and $D$ is a point on side $BC$ such that $2BD=DC$; $AD$ and $BE$ intersect at $F$. The area of quadrilateral $FDCE$ divided by the area of triangle $BDF$ is",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(15,3), D=(5,1), A=7*dir(72)*dir(B--C), E=midpoint(A--C), F=intersectionpoint(A--D, B--E); draw(E--B--A--C--B^^A--D); label(\"$A$\", A, dir(D--A)); label(\"$B$\", B, dir(E--B)); label(\"$C$\", C, dir(0)); label(\"$D$\", D, SE); label(\"$E$\", E, N); label(\"$F$\", F, dir(80));[/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 359,
        "problem": "In $\\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\\angle BAC$, and $BN\\perp AN$. If sides $AB$ and $AC$ have lengths $28$ and $38$, respectively, then find $MN$.",
        "geo_code": "[asy] size(150); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=14*dir(42), C=intersectionpoint(B--(30,0), Circle(A,19)), M=midpoint(B--C), b=A+14*dir(A--C), N=foot(A, B, b); draw(N--B--A--N--M--C--A^^B--M); markscalefactor=0.1; draw(rightanglemark(B,N,A)); pair point=N; label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$M$\", M, dir(point--M)); label(\"$N$\", N, dir(30)); label(rotate(angle(dir(A--C)))*\"$38$\", A--C, dir(A--C)*dir(90)); label(rotate(angle(dir(A--B)))*\"$28$\", A--B, dir(A--B)*dir(90)); [/asy]",
        "answer": "$5$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 360,
        "problem": "In $\\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\\angle BAC$. The lengths of $BD$, $DE$ and $EC$ are $4$, $6$, and $12$, respectively. The length of the shortest side of $\\triangle ABC$ is",
        "geo_code": "[asy] defaultpen(linewidth(.8pt)); pair A = (0,11); pair B = (2,0); pair D = (4,0); pair E = (7,0); pair C = (13,0); label(\"$A$\",A,N); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,S); label(\"$E$\",E,S); label(\"$4$\",midpoint(B--D),N); label(\"$6$\",midpoint(D--E),NW); label(\"$12$\",midpoint(E--C),NW); draw(A--B--C--cycle); draw(A--D); draw(A--E); [/asy]",
        "answer": "$4\\sqrt{10}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 361,
        "problem": "In the adjoining diagram, $BO$ bisects $\\angle CBA$, $CO$ bisects $\\angle ACB$, and $MN$ is parallel to $BC$. If $AB=6, BC=12$, and $AC=9$, then the perimeter of $\\triangle AMN$ is",
        "geo_code": "[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0)); draw(B--M--O--B--C--O--N--C^^N--A--M); label(\"$A$\", A, dir(90)); label(\"$B$\", B, dir(O--B)); label(\"$C$\", C, dir(O--C)); label(\"$M$\", M, dir(90)*dir(B--A)); label(\"$N$\", N, dir(90)*dir(A--C)); label(\"$O$\", O, dir(90));[/asy]",
        "answer": "$15$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 362,
        "problem": "In the adjoining figure, the triangle $ABC$ is a right triangle with $\\angle BCA=90^\\circ$. Median $CM$ is perpendicular to median $BN$, and side $BC=2$. The length of $CN$ is",
        "geo_code": "[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10));real r=54.72; pair B=origin, C=dir(r), A=intersectionpoint(B--(9,0), C--C+4*dir(r-90)), M=midpoint(B--A), N=midpoint(A--C), P=intersectionpoint(B--N, C--M); draw(M--C--A--B--C^^B--N); pair point=P; markscalefactor=0.01; draw(rightanglemark(B,C,N)); draw(rightanglemark(C,P,B)); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$M$\", M, S); label(\"$N$\", N, dir(C--A)*dir(90)); label(\"$2$\", B--C, NW); [/asy]",
        "answer": "$\\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 363,
        "problem": "In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$. If the radius of the largest circle is $18$ and that of the smallest one is $8$, then the radius of the second circle is",
        "geo_code": "[asy] size(250);defaultpen(linewidth(0.7)); real alpha=5.797939254, x=71.191836; int i; for(i=0; i<5; i=i+1) { real r=8*(sqrt(6)/2)^i; draw(Circle((x+r)*dir(alpha), r)); x=x+2r; } real x=71.191836+40+20*sqrt(6), r=18; pair A=tangent(origin, (x+r)*dir(alpha), r, 1), B=tangent(origin, (x+r)*dir(alpha), r, 2); pair A1=300*dir(origin--A), B1=300*dir(origin--B); draw(B1--origin--A1); pair X=(69,-5), X1=reflect(origin, (x+r)*dir(alpha))*X, Y=(200,-5), Y1=reflect(origin, (x+r)*dir(alpha))*Y, Z=(130,0), Z1=reflect(origin, (x+r)*dir(alpha))*Z; clip(X--Y--Y1--X1--cycle); label(\"$L_2$\", Z, S); label(\"$L_1$\", Z1, dir(2*alpha)*dir(90));[/asy]",
        "answer": "$4\\sqrt{6}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 364,
        "problem": "Triangle $\\triangle ABC$ in the figure has area $20$. Points $D, E$ and $F$, all distinct from $A, B$ and $C$, are on sides $AB, BC$ and $CA$ respectively, and $AD = 2, DB = 3$. If triangle $\\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,0), C=(8,7), F=7*dir(A--C), E=(10,0)+4*dir(B--C), D=4*dir(A--B); draw(A--B--C--A--E--F--D); pair point=incenter(A,B,C); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$D$\", D, dir(point--D)); label(\"$E$\", E, dir(point--E)); label(\"$F$\", F, dir(point--F)); label(\"$2$\", (2,0), S); label(\"$3$\", (7,0), S);[/asy]",
        "answer": "$12$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 365,
        "problem": "In an arcade game, the \"monster\" is the shaded sector of a circle of radius $2$ cm, as shown in the figure. The missing piece (the mouth) has central angle $\\usepackage{gensymb} 60\\degree$. What is the perimeter of the monster in cm?",
        "geo_code": "[asy] size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label(\"2\", (sqrt(3)/4, 1/4), NW); label(\"$60^\\circ$\", (1,0));[/asy]",
        "answer": "$\\frac{10}{3}\\pi+4$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 366,
        "problem": "Pegs are put in a board $2$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is",
        "geo_code": "[asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7));[/asy]",
        "answer": "$24$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 367,
        "problem": "Diagonal $DB$ of rectangle $ABCD$ is divided into three segments of length $2$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$. The area of $ABCD$ is",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B)^^rightanglemark(N1,E,B)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label(\"A\", A, NE); label(\"B\", B, NE); label(\"C\", C, dir(0)); label(\"D\", D, dir(180)); label(\"$L$\", (x/2,0), SW); label(\"$L^\\prime$\", C, SW); label(\"2\", D--F, NW); label(\"2\", F--E, SE); label(\"2\", E--B, SE); clip(W--X--Y--Z--cycle);[/asy]",
        "answer": "$12\\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 368,
        "problem": "In $\\triangle ABC$, we have $\\angle C = 3\\angle A$, $a = 27$ and $c = 48$. What is $b$?",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,0), B=(12,0), C=(9,5); draw(A--B--C--cycle); label(\"$A$\", A, SW); label(\"$B$\", B, SE); label(\"$C$\", C, N); label(\"$a$\", B--C, dir(B--C)*dir(-90)); label(\"$b$\", A--C, dir(C--A)*dir(-90)); label(\"$c$\", A--B, dir(A--B)*dir(-90));[/asy]",
        "answer": "$35$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 369,
        "problem": "Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. Length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches. How high is the table?",
        "geo_code": "[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13pt)); path table = origin--(1,0)--(1,6)--(6,6)--(6,0)--(7,0)--(7,7)--(0,7)--cycle; path block = origin--(3,0)--(3,1.5)--(0,1.5)--cycle; path rotblock = origin--(1.5,0)--(1.5,3)--(0,3)--cycle; draw(table^^shift((14,0))*table); filldraw(shift((7,0))*block^^shift((5.5,7))*rotblock^^shift((21,0))*rotblock^^shift((18,7))*block,gray); draw((7.25,1.75)--(8.5,3.5)--(8.5,8)--(7.25,9.75),Arrows(size=5)); draw((21.25,3.25)--(22,3.5)--(22,8)--(21.25,8.25),Arrows(size=5)); unfill((8,5)--(8,6.5)--(9,6.5)--(9,5)--cycle); unfill((21.5,5)--(21.5,6.5)--(23,6.5)--(23,5)--cycle); label(\"$r$\",(8.5,5.75)); label(\"$s$\",(22,5.75)); [/asy]",
        "answer": "$30$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 370,
        "problem": "A triangular corner with side lengths $DB=EB=2$ is cut from equilateral triangle ABC of side length $6$. The perimeter of the remaining quadrilateral is",
        "geo_code": "[asy] draw((0,0)--(2,0)--(2.5,.87)--(1.5,2.6)--cycle, linewidth(1)); draw((2,0)--(3,0)--(2.5,.87)); label(\"6\", (0.75,1.3), NW); label(\"2\", (2.5, 0), S); label(\"2\", (2.75,.44), NE); label(\"A\", (1.5,2.6), N); label(\"B\", (3,0), S); label(\"C\", (0,0), W); label(\"D\", (2.5,.87), NE); label(\"E\", (2,0), S); [/asy]",
        "answer": "$16$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 371,
        "problem": "If $\\angle \\text{CBD}$ is a right angle, then this protractor indicates that the measure of $\\angle \\text{ABD}$ is",
        "geo_code": "[asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label(\"O\",(-2.5,0),W); label(\"A\",A,W); label(\"B\",B,S); label(\"C\",C,W); label(\"D\",D,E); label(\"0\",(-1.8,0),W); label(\"20\",(-1.7,.5),NW); label(\"160\",(1.6,.5),NE); label(\"180\",(1.7,0),E); [/asy]",
        "answer": "$140$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 372,
        "problem": "Four rectangular paper strips of length $10$ and width $2$ are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?",
        "geo_code": "[asy] draw((0,0)--(1,0)--(1,4)--(0,4)--(0,0)--(0,1)--(-1,1)--(-1,2)); draw((-1,2)--(0,2)--(0,4)--(-1,4)--(-1,5)--(1,5)--(1,6)--(0,6)); draw((0,6)--(0,5)--(3,5)--(3,6)--(4,6)--(4,2)--(5,2)); draw((5,2)--(5,1)--(1,1)--(3,1)--(3,0)--(4,0)--(4,1)); draw((1,4)--(3,4)--(3,2)--(1,2)--(4,2)--(3,2)--(3,6)); draw((3,6)--(4,6)--(4,5)--(5,5)--(5,4)--(4,4)); [/asy]",
        "answer": "$64$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 373,
        "problem": "An $8'\\times 10'$ table sits in the corner of a square room, as in Figure $1$ below. The owners desire to move the table to the position shown in Figure $2$. The side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart?",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16); draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N); label(\"S\", (18,8)); label(\"S\", (50,8)); label(\"Figure 1\", (A+B)/2, S); label(\"Figure 2\", (E+F)/2, S); label(\"10'\", (I+J)/2, S); label(\"8'\", (12,12)); label(\"8'\", (L+M)/2, S); label(\"10'\", (42,11)); label(\"table\", (5,12)); label(\"table\", (36,11)); [/asy]",
        "answer": "$9\\sqrt{2}$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 374,
        "problem": "$ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side $BC$ and side $B'C'$ is $\\frac{1}{6}$ the altitude of $\\triangle ABC$. The area of $\\triangle ABC$ divided by the area of $\\triangle A'B'C'$ is",
        "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair H=origin, B=(1,-(1/sqrt(3))), C=(-1,-(1/sqrt(3))), A=(0,(2/sqrt(3))), E=(2,-(2/sqrt(3))), F=(-2,-(2/sqrt(3))), D=(0,(4/sqrt(3))); draw(A--B--C--A^^D--E--F--D); label(\"A'\", A, N); label(\"B'\", B, SE); label(\"C'\", C, SW); label(\"A\", D, E); label(\"B\", E, E); label(\"C\", F, W); [/asy]",
        "answer": "$4$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 375,
        "problem": "Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$.",
        "geo_code": "[asy] defaultpen(fontsize(10)+0.8); size(175); pair A,B,C,D,M,P,Q; C=origin; B=(8,0); D=IP(CR(C,6.5),CR(B,8)); A=(4,-3); P=midpoint(A--B); Q=midpoint(C--D); draw(B--C--D--B--A--C^^A--D); draw(D--P--C^^P--Q, gray+dashed+0.5); pen p=fontsize(12)+linewidth(3); dot(\"$A$\",A,down,p); dot(\"$B$\",B,right,p); dot(\"$C$\",C,left,p); dot(\"$D$\",D,up,p); dot(\"$M$\",P,dir(-45),p); dot(\"$N$\",Q,0.2*(Q-P),p); label(\"$27$\",B--D,2*dir(30),fontsize(10)); label(\"$7$\",A--C,2*dir(210),fontsize(10)); label(\"$18$\",A--D,1.5*dir(30),fontsize(10)); label(\"$36$\",(3,0),up,fontsize(10)); [/asy]",
        "answer": "$137$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 376,
        "problem": "The side length of the square is 4, what is the shaded area of the square?",
        "geo_code": "[asy] draw((0,0)--(0,3)--(3,3)--(3,0)--cycle); draw((0,2)--(2,2)--(2,0)); draw((0,1)--(1,1)--(1,0)); draw((0,0)--(3,3)); fill((0,0)--(0,1)--(1,1)--cycle,grey); fill((1,0)--(1,1)--(2,2)--(2,0)--cycle,grey); fill((0,2)--(2,2)--(3,3)--(0,3)--cycle,grey); [/asy]",
        "answer": "$8$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 377,
        "problem": "The area of this figure is $400\\text{ cm}^2$. Its perimeter is",
        "geo_code": "[asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed); [/asy]",
        "answer": "$100$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 378,
        "problem": "Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?  <i>Assume that the planes cutting the prism do not intersect anywhere in or on the prism.</i>",
        "geo_code": "[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1)); [/asy]",
        "answer": "$36$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 379,
        "problem": "Let $ABCD$ be a parallelogram with $\\angle{ABC}=120^\\circ, AB=16$ and $BC=10.$ Extend $\\overline{CD}$ through $D$ to $E$ so that $DE=4.$ If $\\overline{BE}$ intersects $\\overline{AD}$ at $F$, then $FA$ is",
        "geo_code": "[asy] draw((0,0)--(16,0)--(21,5*sqrt(3))--(5,5*sqrt(3))--cycle,dot); draw((5,5*sqrt(3))--(1,5*sqrt(3))--(16,0),dot); MP(\"A\",(0,0),S);MP(\"B\",(16,0),S);MP(\"C\",(21,5sqrt(3)),NE);MP(\"D\",(5,5sqrt(3)),N);MP(\"E\",(1,5sqrt(3)),N); MP(\"16\",(9,0),S);MP(\"10\",(18.5,5sqrt(3)/2),E);MP(\"4\",(3,5sqrt(3)),N); dot((4,4sqrt(3))); MP(\"F\",(4,4sqrt(3)),dir(210)); [/asy]",
        "answer": "$8$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 380,
        "problem": "An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After four changes, what fractional part of the original area of the black triangle remains black?",
        "geo_code": "[asy] unitsize(36); fill((0,0)--(2,0)--(1,sqrt(3))--cycle,gray); draw((0,0)--(2,0)--(1,sqrt(3))--cycle,linewidth(1)); fill((4,0)--(6,0)--(5,sqrt(3))--cycle,gray); fill((5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--cycle,white); draw((5,sqrt(3))--(4,0)--(5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--(5,0)--(6,0)--cycle,linewidth(1)); fill((8,0)--(10,0)--(9,sqrt(3))--cycle,gray); fill((9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--cycle,white); fill((17/2,0)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--cycle,white); fill((9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--cycle,white); fill((19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--cycle,white); draw((9,sqrt(3))--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--(9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--(17/2,0)--(33/4,sqrt(3)/4)--(8,0)--(9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--(9,0)--(19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--(19/2,0)--(10,0)--cycle,linewidth(1)); label(\"Change 1\",(3,3*sqrt(3)/4),N); label(\"$\\Longrightarrow $\",(3,5*sqrt(3)/8),S); label(\"Change 2\",(7,3*sqrt(3)/4),N); label(\"$\\Longrightarrow $\",(7,5*sqrt(3)/8),S); [/asy]",
        "answer": "$\\frac{81}{256}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 381,
        "problem": "In the arrow-shaped polygon [see figure], the angles at vertices $A,C,D,E$ and $F$ are right angles, $BC=FG=5, CD=FE=30, DE=10$, and $AB=AG$. The area of the polygon is",
        "geo_code": "[asy] draw((0,0)--(2,2)--(2,1)--(5,1)--(5,-1)--(2,-1)--(2,-2)--cycle,dot); MP(\"A\",(0,0),W);MP(\"B\",(2,2),N);MP(\"C\",(2,1),S);MP(\"D\",(5,1),NE);MP(\"E\",(5,-1),SE);MP(\"F\",(2,-1),NW);MP(\"G\",(2,-2),S); MP(\"5\",(2,1.5),E);MP(\"5\",(2,-1.5),E);MP(\"30\",(3.5,1),N);MP(\"30\",(3.5,-1),S);MP(\"10\",(5,0),E); [/asy]",
        "answer": "$400$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 382,
        "problem": "An isosceles right triangle with legs of length $16$ is partitioned into $16$ congruent triangles as shown. The shaded area is",
        "geo_code": "[asy] for (int a=0; a <= 3; ++a) { for (int b=0; b <= 3-a; ++b) { fill((a,b)--(a,b+1)--(a+1,b)--cycle,grey); } } for (int c=0; c <= 3; ++c) { draw((c,0)--(c,4-c),linewidth(1)); draw((0,c)--(4-c,c),linewidth(1)); draw((c+1,0)--(0,c+1),linewidth(1)); } label(\"$16$\",(2,0),S); label(\"$16$\",(0,2),W); [/asy]",
        "answer": "$80$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 383,
        "problem": "Five equilateral triangles, each with side $2$, are arranged so they are all on the same side of a line containing one side of each vertex. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is",
        "geo_code": "[asy] draw((-7,0)--(7,0),black+linewidth(.75)); draw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75)); draw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt(3),0),black+linewidth(.75)); [/asy]",
        "answer": "$4\\sqrt{3}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 384,
        "problem": "The ratio of the radii of two concentric circles is $1:3$. If $\\overline{AC}$ is a diameter of the larger circle, $\\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$, then the length of $BC$ is",
        "geo_code": "[asy] draw(circle((0,0),18),black+linewidth(.75)); draw(circle((0,0),6),black+linewidth(.75)); draw((-18,0)--(18,0)--(-14,8*sqrt(2))--cycle,black+linewidth(.75)); dot((-18,0));dot((18,0));dot((-14,8*sqrt(2))); MP(\"A\",(-18,0),W);MP(\"C\",(18,0),E);MP(\"B\",(-14,8*sqrt(2)),W); [/asy]",
        "answer": "$6\\sqrt{5}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "length"
    },
    {
        "index": 385,
        "problem": "Part of an \"n-pointed regular star\" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_1,A_2,\\cdots,A_n$ are congruent, and angles $B_1,B_2,\\cdots,B_n$ are congruent. If the acute angle at $A_1$ is $20^\\circ$ less than the acute angle at $B_1$, then $n=$",
        "geo_code": "[asy] draw((1,0)--(2*cos(pi/8),2*sin(pi/8))--(cos(pi/4),sin(pi/4))--(2*cos(3*pi/8),2*sin(3*pi/8))--(cos(pi/2),sin(pi/2))--(2*cos(5*pi/8),2*sin(5*pi/8))--(cos(3*pi/4),sin(3*pi/4))--(2*cos(7*pi/8),2*sin(7*pi/8))--(-1,0),black+linewidth(.75)); MP(\"A_1\",(2*cos(5*pi/8),2*sin(5*pi/8)),N);MP(\"A_2\",(2*cos(3*pi/8),2*sin(3*pi/8)),N);MP(\"A_3\",(2*cos(1*pi/8),2*sin(1*pi/8)),N); MP(\"A_n\",(2*cos(7*pi/8),2*sin(7*pi/8)),N); MP(\"B_1\",(cos(4*pi/8),sin(4*pi/8)),S);MP(\"B_2\",(cos(2*pi/8),sin(2*pi/8)),S);MP(\"B_n\",(cos(6*pi/8),sin(6*pi/8)),S); [/asy]",
        "answer": "$18$",
        "category": "Global Abstract Integration",
        "source": "HARP",
        "problem_type": "count"
    },
    {
        "index": 386,
        "problem": "Square corners, 5 units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is",
        "geo_code": "[asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy]",
        "answer": "$500$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 387,
        "problem": "The rectangle shown has length $AC=34$, width $AE=18$, and $B$ and $F$ are midpoints of $\\overline{AC}$ and $\\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is",
        "geo_code": "[asy] pair A,B,C,D,EE,F; A = (0,20); B = (16,20); C = (32,20); D = (32,0); EE = (0,0); F = (0,10); draw(A--C--D--EE--cycle); draw(B--D--F); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); label(\"$A$\",A,NW); label(\"$B$\",B,N); label(\"$C$\",C,NE); label(\"$D$\",D,SE); label(\"$E$\",EE,SW); label(\"$F$\",F,W); [/asy]",
        "answer": "$306$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 388,
        "problem": "In $\\triangle ABC$, $\\angle A=55^\\circ$, $\\angle C=75^\\circ, D$ is on side $\\overline{AB}$ and $E$ is on side $\\overline{BC}$. If $DB=BE$, then $\\angle{ADE} =$",
        "geo_code": "[asy] draw((-5,0)--(5,0)--(2,14)--cycle,black+linewidth(.75)); draw((-2.25,5.5)--(4,14/3),black+linewidth(.75)); MP(\"A\",(-5,0),S);MP(\"C\",(5,0),S);MP(\"B\",(2,14),N);MP(\"E\",(4,14/3),E);MP(\"D\",(-2.25,5.5),W); MP(\"55^\\circ\",(-4.5,0),NE);MP(\"75^\\circ\",(5,0),NW); [/asy]",
        "answer": "$115$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 389,
        "problem": "The convex pentagon $ABCDE$ has $\\angle{A}=\\angle{B}=120^\\circ,EA=AB=BC=2$ and $CD=DE=4$. How many times larger is the area of quadrilateral EABC than that of $\\triangle DCE$?",
        "geo_code": "[asy] draw((-1,0)--(1,0)--(1+sqrt(2),sqrt(2))--(0,sqrt(2)+sqrt(13-2*sqrt(2)))--(-1-sqrt(2),sqrt(2))--cycle,black+linewidth(.75)); MP(\"A\",(-1,0),SW);MP(\"B\",(1,0),SE);MP(\"C\",(1+sqrt(2),sqrt(2)),E);MP(\"D\",(0,sqrt(2)+sqrt(13-2*sqrt(2))),N);MP(\"E\",(-1-sqrt(2),sqrt(2)),W); dot((-1,0));dot((1,0));dot((1+sqrt(2),sqrt(2)));dot((-1-sqrt(2),sqrt(2)));dot((0,sqrt(2)+sqrt(13-2*sqrt(2)))); [/asy]",
        "answer": "$\\frac{3}{4}$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "ratio"
    },
    {
        "index": 390,
        "problem": "Amy painted a dartboard over a square clock face using the \"hour positions\" as boundaries.[See figure.] If $t$ is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock. If the area of $t$ is 1, find the value of $q$.",
        "geo_code": "[asy] draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75)); draw((0,-1)--(0,1), black+linewidth(.75)); draw((-1,0)--(1,0), black+linewidth(.75)); draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75)); draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75)); draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75)); draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75)); [/asy]",
        "answer": "$2\\sqrt{3}-2$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 391,
        "problem": "If $\\angle A = 60^\\circ$, $\\angle E = 40^\\circ$ and $\\angle C = 30^\\circ$, then $\\angle EDC =$",
        "geo_code": "[asy] pair A,B,C,D,EE; A = origin; B = (2,0); C = (5,0); EE = (1.5,3); D = (1.75,1.5); draw(A--C--D); draw(B--EE--A); dot(A); dot(B); dot(C); dot(D); dot(EE); label(\"$A$\",A,SW); label(\"$B$\",B,S); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$E$\",EE,N); [/asy]",
        "answer": "$130$",
        "category": "Primitive Recognition",
        "source": "HARP",
        "problem_type": "angle"
    },
    {
        "index": 392,
        "problem": "Around the outside of a $8$ by $8$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles. The area of the square $ABCD$ is",
        "geo_code": "[asy] pair A,B,C,D; A = origin; B = (4,0); C = (4,4); D = (0,4); draw(A--B--C--D--cycle); draw(arc((2,1),(1,1),(3,1),CCW)--arc((3,2),(3,1),(3,3),CCW)--arc((2,3),(3,3),(1,3),CCW)--arc((1,2),(1,3),(1,1),CCW)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); dot(A); dot(B); dot(C); dot(D); dot((1,1)); dot((3,1)); dot((1,3)); dot((3,3)); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,NE); label(\"$D$\",D,NW); [/asy]",
        "answer": "$256$",
        "category": "Local Relation Composition",
        "source": "HARP",
        "problem_type": "area"
    },
    {
        "index": 393,
        "problem": "Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16$,$AB=107$,$FG=17$, and $EF=184$, what is the length of $CE$?",
        "geo_code": "[asy] import graph; unitsize(0.1cm); pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pair F = (90,16);pair G = (90,33);pair H = (3,33); dot(A^^B^^C^^D^^E^^F^^G^^H); label(\"$A$\", A, S);label(\"$B$\", B, S);label(\"$C$\", C, N);label(\"$D$\", D, N);label(\"$E$\", E, S);label(\"$F$\", F, S);label(\"$G$\", G, N);label(\"$H$\", H, N); draw(E--D--A--B--C--E--H--G--F--C); [/asy]",
        "answer": "$104$",
        "category": "Local Relation Composition",
        "source": "AIME-24",
        "problem_type": "length"
    },
    {
        "index": 394,
        "problem": "Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\\times 8$ grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.",
        "geo_code": "[asy] size(10cm); usepackage(\"tikz\");label(\"\\begin{tikzpicture}[scale=.5]\\draw(0,0)grid(8,8);\\draw[line width=2,red](0,0)--(2,0)--(2,3)--(5,3)--(5,8)--(8,8);\\end{tikzpicture}\",origin); label(\"\\begin{tikzpicture}[scale=.5]\\draw(0,0)grid(8,8);\\draw[line width=2,red](0,0)--(0,3)--(3,3)--(3,5)--(8,5)--(8,8);\\end{tikzpicture}\",E); [/asy]",
        "answer": "$294$",
        "category": "Local Relation Composition",
        "source": "AIME-24",
        "problem_type": "count"
    },
    {
        "index": 395,
        "problem": "Eight circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ and $BC$ of triangle $ABC$, respectively. $2024$ circles of radius $1$ can be arranged in the same manner. The inradius of triangle $ABC$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",
        "geo_code": "[asy] pair A = (2,1); pair B = (0,0); pair C = (3,0); dot(A^^B^^C); label(\"$A$\", A, N); label(\"$B$\", B, S); label(\"$C$\", C, S); draw(A--B--C--cycle); for(real i=0.62; i<2.7; i+=0.29){ draw(circle((i,0.145), 0.145)); } [/asy]",
        "answer": "$197$",
        "category": "Global Abstract Integration",
        "source": "AIME-24",
        "problem_type": "count"
    },
    {
        "index": 396,
        "problem": "Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$. When $T$ rests on the inside of $S$, it is internally tangent to $S$ along a circle with radius $r_i$, and when $T$ rests on the outside of $S$, it is externally tangent to $S$ along a circle with radius $r_o$. The difference $r_i-r_o$ can be written as $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",
        "geo_code": "[asy] unitsize(0.3 inch); draw(ellipse((0,0), 3, 1.75)); draw((-1.2,0.1)..(-0.8,-0.03)..(-0.4,-0.11)..(0,-0.15)..(0.4,-0.11)..(0.8,-0.03)..(1.2,0.1)); draw((-1,0.04)..(-0.5,0.12)..(0,0.16)..(0.5,0.12)..(1,0.04)); draw((0,2.4)--(0,-0.15)); draw((0,-0.15)--(0,-1.75), dashed); draw((0,-1.75)--(0,-2.25)); draw(ellipse((2,0), 1, 0.9)); draw((2.03,-0.02)--(2.9,-0.4)); [/asy]",
        "answer": "$127$",
        "category": "Global Abstract Integration",
        "source": "AIME-24",
        "problem_type": "count"
    },
    {
        "index": 397,
        "problem": "Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.",
        "geo_code": "[asy] unitsize(0.6 inch); for(int i=0; i<360; i+=30) { dot(dir(i), 4+black); draw(dir(i)--dir(i+30)); } draw(dir(120)--dir(330)); filldraw(dir(210)--dir(240)--dir(30)--dir(60)--cycle, mediumgray, linewidth(1.5)); draw((0,0.366)--(0.366,0), linewidth(1.5)); [/asy]",
        "answer": "$315$",
        "category": "Global Abstract Integration",
        "source": "AIME-24",
        "problem_type": "count"
    },
    {
        "index": 398,
        "problem": "The results of a cross-country team's training run are graphed below. What is the ID number of the student who has the greatest average speed?",
        "geo_code": "[asy]\nfor ( int i = 1; i <= 7; ++i )\n{\n\ndraw((i,0)--(i,6));\n}\n\nfor ( int i = 1; i <= 5; ++i )\n{\n\ndraw((0,i)--(8,i));\n}\ndraw((-0.5,0)--(8,0), linewidth(1));\ndraw((0,-0.5)--(0,6), linewidth(1));\nlabel(\"$O$\", (0,0), SW);\nlabel(scale(.85)*rotate(90)*\"distance\", (0, 3), W);\nlabel(scale(.85)*\"time\", (4, 0), S);\ndot((1.25, 4.5));\nlabel(scale(.85)*\"1\", (1.25, 4.8), N);\ndot((2.5, 2.2));\nlabel(scale(.85)*\"2\", (2.5, 2.2), S);\ndot((4.25,5.2));\nlabel(scale(.85)*\"3\", (4.25, 5.2), SE);\ndot((5.6, 2.8));\nlabel(scale(.85)*\"4\", (5.6, 2.8), N);\ndot((6.8, 1.4));\nlabel(scale(.85)*\"5\", (6.8, 1.4), E);\n[/asy]",
        "answer": "$1$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "count"
    },
    {
        "index": 399,
        "problem": "The volume of the cylinder shown is $45\\pi$ cubic cm. What is the height in centimeters of the cylinder?",
        "geo_code": "[asy]\nsize(120);\ndraw(shift(2.2,0)*yscale(0.3)*Circle((0,0), 1.2));\n\ndraw((1,0)--(1,-2));\ndraw((3.4,0)--(3.4,-2));\n\ndraw((1,-2)..(2.2,-2.36)..(3.4,-2));\n\nlabel(\"$h$\",midpoint((3.4,0)--(3.4,-2)),E);\n\ndraw (((2.2,0)--(3.4,0)));\n\nlabel(\"$r=3$\",midpoint((2.2,0)--(3.4,0)),N);\n\n[/asy]",
        "answer": "$5$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 400,
        "problem": "Suppose $\\sin D = 0.7$ in the diagram below. What is $DE$?",
        "geo_code": "[asy]\npair D,E,F;\nF = (0,0);\nD = (7,7);\nE = (0,7);\ndraw(D--E--F--D);\ndraw(rightanglemark(D,E,F,15));\nlabel(\"$D$\",D,NE);\nlabel(\"$E$\",E,NW);\nlabel(\"$F$\",F,SW);\nlabel(\"$7$\",(E+F)/2,W);\n[/asy]",
        "answer": "$\\sqrt{51}$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 401,
        "problem": "Let $z = 2 + \\sqrt{2} - (3 + 3 \\sqrt{2})i$, and let $c = 2 - 3i$.  Let $w$ be the result when $z$ is rotated around $c$ by $\\frac{\\pi}{4}$ counter-clockwise.\n\n\n\nFind $w.$",
        "geo_code": "[asy]\nunitsize(0.6 cm);\n\npair C, W, Z;\n\nZ = (2 + sqrt(2), -3 - 3*sqrt(2));\nC = (2,-3);\nW = rotate(45,C)*(Z);\n\ndraw(Z--C--W);\n\ndot(\"$c$\", C, N);\ndot(\"$w$\", W, SE);\ndot(\"$z$\", Z, S);\nlabel(\"$\\frac{\\pi}{4}$\", C + (0.6,-1));\n[/asy]",
        "answer": "$6 - 5i$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 402,
        "problem": "Below is the graph of $y = a \\sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$  Find the smallest possible value of $c.$",
        "geo_code": "[asy]import TrigMacros;\n\nsize(400);\n\nreal f(real x)\n{\n\treturn 2*sin(3*x + pi) + 1;\n}\n\ndraw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-4,4,pi/2,1);\nlayer();\nrm_trig_labels(-5,5, 2);\n\nlabel(\"$1$\", (0,1), E);\nlabel(\"$2$\", (0,2), E);\nlabel(\"$3$\", (0,3), E);\nlabel(\"$-1$\", (0,-1), E);\nlabel(\"$-2$\", (0,-2), E);\nlabel(\"$-3$\", (0,-3), E);\n[/asy]",
        "answer": "$\\pi$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "count"
    },
    {
        "index": 403,
        "problem": "$\\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?",
        "geo_code": "[asy]\ndraw((0,0)--(10,0));\ndraw((0,3)--(10,3));\ndraw((2,3)--(8,0));\ndraw((2,3)--(4,0));\nlabel(\"$A$\",(2,3),N);\nlabel(\"$B$\",(4,0),S);\nlabel(\"$C$\",(8,0),S);\nlabel(\"$124^{\\circ}$\",(2,3),SW);\nlabel(\"$x^{\\circ}$\",(4.5,3),S);\n[/asy]",
        "answer": "$28$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 404,
        "problem": "A regular octagon has the same perimeter as the regular hexagon shown here with side length 16 cm.  How long is each side of the octagon?",
        "geo_code": "[asy]size(80); pair A = dir(120); pair B=dir(60); pair M=(A+B)/2; draw(dir(360)--B--A--dir(180)--dir(240)--dir(300)--cycle); label(\"16 cm\", M, N);[/asy]",
        "answer": "$12$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 405,
        "problem": "In quadrilateral $ABCD$, angle $BAD$ and angle $CDA$ are trisected as shown. What is the degree measure of angle $AFD$?",
        "geo_code": "[asy]\nsize(150);\npair A , B, C, D; A = (0,0); B = (2, 4); C = (7,4); D = (7, -2);\ndraw( (0,0)--(2,4) -- (7,4) -- (7, -2)-- cycle);\nlabel(\"$A$\", A, SW);\nlabel(\"$B$\", B, NW);\nlabel(\"$C$\", C, NE);\nlabel(\"$D$\", D, SE);\npair E, F;\nE = (4.5-.2,1-.2); F = (5, 3);\ndraw(A--E--D); draw(A--F--D);\nlabel(\"$E$\", E, N); label(\"$F$\", F, NW);\ndot(A);dot(B);dot(C);dot(D);dot(E);dot(F);\nlabel(\"$x$\", (1, 1.5), S); label(\"$x$\", (2, 1), S+W); label(\"$x$\", (2, -1), N+N+N+W);\nlabel(\"$y$\", (5.5+.3, .5-.3), S); label(\"$y$\", (6.5+.3, 0)); label(\"$y$\", (5+.5, -1.5+.3));\nlabel(\"$110^{\\circ}$\",(2.5,3.5)); label(\"$100^{\\circ}$\",(6.5-.2,3.5));\n[/asy]",
        "answer": "$80$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 406,
        "problem": "A hexagon is inscribed in a circle:  What is the measure of $\\alpha$, in degrees?",
        "geo_code": "[asy]\npair pA, pB, pC, pD, pE, pF, pO;\npO = (0, 0);\npA = pO + dir(-10);\npB = pO + dir(60);\npC = pO + dir(130);\npD = pO + dir(170);\npE = pO + dir(-160);\npF = pO + dir(-80);\ndraw(pA--pB--pC--pD--pE--pF--pA);\nlabel(\"$105^\\circ$\", pF, N * 2);\nlabel(\"$110^\\circ$\", pB, SW * 1.5);\nlabel(\"$\\alpha$\", pD, E);\ndraw(circle(pO, 1));\n[/asy]",
        "answer": "$145$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 407,
        "problem": "In the diagram, $D$ and $E$ are the midpoints of $\\overline{AB}$ and $\\overline{BC}$ respectively.  Determine the area of quadrilateral $DBEF$.",
        "geo_code": "[asy]\nsize(180); defaultpen(linewidth(.7pt)+fontsize(10pt));\npair A, B, C, D, E, F;\nA=(0,6);\nB=(0,0);\nC=(9,0);\nD=(0,3);\nE=(4,0);\nF=(3,2);\ndraw(E--A--C--D);\ndraw((-1,0)--(10,0), EndArrow);\ndraw((0,-1)--(0,8), EndArrow);\nlabel(\"$A(0,6)$\", A, W);\nlabel(\"$B(0,0)$\", B, SW);\nlabel(\"$C(8,0)$\", C, S);\nlabel(\"$D$\", D, W);\nlabel(\"$E$\", E, S);\nlabel(\"$F$\", F, SW);\nlabel(\"$x$\", (10,0), dir(0));\nlabel(\"$y$\", (0,8), dir(90));\n[/asy]",
        "answer": "$8$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 408,
        "problem": "A set of three points is chosen randomly from the grid shown. Each three-point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?",
        "geo_code": "[asy]\nsize(50);\nfor (int i=0; i<3; ++i) {\nfor (int j=0; j<3; ++j) {\ndot((i,j));};}\n[/asy]",
        "answer": "$\\frac{2}{21}$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "ratio"
    },
    {
        "index": 409,
        "problem": "Two runners, $A$ and $B,$ start at a point $O$ on a linear track, and start running in the same direction.  Runner $B$ runs three times as fast as runner $A.$  An observer stands at point $P$ so that $\\overline{OP}$ is perpendicular to the track.  Find the maximum of $\\angle APB,$ in degrees.",
        "geo_code": "[asy]\nunitsize(2 cm);\n\npair A, B, O, P;\n\nA = (0.4,0);\nB = (1.2,0);\nO = (0,0);\nP = (0,1);\n\ndraw((-0.5,0)--(2,0));\ndraw(O--P);\ndraw(P--A);\ndraw(P--B);\n\nlabel(\"$A$\", A, S);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, S);\nlabel(\"$P$\", P, N);\n[/asy]",
        "answer": "$30$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 410,
        "problem": "A square and a regular heptagon  are coplanar and share a common side $\\overline{AD}$, as shown. What is the degree measure of angle $BAC$?  Express your answer as a common fraction.",
        "geo_code": "[asy]\nfor(int i=0; i <=7; ++i) {\ndraw(dir(360*i/7+90)--dir(360*(i+1)/7+90));\n}\npair A = dir(360*3/7+90);\npair F = dir(360*4/7+90);\npair C = A+dir(90)*(F-A);\npair D = C+F-A;\npair B = dir(360*2/7+90);\n\ndraw(A--C--D--F);\n\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,W);\nlabel(\"$C$\",C,SE);\nlabel(\"$D$\",F,S);\n\n[/asy]",
        "answer": "$\\frac{270}7$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 411,
        "problem": "There is a total of 70 squares of three sizes whose vertices are points on this rectangular $3\\times n$ grid of points. What is the value of $n$?",
        "geo_code": "[asy]\n\nunitsize(0.4 inch);\n\ndot((0,0),linewidth(9bp));\ndot((1,0),linewidth(9bp));\ndot((2,0),linewidth(9bp));\ndot((0,1),linewidth(9bp));\ndot((0,2),linewidth(9bp));\ndot((1,1),linewidth(9bp));\ndot((2,1),linewidth(9bp));\ndot((1,2),linewidth(9bp));\ndot((2,2),linewidth(9bp));\n\nfilldraw((2.95,-0.05)--(3.05,-0.05)--(3.05,0.05)--(2.95,0.05)--cycle,black);\nfilldraw((2.45,-0.05)--(2.55,-0.05)--(2.55,0.05)--(2.45,0.05)--cycle,black);\nfilldraw((3.45,-0.05)--(3.55,-0.05)--(3.55,0.05)--(3.45,0.05)--cycle,black);\n\nfilldraw((2.95,0.95)--(3.05,0.95)--(3.05,1.05)--(2.95,1.05)--cycle,black);\nfilldraw((2.45,0.95)--(2.55,0.95)--(2.55,1.05)--(2.45,1.05)--cycle,black);\nfilldraw((3.45,0.95)--(3.55,0.95)--(3.55,1.05)--(3.45,1.05)--cycle,black);\n\nfilldraw((2.95,1.95)--(3.05,1.95)--(3.05,2.05)--(2.95,2.05)--cycle,black);\nfilldraw((2.45,1.95)--(2.55,1.95)--(2.55,2.05)--(2.45,2.05)--cycle,black);\nfilldraw((3.45,1.95)--(3.55,1.95)--(3.55,2.05)--(3.45,2.05)--cycle,black);\n\ndot((4,0),linewidth(9bp));\ndot((5,0),linewidth(9bp));\ndot((4,1),linewidth(9bp));\ndot((5,1),linewidth(9bp));\ndot((4,2),linewidth(9bp));\ndot((5,2),linewidth(9bp));\n\n[/asy]",
        "answer": "$19$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "count"
    },
    {
        "index": 412,
        "problem": "In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\\triangle ABC$, as shown. \n\n\nWhat is the degree measure of the smallest angle in triangle $PQS$?",
        "geo_code": "[asy]\nsize(200);\npair A, B, C, P, Q, R, S;\nR=(0,0);\nQ=(-2,0);\nS=(2,0);\nP=(1,1.732);\nB=(-5.73,-1);\nC=(3.732,-1);\nA=(1.366,3.098);\ndraw(A--B--C--A);\ndraw(circle(P, 1));\ndraw(circle(Q, 1));\ndraw(circle(R, 1));\ndraw(circle(S, 1));\nlabel(\"A\", A, N);\nlabel(\"B\", B, SW);\nlabel(\"C\", C, SE);\ndot(P);\ndot(Q);\ndot(R);\ndot(S);\nlabel(\"P\", P, N);\nlabel(\"Q\", Q, SW);\nlabel(\"R\", R, SW);\nlabel(\"S\", S, SE);\n[/asy]",
        "answer": "$30$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 413,
        "problem": "Five points $A$, $B$, $C$, $D$, and $O$ lie on a flat field.  $A$ is directly north of $O$, $B$ is directly west of $O$, $C$ is directly south of $O$, and $D$ is directly east of $O$. The  distance between $C$ and $D$ is 140 m.  A hot-air balloon is positioned in the air at $H$ directly above $O$. The balloon is held in place by four ropes $HA$, $HB$, $HC$, and $HD$.  Rope $HC$ has length 150 m and rope $HD$ has length 130 m. \n\nTo reduce the total length of rope used, rope $HC$ and rope $HD$ are to be replaced by a single rope $HP$ where $P$ is a point on the straight line between $C$ and $D$. (The balloon remains at the same position $H$ above $O$ as described above.)  Determine the greatest length of rope that can be saved.",
        "geo_code": "[asy]\nsize(250);\npair A, B, C, D, O, H, W, X, Y, Z;\nO=(0,0);\nA=(1,1);\nD=(1.5,-.3);\nB=(-1.5,.3);\nC=(-1,-1);\nH=(0,2.5);\nW=(5/3)*(A+D);\nX=(5/3)*(A+B);\nY=(-1)*(W);\nZ=(-1)*(X);\ndraw(W--X--Y--Z--W);\ndraw(A--C);\ndraw(B--D);\ndraw(O--H, linewidth(1));\ndraw(A--H, dashed);\ndraw(B--H, dashed);\ndraw(C--H, dashed);\ndraw(D--H, dashed);\ndot(A);\ndot(B);\ndot(C);\ndot(D);\ndot(O);\ndot(H);\nlabel(\"A\", A, NE);\nlabel(\"B\", B, SW);\nlabel(\"C\", C, SE);\nlabel(\"D\", D, NE);\nlabel(\"O\", O, SE);\nlabel(\"H\", H, NW);\n[/asy]",
        "answer": "$160$",
        "category": "Global Abstract Integration",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 414,
        "problem": "Twelve 1 by 1 squares form a rectangle, as shown. What is the total area of the shaded region?",
        "geo_code": "[asy]\nsize(4cm);defaultpen(linewidth(0.75));\n\n// Filled portions\nfill((0, 4)--(0, 0)--(2, 0)--cycle, gray(0.75));\nfill((0, 4)--(3, 4)--(3, 0)--cycle, gray(0.75));\n\n// grid\nint j;\nfor (j = 0; j < 4; ++j) {draw((j, 0)--(j, 4));}\nfor (j = 0; j < 5; ++j) {draw((0, j)--(3, j));}\n\n//diagonals\ndraw((0, 4)--(3, 0)); draw((0, 4)--(2, 0));\n[/asy]",
        "answer": "$10$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 415,
        "problem": "The wheel shown is spun twice, so that the numbers indicated by the pointer are randomly determined (with each number on the wheel being equally likely). The two numbers determined in this way are recorded. The first number is divided by 4, determining one of the remainders 1,2,3 marking the columns of the checkerboard shown. The second number is divided by 5, determining one of the remainders 1,2,3,4 marking the rows of the checkerboard. Finally, a checker is placed on the square where this column and row meet. What is the probability that the checker is placed on a shaded square of the checkerboard?",
        "geo_code": "[asy]\nunitsize(1cm);\ndraw(Circle((0,0),2),linewidth(0.7));\ndraw((1.7,1)--(-1.7,-1),linewidth(0.7));\ndraw((1.7,-1)--(-1.7,1),linewidth(0.7));\ndraw((0,2)--(0,-2));\nlabel(\"1\",(0.8,0.5),NW);\nlabel(\"2\",(0.8,-0.5),SW);\nlabel(\"6\",(-0.8,0.5),NE);\nlabel(\"9\",(-0.8,-0.5),SE);\nlabel(\"3\",(-0.7,0),W);\nlabel(\"7\",(0.7,0),E);\ndraw((-2.8,0)--(-2.1,0),Arrow);\nlabel(\"Pointer\",(-2.8,0),W);\nfill((3,0)--(3,1)--(4,1)--(4,0)--cycle,gray(0.7));\nfill((3,-2)--(3,-1)--(4,-1)--(4,-2)--cycle,gray(0.7));\nfill((4,1)--(4,2)--(5,2)--(5,1)--cycle,gray(0.7));\nfill((4,-1)--(4,0)--(5,0)--(5,-1)--cycle,gray(0.7));\nfill((5,0)--(5,1)--(6,1)--(6,0)--cycle,gray(0.7));\nfill((5,-2)--(5,-1)--(6,-1)--(6,-2)--cycle,gray(0.7));\ndraw((3,-2)--(3,2)--(6,2)--(6,-2)--cycle,linewidth(0.7));\ndraw((3,-1)--(6,-1),linewidth(0.7));\ndraw((3,0)--(6,0),linewidth(0.7));\ndraw((3,1)--(6,1),linewidth(0.7));\ndraw((4,-2)--(4,2),linewidth(0.7));\ndraw((5,-2)--(5,2),linewidth(0.7));\nlabel(\"1\",(3.5,-2),S);\nlabel(\"2\",(4.5,-2),S);\nlabel(\"3\",(5.5,-2),S);\nlabel(\"1\",(3,-1.5),W);\nlabel(\"2\",(3,-0.5),W);\nlabel(\"3\",(3,0.5),W);\nlabel(\"4\",(3,1.5),W);\n[/asy]",
        "answer": "$\\frac{1}{2}$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "ratio"
    },
    {
        "index": 416,
        "problem": "The area of $\\triangle ABC$ is 6 square centimeters. $\\overline{AB}\\|\\overline{DE}$. $BD=4BC$. What is the number of square centimeters in the area of $\\triangle CDE$?",
        "geo_code": "[asy]\ndraw((-.3,-3)--(.1,1)--(-1,0)--(3,0)--cycle);\nlabel(\"$A$\",(.1,1),N);\nlabel(\"$B$\",(-1,0),W);\nlabel(\"$C$\",(0,0),NE);\nlabel(\"$D$\",(3,0),E);\nlabel(\"$E$\",(-.3,-3),S);\n[/asy]",
        "answer": "$54$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 417,
        "problem": "In the diagram, $PT$ is parallel to $QR.$ What is the measure of $\\angle PQR$ in degrees?",
        "geo_code": "[asy]\ndraw((0,0)--(10,0),black+linewidth(1));\ndraw((0,0)--(10,0),MidArrow);\ndraw((10,0)--(20,0),black+linewidth(1));\ndraw((0,0)--(-7,10)--(7,10)--(10,0),black+linewidth(1));\ndraw((-5,10)--(7,10),MidArrow);\nlabel(\"$x^{\\circ}$\",(-6,10),SE);\nlabel(\"$2x^{\\circ}$\",(7,10),SW);\nlabel(\"$128^{\\circ}$\",(10,0),NE);\nlabel(\"$P$\",(-7,10),N);\nlabel(\"$T$\",(7,10),N);\nlabel(\"$R$\",(10,0),S);\nlabel(\"$Q$\",(0,0),S);\n[/asy]",
        "answer": "$116$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 418,
        "problem": "In the figure below, quadrilateral $CDEG$ is a square with $CD = 3$, and quadrilateral $BEFH$ is a rectangle. If $BE = 5$, how many units is $BH$? Express your answer as a mixed number.",
        "geo_code": "[asy]\nunitsize(5mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\n\npair A=(0,0), B=(3,0), C=(6,0), D=(9,0), Ep=(9,3), G=(6,3);\npair F0=bisectorpoint(B,2*Ep-B), H0=bisectorpoint(Ep,2*B-Ep);\npair H=extension(B,H0,A,G);\npair F=extension(Ep,F0,A,G);\n\ndraw(H--B--Ep--F--A--D--Ep--G--C);\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,S);\nlabel(\"$D$\",D,S);\nlabel(\"$E$\",Ep,E);\nlabel(\"$F$\",F,N);\nlabel(\"$G$\",G,NW);\nlabel(\"$H$\",H,NW);\n[/asy]",
        "answer": "$1\\frac{4}{5}$",
        "category": "Global Abstract Integration",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 419,
        "problem": "In circle $J$, $HO$ and $HN$ are tangent to the circle at $O$ and $N$. Find the number of degrees in the sum of $m\\angle J$ and $m\\angle H$.",
        "geo_code": "[asy]\nsize(150);\nimport graph;\npair J = (0,0), H = (6,0), O, N;\npath circ = Circle(J,3);\npair M = midpoint(J--H);\npath secCirc = Circle(M,3);\npair[] tangentPoints = intersectionpoints(circ,secCirc);\nO = tangentPoints[0]; N = tangentPoints[1];\ndraw(J--N--H--O--cycle);\ndraw(circ);\nlabel(\"$H$\",H,E);\nlabel(\"$J$\",J,W);\nlabel(\"$N$\",N,S);\nlabel(\"$O$\",O,NE);\n[/asy]",
        "answer": "$180$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 420,
        "problem": "In the diagram, the three concentric circles have radii of $4,$ $6,$ and $7.$ Three regions are labeled $X,$ $Y,$ or $Z$ below. Of these three regions, what is the difference between the area of the region with the greatest area and the area of the region with the smallest area? Express your answer in exact form.",
        "geo_code": "[asy]\nimport graph;\nfilldraw(circle((0,0),7), lightgray, black+linewidth(1));\nfilldraw(circle((0,0),6), gray, black+linewidth(1));\nfilldraw(circle((0,0),4), white, black+linewidth(1));\ndot((0,0));\nlabel(\"$X$\",(2,0));\nlabel(\"$Y$\",(5,0));\nlabel(\"$Z$\",(6.5,0));\n[/asy]",
        "answer": "$7\\pi$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 421,
        "problem": "In regular pentagon $FGHIJ$, extending the sides of the pentagon, as shown, forms a star. What is the measure of angle $A$ in the figure?",
        "geo_code": "[asy]\ndraw((-42.4,30.8)--(-10,30.8)--(0,63.2)--(10,30.8)--(42.4,30.8)--(16.2,11.8)--(24.9,-18.1)--(0,0)--(-24.9,-18.1)--(-16.2,11.8)--cycle,linewidth(1));\ndraw((-10,30.8)--(10,30.8)--(16.2,11.8)--(0,0)--(-16.2,11.8)--cycle,linewidth(1));\nlabel(\"$A$\",(-42.4,30.8),W);\nlabel(\"$F$\",(-10,30.8),NW);\ndot((-10,30.8));\nlabel(\"$G$\",(10,30.8),NE);\ndot((10,30.8));\nlabel(\"$H$\",(16.2,11.8),E);\ndot((16.2,11.8));\nlabel(\"$I$\",(0,0),S);\ndot((0,0));\nlabel(\"$J$\",(-16.2,11.8),WSW);\ndot((-16.2,11.8));\n[/asy]",
        "answer": "$36$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 422,
        "problem": "$\\textbf{Juan's Old Stamping Grounds}$\n\nJuan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)  What was the average price, in cents, of his $70\\text{'s}$ stamps? Round your answer to the nearest tenth of a cent.",
        "geo_code": "[asy]\n/* AMC8 2002 #8, 9, 10 Problem */\nsize(3inch, 1.5inch);\nfor ( int y = 0; y <= 5; ++y )\n{\n\ndraw((0,y)--(18,y));\n}\ndraw((0,0)--(0,5));\ndraw((6,0)--(6,5));\ndraw((9,0)--(9,5));\ndraw((12,0)--(12,5));\ndraw((15,0)--(15,5));\ndraw((18,0)--(18,5));\n\nlabel(scale(0.8)*\"50s\", (7.5,4.5));\nlabel(scale(0.8)*\"4\", (7.5,3.5));\nlabel(scale(0.8)*\"8\", (7.5,2.5));\nlabel(scale(0.8)*\"6\", (7.5,1.5));\nlabel(scale(0.8)*\"3\", (7.5,0.5));\n\nlabel(scale(0.8)*\"60s\", (10.5,4.5));\nlabel(scale(0.8)*\"7\", (10.5,3.5));\nlabel(scale(0.8)*\"4\", (10.5,2.5));\nlabel(scale(0.8)*\"4\", (10.5,1.5));\nlabel(scale(0.8)*\"9\", (10.5,0.5));\n\nlabel(scale(0.8)*\"70s\", (13.5,4.5));\nlabel(scale(0.8)*\"12\", (13.5,3.5));\nlabel(scale(0.8)*\"12\", (13.5,2.5));\nlabel(scale(0.8)*\"6\", (13.5,1.5));\nlabel(scale(0.8)*\"13\", (13.5,0.5));\n\nlabel(scale(0.8)*\"80s\", (16.5,4.5));\nlabel(scale(0.8)*\"8\", (16.5,3.5));\nlabel(scale(0.8)*\"15\", (16.5,2.5));\nlabel(scale(0.8)*\"10\", (16.5,1.5));\nlabel(scale(0.8)*\"9\", (16.5,0.5));\n\nlabel(scale(0.8)*\"Country\", (3,4.5));\nlabel(scale(0.8)*\"Brazil\", (3,3.5));\nlabel(scale(0.8)*\"France\", (3,2.5));\nlabel(scale(0.8)*\"Peru\", (3,1.5));\nlabel(scale(0.8)*\"Spain\", (3,0.5));\n\nlabel(scale(0.9)*\"Juan's Stamp Collection\", (9,0), S);\nlabel(scale(0.9)*\"Number of Stamps by Decade\", (9,5), N);\n[/asy]",
        "answer": "$5.4$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "count"
    },
    {
        "index": 423,
        "problem": "In the diagram below, we have $\\sin \\angle RPQ = \\frac{7}{25}$.  What is $\\cos \\angle RPS$?",
        "geo_code": "[asy]\n\npair R,P,Q,SS;\n\nSS = (-2,0);\n\nP = (0,0);\n\nQ = (2,0);\n\nR = rotate(aSin(7/25))*(1.5,0);\n\ndot(\"$S$\",SS,S);\n\ndot(\"$Q$\",Q,S);\n\ndot(\"$R$\",R,N);\n\ndot(\"$P$\",P,S);\n\ndraw(Q--SS);\n\ndraw(P--R);\n\n[/asy]",
        "answer": "$-\\frac{24}{25}$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "count"
    },
    {
        "index": 424,
        "problem": "In the diagram, two circles, each with center $D$, have radii of $1$ and $2$. The total area of the shaded region is $\\frac5{12}$ of the area of the larger circle. How many degrees are in the measure of (the smaller) $\\angle ADC$?",
        "geo_code": "[asy]\nsize(4cm);\ndefaultpen(linewidth(0.75));\nreal adc = 100;\npair d = (0, 0); pair a = 2 * dir(100); pair c = (2, 0);\npath inner = arc(d, a/2, c/2, CW);\n\npath outer = arc(d, c, a, CCW);\nguide region1 = (a--a/2)..inner..(c/2--c)..outer..cycle;\nguide region2 = arc(d, a/2, c/2, CCW)..(c/2--d--a/2)..cycle;\nfill(region1, gray(0.75));\nfill(region2, gray(0.75));\ndraw(unitcircle); draw(scale(2) * unitcircle);\ndraw(a--d--c);\nlabel(\"$A$\", a, N); label(\"$C$\", c, E); label(\"$D$\", d, NE);\n[/asy]",
        "answer": "$120$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 425,
        "problem": "Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\\overline{AM}$ with $AD=10$ and $\\angle BDC=3\\angle BAC.$  Find the perimeter of triangle $ABC.$",
        "geo_code": "[asy]\nunitsize(0.4 cm);\n\npair A, B, C, D, M;\n\nA = (0,11);\nD = (0,1);\nB = (-11/2,0);\nC = (11/2,0);\nM = (B + C)/2;\n\ndraw(A--B--C--cycle);\ndraw(A--M);\ndraw(B--D--C);\n\nlabel(\"$A$\", A, N);\nlabel(\"$B$\", B, SW);\nlabel(\"$C$\", C, SE);\nlabel(\"$D$\", D, NW);\nlabel(\"$M$\", M, S);\n[/asy]",
        "answer": "$11 \\sqrt{5} + 11$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 426,
        "problem": "The following line is parameterized, so that its direction vector is of the form $\\begin{pmatrix} -7 \\\\ b \\end{pmatrix}.$  Find $b.$",
        "geo_code": "[asy]\nunitsize(0.4 cm);\n\npair A, B, L, R;\nint i, n;\n\nfor (i = -8; i <= 8; ++i) {\n  draw((i,-8)--(i,8),gray(0.7));\n  draw((-8,i)--(8,i),gray(0.7));\n}\n\ndraw((-8,0)--(8,0),Arrows(6));\ndraw((0,-8)--(0,8),Arrows(6));\n\nA = (-5,4);\nB = (-1,3);\nL = extension(A, B, (-8,0), (-8,1));\nR = extension(A, B, (8,0), (8,1));\n\ndraw(L--R, red);\n\nlabel(\"$x$\", (8,0), E);\nlabel(\"$y$\", (0,8), N);\n[/asy]",
        "answer": "$\\frac{7}{4}$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "count"
    },
    {
        "index": 427,
        "problem": "What is the number of square centimeters in the area of this trapezoid?",
        "geo_code": "[asy]\ndefaultpen(linewidth(0.7));\ndraw((0,0)--(27,0)--(15,9)--(0,9)--cycle);\nlabel(\"5 cm\",(19,4.5),NE);\nlabel(\"5 cm\",(7.5,9),N);\nlabel(\"3 cm\",(0,4.5),W);\nlabel(\"9 cm\",(13.5,0),S);\ndraw(rightanglemark((0,9),(0,0),(27,0),35));\ndraw(rightanglemark((0,0),(0,9),(15,9),35));\n[/asy]",
        "answer": "$21$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 428,
        "problem": "In the diagram, $AD=BD=CD$ and $\\angle BCA = 40^\\circ.$ What is the measure of $\\angle BAC?$",
        "geo_code": "[asy]\ndraw((0,0)--(10,0)--(8.2635,9.8481)--cycle,black+linewidth(1));\ndraw((10,0)--(20,0)--(8.2635,9.8481),black+linewidth(1));\ndraw((5,-0.5)--(5,0.5),black+linewidth(1));\ndraw((15,-0.5)--(15,0.5),black+linewidth(1));\ndraw((8.6318,4.8359)--(9.6317,5.0122),black+linewidth(1));\nlabel(\"$A$\",(8.2635,9.8481),N);\nlabel(\"$B$\",(0,0),SW);\nlabel(\"$C$\",(20,0),SE);\nlabel(\"$D$\",(10,0),S);\n[/asy]",
        "answer": "$90$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 429,
        "problem": "A rectangular field is enclosed in an athletic track, as shown below.  The track consists of two edges of the field, and two semicircles.  The length of the track is 400 meters.  What is the largest possible area of the field, in square meters?",
        "geo_code": "[asy]\nunitsize(1 cm);\n\nfilldraw((0,0)--(3,0)--(3,2)--(0,2)--cycle,lightgreen);\ndraw((0,0)--(3,0),linewidth(2*bp));\ndraw((0,2)--(3,2),linewidth(2*bp));\ndraw(arc((3,1),1,-90,90),linewidth(2*bp));\ndraw(arc((0,1),1,90,270),linewidth(2*bp));\n[/asy]",
        "answer": "$\\frac{20000}{\\pi}$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 430,
        "problem": "In $\\triangle{RST}$, shown, $\\sin{R}=\\frac{2}{5}$.  What is $\\sin{T}$?",
        "geo_code": "[asy]\npair R,S,T;\nT = (0,0);\nS = (2,0);\nR = (2,sqrt(21));\ndraw(R--S--T--R);\ndraw(rightanglemark(R,S,T,10));\nlabel(\"$T$\",T,SW);\nlabel(\"$S$\",S,SE);\nlabel(\"$R$\",R,NE);\nlabel(\"$5$\",(R+T)/2,NW);\n[/asy]",
        "answer": "$\\frac{\\sqrt{21}}{5}$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "count"
    },
    {
        "index": 431,
        "problem": "The distances from a point $P$ to five of the vertices of a regular octahedron are 3, 7, 8, 9, and 11.  Find the distance from $P$ to the sixth vertex.",
        "geo_code": "[asy]\nimport three;\n\nsize(125);\ncurrentprojection = perspective(6,3,1);\n\ntriple A, B, C, D, E, F, P;\n\nA = (1,0,0);\nB = (-1,0,0);\nC = (0,1,0);\nD = (0,-1,0);\nE = (0,0,1);\nF = (0,0,-1);\nP = (1.2,1.5,1);\n\ndraw(A--P,red);\ndraw(B--P,red);\ndraw(C--P,red);\ndraw(D--P,red);\ndraw(E--P,red);\ndraw(F--P,red);\n\ndraw(A--C);\ndraw(A--D);\ndraw(A--E);\ndraw(A--F);\ndraw(C--E--D--F--cycle);\ndraw(D--B--C,dashed);\ndraw(B--C,dashed);\ndraw(B--D,dashed);\ndraw(B--E,dashed);\ndraw(B--F,dashed);\n\nlabel(\"$P$\", P, NE);\n[/asy]",
        "answer": "$\\sqrt{66}$",
        "category": "Global Abstract Integration",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 432,
        "problem": "Quadrilateral $ABCD$ is a square with area 16 square inches. The figure represents the pieces of a Chinese tangram in which all the triangles are isosceles and piece \"e'' is a square. What is the area of the gray piece, in square inches?",
        "geo_code": "[asy]\nfill((6,0)--(12,6)--(12,0)--cycle,gray(.7));\ndraw((0,0)--(0,12)--(12,12)--(12,0)--cycle,linewidth(1));\ndraw((0,0)--(12,12),linewidth(1));\ndraw((3,3)--(6,0)--(12,6),linewidth(1));\ndraw((0,12)--(9,3)--(9,9),linewidth(1));\nlabel(\"$A$\",(0,12),W);\nlabel(\"$B$\",(12,12),E);\nlabel(\"$C$\",(12,0),E);\nlabel(\"$D$\",(0,0),W);\nlabel(\"e\",(6,3));\n\n[/asy]",
        "answer": "$2$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 433,
        "problem": "A square and an equilateral triangle have\tequal\tperimeters.\tThe area of the triangle is $16\\sqrt{3}$ square centimeters. How long, in centimeters, is a diagonal of the square? Express your answer in simplest radical form.",
        "geo_code": "[asy]\ndefaultpen(linewidth(1));\ndraw((0,0)--(1,0)--(1,1)--(0,1)--cycle);\npair a = (1.25,0)+1.25*dir(60);\npair b = a+1.25*dir(-60);\ndraw((1.25,0)--a--b--cycle);\n\n[/asy]",
        "answer": "$6\\sqrt{2}$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 434,
        "problem": "Rectangle $ABCD$ has center $O$ and $AB/AD=k$. A point is randomly chosen from the interior of rectangle $ABCD$. What is the probability that it is closer to $O$ than to any of the four vertices?",
        "geo_code": "[asy]\nsize(200);\ndraw((-250,100)--(250,100)--(250,-100)--(-250,-100)--cycle);\ndot((0,0));\nlabel(\"$O$\",(0,0),N);\nlabel(\"$A$\",(-250,100),NW); label(\"$B$\",(250,100),NE); label(\"$C$\",(250,-100),SE); label(\"$D$\",(-250,-100),SW);[/asy]",
        "answer": "$\\frac{1}{2}$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "ratio"
    },
    {
        "index": 435,
        "problem": "The circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ have two common tangents, forming four points of tangency.  Find the area of the quadrilateral formed by the four points of tangency.",
        "geo_code": "[asy]\nunitsize(0.8 cm);\n\nreal upperparab (real x) {\n  return (sqrt(8*x));\n}\n\nreal lowerparab (real x) {\n  return (-sqrt(8*x));\n}\n\npair A, B, C, D;\n\nA = (-1,1);\nB = (2,4);\nC = (-1,-1);\nD = (2,-4);\n\ndraw(graph(upperparab,0,3));\ndraw(graph(lowerparab,0,3));\ndraw(Circle((0,0),sqrt(2)));\ndraw(interp(A,B,-0.2)--interp(A,B,1.2));\ndraw(interp(C,D,-0.2)--interp(C,D,1.2));\ndraw(A--C);\ndraw(B--D);\n\ndot(A);\ndot(B);\ndot(C);\ndot(D);\n[/asy]",
        "answer": "$15$",
        "category": "Local Relation Composition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 436,
        "problem": "In the diagram below, we have $\\overline{ST}\\parallel\\overline{QR}$, $\\angle P= 40^\\circ$, and $\\angle Q =35^\\circ$. Find the measure of $\\angle STR$ in degrees.",
        "geo_code": "[asy]\npair P,Q,R,SS,T;\nQ = (0,0);\nR = (1,0);\nP = (1.1,0.5);\nSS = 0.6*P;\nT = R + 0.6*(P-R);\ndraw(T--SS--P--R--Q--SS);\nlabel(\"$P$\",P,N);\nlabel(\"$S$\",SS,NW);\nlabel(\"$Q$\",Q,S);\nlabel(\"$R$\",R,S);\nlabel(\"$T$\",T,ENE);\n[/asy]",
        "answer": "$75$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "angle"
    },
    {
        "index": 437,
        "problem": "In the diagram, what is the area of the shaded triangle?",
        "geo_code": "[asy]\nfill((0,0)--(2,3)--(10,0)--cycle,gray);\ndraw((0,0)--(10,0)--(10,3)--(0,3)--cycle,linewidth(1));\ndraw((0,0)--(2,3)--(10,0),linewidth(1));\nlabel(\"10 cm\",(5,3),N);\nlabel(\"10 cm\",(5,0),S);\nlabel(\"3 cm\",(0,1.5),W);\nlabel(\"3 cm\",(10,1.5),E);\ndraw((0,2.5)--(.5,2.5)--(.5,3));\ndraw((10,2.5)--(9.5,2.5)--(9.5,3));\n[/asy]",
        "answer": "$15$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "area"
    },
    {
        "index": 438,
        "problem": "Below is a magic square, meaning that the sum of the numbers in each row, in each column, and in each of the $2$ main diagonals are equal. What is the value of $n$?",
        "geo_code": "[asy]size(125);\nfor(int i = 0; i<4; ++i)\n{\n\ndraw((0,i)--(3,i),linewidth(1));\n}\n\nfor(int j = 0; j<4; ++j)\n{\n\ndraw((j,0)--(j,3),linewidth(1));\n}\n\nlabel(\"$n-3$\",(.5,.5));\nlabel(\"3\",(.5,1.5));\nlabel(\"$n+1$\",(.5,2.5));\n\nlabel(\"$n+2$\",(1.5,.5));\nlabel(\"$2n-9$\",(1.5,1.5));\nlabel(\"$1$\",(1.5,2.5));\n\nlabel(\"$2$\",(2.5,.5));\nlabel(\"$n$\",(2.5,1.5));\nlabel(\"$n-1$\",(2.5,2.5));\n[/asy]",
        "answer": "$7$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "count"
    },
    {
        "index": 439,
        "problem": "What is the shortest distance that can be travelled if you start at any point, $A$, $B$, $C$, or $D$, and visit the other three points once?",
        "geo_code": "[asy]\npair A,B,C, D;\nA = (-3,3); B = (3,0); C = (0, -4); D = (0,0);\ndraw(D--B--C--cycle); draw(D--A--B--cycle);draw(D--A--C--cycle);\nlabel(\"$A$\", A, NW);label(\"$B$\", B, E); label(\"$C$\", C, S);label(\"$D$\", D, NE);\nlabel(\"3\", D--B, S); label(\"6\", A--B, NE); label(\"6\", A--C, SW); label(\"4\", D--C, NW+N);\nlabel(\"5\", A--D, SE+NE); label(\"5\", C--B, E);\n[/asy]",
        "answer": "$13$",
        "category": "Primitive Recognition",
        "source": "MATH-500",
        "problem_type": "length"
    },
    {
        "index": 440,
        "problem": "The perpendicular height of the solid figure is 12m.\n\nHence, find the length of the diameter of the solid figure's base.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n h = 9 \n l = 11\n r = np.sqrt(l**2 - h**2)  \n \n \n plt.plot([0, -r], [0, -h], 'k', lw=3)\n plt.plot([0, r], [0, -h], 'k', lw=3)\n \n plt.plot([0, 0], [0, -h], 'k--', lw=2)\n \n \n theta = np.linspace(0, 2 * np.pi, 100)\n x = r * np.cos(theta)\n y = (r/2) * np.sin(theta) - h\n plt.plot(x, y, 'navy', lw=3)\n \n \n plt.plot([-1.5*r, -1.5*r], [0, -h], 'k', lw=2)\n plt.plot([-1.5*r-0.2, -1.5*r+0.2], [0, 0], 'k', lw=2)\n plt.plot([-1.5*r-0.2, -1.5*r+0.2], [-h, -h], 'k', lw=2)\n plt.text(-1.65*r, -h/2, '12 m', fontsize=14, va='center')\n \n \n plt.plot([0.5, 1.1*r], [0, -h], 'k', lw=3)\n plt.plot([0, 0.2*r-0.1], [0, 0.1*r], 'k', lw=3)\n plt.plot([1*r, 1.2*r], [-h-0.2, -h+0.2], 'k', lw=3)\n plt.text(r/2 + 0.6, -h/2, '13 m', fontsize=14)\n \n \n plt.plot([0, r], [-h, -h], 'k--', lw=2)\n plt.plot([0.1*r, 0.5, 0.1], [-h, -h+0.5, -h+0.5], 'teal', lw=4)\n plt.text(r/2, -h - 0.9, 'r', fontsize=14, ha='center')\n \n \n plt.plot([0], [-h], 'o', color='black')\n \n plt.axis('equal')\n plt.axis('off')\n plt.ylim(-15, 2)\n plt.xlim(-15, 15) \n[/python]",
        "answer": "$10$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "length"
    },
    {
        "index": 441,
        "problem": "The diagram shows a solid figure with a slant height of 13 m. The radius of the base of the solid figure is denoted by $r$.\n\nFind the value of $r$.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n h = 9\n s = 10\n \n r = (s**2 - h**2)**0.5\n \n theta = np.linspace(0, 2*np.pi, 200)\n x_circ = r * np.cos(theta)\n y_circ = r/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(6,10))\n \n ax.plot(x_circ, y_circ, 'k')\n \n ax.plot([0, -r], [h, 0], 'k')\n ax.plot([0, r], [h, 0], 'k')\n \n ax.plot([0, 0], [0, h], 'k--')\n \n ax.plot([-r, r], [0, 0], 'b', linewidth=1)\n \n ax.plot([0.4, 0.4, 0], [0, 0.4, 0.4], 'b', linewidth=1)\n \n ax.text(-0.2, h/2, \"$12m$\", fontsize=14, color=\"k\")\n ax.text(r-2.5, h/2, \"$13m$\", fontsize=14, color=\"k\")\n ax.text(r/2-0.1, -0.7, \"$r$\", fontsize=14, color=\"b\")\n ax.text(0-0.4, -0.5, \"$O$\", fontsize=14, color=\"b\")\n \n ax.set_aspect('equal')\n ax.axis('off')\n plt.ylim(-r-1, h+1)\n plt.xlim(-r-2, r+2) \n[/python]",
        "answer": "$5$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "length"
    },
    {
        "index": 442,
        "problem": "A solid figure has a side $GF$ of length 15cm as shown.\n\nNow, we want to find $y$, the length of the diagonal $DF$.\n\nCalculate $y$ to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig = plt.figure(figsize=(9,6))\n ax = fig.add_subplot(121, projection='3d')\n \n # Vertices as (x, y, z)\n A = [0, 0, 12]\n B = [0, 8, 12]\n C = [17, 8, 12]\n D = [17, 0, 12]\n E = [0, 0, 0]\n F = [0, 8, 0]\n G = [17, 8, 0]\n H = [17, 0, 0]\n \n edges = [\n     (A,D), (D,C), (C,B), (B,A),\n     (H,G), (E,H),\n     (A,E), (D,H), (C,G)\n ]\n \n for e in edges:\n     x, y, z = zip(*e)\n     ax.plot(x, y, z, color='black')\n \n ax.plot([B[0],F[0]], [B[1],F[1]], [B[2],F[2]], ls='dashed', color='black')\n ax.plot([E[0],F[0]], [E[1],F[1]], [E[2],F[2]], ls='dashed', color='black')\n ax.plot([F[0],G[0]], [F[1],G[1]], [F[2],G[2]], ls='dashed', color='black')\n \n ax.plot([D[0],F[0]], [D[1],F[1]], [D[2],G[2]], ls='dotted', color='grey')\n ax.plot([H[0],F[0]], [H[1],F[1]], [H[2],G[2]], ls='dotted', color='grey')\n \n verts = {'A':A, 'B':B, 'C':C, 'D':D, 'E':E, 'F':F, 'G':G, 'H':H}\n for k, v in verts.items():\n     ax.scatter(*v, color='navy')\n     d = dict(A=(-0.5,-0.5,0.5), D=(0.5,-0.5,0.5), C=(0.5,0.5,0.5), B=(-0.5,0.5,0.5),\n              H=(0.5,-0.5,-0.25), G=(0.7,0.7,-0.2), F=(-0.5,0.5,-0.2), E=(-0.8,-0.7,-0.2))\n     o = d.get(k, (0,0,0))\n     ax.text(v[0]+o[0], v[1]+o[1], v[2]+o[2], f'{k}', color='navy', fontsize=17, fontweight='bold')\n \n ax.text(1, 6, (A[2]+D[2])/2, r\"$11\\,\\mathit{cm}$\", fontsize=17, color='black', va='center', ha='right', rotation=90)\n ax.text(17, 6, 6, r\"$11\\,\\mathit{cm}$\", fontsize=17, color='black', va='center', ha='left', rotation=90)\n \n ax.text(7.5, 8.7, 0, r\"$15\\,\\mathit{cm}$\", fontsize=17, color='black', ha='center', va='bottom')\n \n ax.text((D[0]+F[0])/2, (D[1]+F[1])/2+0.5, (D[2]+F[2])/2+0.9, r\"$y\\,cm$\", fontsize=17, color='black')\n ax.text((E[0]+G[0])/2-1, (E[1]+G[1])/2-1, (E[2]+G[2])/2, r\"$z\\,cm$\", fontsize=17, color='black')\n \n ax.set_xlim(-3, 19)\n ax.set_ylim(-3, 12)\n ax.set_zlim(-3, 16)\n ax.set_xticks([])\n ax.set_yticks([])\n ax.set_zticks([])\n ax.set_box_aspect([2,1,1])\n ax.view_init(azim=-60, elev=22)\n plt.tight_layout()\n ax.axis('off') \n[/python]",
        "answer": "$21.61$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "length"
    },
    {
        "index": 443,
        "problem": "The length of CD side is 12cm. \n\nCalculate the area of the triangular divider correct to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n import numpy as np\n \n A = np.array([0, 0, 0])\n B = np.array([20, 0, 0])\n C = np.array([20, 15, 0])\n D = np.array([20, 15, 15])\n E = np.array([0, 0, 15])\n F = np.array([20, 0, 15])\n G = np.array([0, 15, 0])\n H = np.array([0, 15, 15])\n \n fig = plt.figure(figsize=(8, 6))\n ax = fig.add_subplot(111, projection='3d')\n \n edges = [\n     [A, B], [B, C], [C, G], [G, A],  # bottom\n     [E, F], [F, D], [D, H], [H, E],  # top\n     [A, E], [B, F], [C, D], [G, H]   # vertical\n ]\n for edge in edges:\n     ax.plot(*zip(*edge), c='k')\n \n face = [A, C, D]\n ax.add_collection3d(Poly3DCollection([[A, C, D]], facecolors='cyan', linewidths=1, edgecolors='cyan', alpha=0.4))\n \n ax.plot([A[0], C[0]], [A[1], C[1]], [A[2], C[2]], color='r', linestyle='--', linewidth=2)\n \n def annotate_point(point, label, offset=(0,0.5,0)):\n     ax.text(point[0]+offset[0], point[1]+offset[1], point[2]+offset[2], label, fontsize=15, color='navy')\n \n annotate_point(A, 'A')\n annotate_point(B, 'B')\n annotate_point(C, 'C')\n annotate_point(D, 'D')\n \n ax.text(8.5, -1.5, 0, r'$17cm$', fontsize=13)\n ax.text(8.5, 6, 0, r'$zcm$', fontsize=13)\n ax.text(20.8, 6, 0, r'$12cm$', fontsize=13)\n ax.text(20.8, 15, 6, r'$12cm$', fontsize=13)\n \n ax.set_xlim([0, 20])\n ax.set_ylim([0, 15])\n ax.set_zlim([0, 15])\n ax.set_box_aspect([20, 15, 15])\n \n ax.axis('off') \n[/python]",
        "answer": "$124.85$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 444,
        "problem": "The figure shows a solid figure with radius of 6 centimeters.\n\nFind the volume of the solid figure, rounding your answer to two decimal places.",
        "geo_code": "[python]\n import numpy as np\n import matplotlib.pyplot as plt\n \n r = 10\n h = 12\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = r * np.cos(theta)\n y = r/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(6,6))\n ax.plot(x, y + h, 'k')\n ax.plot(x[50:], y[50:], 'k')\n \n ax.plot([r, r], [h, 0], 'k')\n ax.plot([-r, -r], [h, 0], 'k')\n \n ax.plot(x[(theta > np.pi)], y[(theta > np.pi)] + h, 'k--', alpha=0.5)\n \n ax.plot([0, r], [h, h], 'k')\n ax.text(r/2, h+0.5, '6 cm', fontsize=12, ha='center')\n \n ax.text(r+0.5, h/2, '8 cm', fontsize=12, va='center')\n \n ax.axis('equal')\n ax.axis('off') \n[/python]",
        "answer": "$904.78$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 445,
        "problem": "Find the volume of the solid figure shown. The perpendicular height is 6 centimeters.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(10, 10))\n \n center = [0, 0]\n radius = 1\n height = 5\n \n theta = np.linspace(0, 2 * np.pi, 200)\n x = radius * np.cos(theta)\n y = 0.5 * radius * np.sin(theta)\n \n ax.plot(x, y, 'k')\n \n ax.plot([0, -radius], [height, 0], 'k')\n ax.plot([0, radius], [height, 0], 'k')\n \n ax.plot([0, 0], [0, height], 'k')\n \n ax.plot([0, radius], [0, 0], 'k')\n \n ax.plot([0.2, 0.2, 0], [0, 0.2, 0.2], 'k')\n \n ax.text(0.1, height/2, r\"$6cm$\", fontsize=14, ha='left', va='center')\n ax.text(radius/2, -0.3, r\"$2cm$\", fontsize=14, ha='center')\n \n ax.set_xlim(-3, 3)\n ax.set_ylim(-1, 7)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$25.13$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 446,
        "problem": "Find the volume of the solid figure shown. The slant height is 8 centimeters.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import numpy as np\n import matplotlib.pyplot as plt\n \n radius = 3\n slant_height = 9\n height = np.sqrt(slant_height**2 - radius**2)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x_circle = radius * np.cos(theta)\n y_circle = radius/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(6, 6))\n \n ax.plot([0, -radius], [height, 0], 'k')\n ax.plot([0, radius], [height, 0], 'k')\n \n ax.plot(x_circle, y_circle, 'k')\n \n ax.plot([0, radius], [0, 0], 'k', lw=1)\n \n ax.text(radius * 0.5, height/2 + 0.1, '8 cm', fontsize=12, style='italic')\n ax.text(radius/4, -0.4, '2 cm', fontsize=12, style='italic')\n \n ax.plot(0, height, 'ko')\n \n ax.set_aspect('equal')\n ax.axis('off')\n plt.ylim(-2.5, height+1)\n plt.xlim(-3, 3) \n[/python]",
        "answer": "$32.45$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 447,
        "problem": "The diameter is 3 centimeters.\n\nFind the volume of the solid, correct to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n A = np.array([0, 0])\n B = np.array([4, 0]) \n \n mid_BC = (A + B) / 2\n height = 9\n C = mid_BC + np.array([0, height])\n \n plt.plot([A[0], B[0]], [A[1], B[1]], 'k')\n plt.plot([A[0], C[0]], [A[1], C[1]], 'k')\n plt.plot([B[0], C[0]], [B[1], C[1]], 'k')\n \n plt.plot([mid_BC[0], C[0]], [mid_BC[1], C[1]], 'k--')\n \n theta = np.linspace(np.pi, 2 * np.pi, 100)\n r = 2\n x_semicircle = mid_BC[0] + r * np.cos(theta)\n y_semicircle = mid_BC[1] + r * np.sin(theta)\n plt.plot(x_semicircle, y_semicircle, 'k')\n \n plt.text(1, -0.5, r'$3cm$', fontsize=14)\n plt.text(3.75, 4, r'$9cm$', fontsize=14)\n plt.plot([2, 2.5], [0.5, 0.5], 'k')\n plt.plot([2.5, 2.5], [0.5, 0], 'k')\n \n plt.axis('equal')\n plt.axis('off') \n[/python]",
        "answer": "$10.45$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 448,
        "problem": "Find the volume of the solid figure pictured here. The radius is 4.1 centimeters.\n\n(Give your answer correct to 1 decimal place.)",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,6))\n \n ax.fill([0, 4.3, -4.3, 0], [0, -13.5, -13.5, 0], color='olive')\n \n theta = np.linspace(0, 2*np.pi, 100)\n x_ellipse = 4.3 * np.cos(theta)\n y_ellipse = -13.5 + 1 * np.sin(theta)\n ax.plot(x_ellipse[50:], y_ellipse[50:], color='black')\n ax.plot(x_ellipse[:50], y_ellipse[:50], color='black', linestyle=\"--\")\n ax.fill(x_ellipse, y_ellipse, color=\"olive\")\n \n ax.plot([0, 0], [0, -13.5], 'k--', lw=1)\n ax.plot([0, 4.3], [-13.5, -13.5], 'k--', lw=1)\n \n offset = 0.3\n ax.plot([0, offset], [-13.5, -13.5], color='black')\n ax.plot([offset, offset], [-13.5, -13.5+offset], color='black')\n ax.plot([0, offset], [-13.5+offset, -13.5+offset], color='black')\n \n ax.text(-1.18, -6.25, r\"$12.5cm$\", fontsize=14, va='center', color='black', style='italic')\n ax.text(2.05, -13.1, r\"4.1cm\", fontsize=14, ha='center', color='black', style='italic')\n \n ax.set_aspect('equal')\n plt.axis('off') \n[/python]",
        "answer": "$220.0$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 449,
        "problem": "Find the volume of the solid figure pictured. The perpendicular height is 6.\n\n(Give your answer correct to 2 decimal places.)",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,6))\n \n ax.fill([0, 4.1, -4.1, 0], [0, -12.5, -12.5, 0], color='olive')\n \n theta = np.linspace(0, 2*np.pi, 100)\n x_ellipse = 4.1 * np.cos(theta)\n y_ellipse = -12.5 + 1 * np.sin(theta)\n ax.plot(x_ellipse[50:], y_ellipse[50:], color='black')\n ax.plot(x_ellipse[:50], y_ellipse[:50], color='black', linestyle=\"--\")\n ax.fill(x_ellipse, y_ellipse, color=\"olive\")\n \n \n ax.plot([0, 0], [0, -12.5], 'k--', lw=1)\n ax.plot([-4.1, 0], [-12.5, -12.5], 'k--', lw=1)\n \n offset = 0.3\n #ax.plot([0, offset], [-12.5, -12.5], color='black')\n ax.plot([-offset, -offset], [-12.5, -12.5+offset], color='black')\n ax.plot([-offset, 0], [-12.5+offset, -12.5+offset], color='black')\n \n ax.text(0.2, -6.25, r\"$6$\", fontsize=14, va='center', color='black', style='italic')\n ax.text(-2.05, -13.1, r\"$3$\", fontsize=14, ha='center', color='black', style='italic')\n \n ax.set_aspect('equal')\n plt.axis('off') \n[/python]",
        "answer": "$56.55$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 450,
        "problem": "Find the volume of the solid figure shown.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(8,9))\n \n circle = plt.Circle((0, 0), 3.5, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = 3.5 * np.cos(theta)\n y = 1.75 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 3.5], [0, 0], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(1.75, 0.1, '3 cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]",
        "answer": "$113.10$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 451,
        "problem": "Find the volume of the solid figure shown.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n y = 3 * np.cos(theta)\n x = 1.1 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 0], [3, -3], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(0.2, 0.1, '4 cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]",
        "answer": "$33.51$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 452,
        "problem": "The diameter of large solid figure is 12 centimeters. Find the volume of the solid.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(8,8))\n \n R = 9\n r = 4.5\n \n theta = np.linspace(0, np.pi, 500)\n x_outer = R * np.cos(theta)\n y_outer = R * np.sin(theta)\n \n theta_inner = np.linspace(0, np.pi, 200)\n x_inner = r * np.cos(theta_inner)\n y_inner = r * np.sin(theta_inner) + (R-r)\n \n ax.plot(x_outer, -y_outer, color='navy', lw=3, zorder=1)\n \n ellipse_x = R * np.cos(np.linspace(0, 2*np.pi, 200))\n ellipse_y = 0.33 * R * np.sin(np.linspace(0, 2*np.pi, 200))\n ax.plot(ellipse_x, ellipse_y, color='navy', lw=3)\n \n dash_theta = np.linspace(0, 2*np.pi, 400)\n dash_x = r * np.cos(dash_theta)\n dash_y = 0.33 * r * np.sin(dash_theta)\n ax.plot(dash_x[:200], dash_y[:200], 'k--', lw=2)\n ax.plot(dash_x[200:], dash_y[200:], 'k', lw=2)\n \n ax.plot(x_inner, y_inner-R+r, color='navy', lw=3, zorder=2)\n \n ax.annotate('', xy=(-r, 5), xytext=(r, 5), arrowprops=dict(arrowstyle='<->', lw=1.8))\n ax.text(0, 5.25, \"6 cm\", ha='center', va='bottom', fontsize=14)\n \n ax.annotate('', xy=(-R, -9.2), xytext=(R, -9.2), arrowprops=dict(arrowstyle='<->', lw=1.8))\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$508.94$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 453,
        "problem": "Find the volume of the solid shown, giving your answer correct to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = 3 * np.cos(theta)\n y = 1.5 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 3], [0, 0], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(1, 0.1, '8.8 cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]",
        "answer": "$2854.54$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 454,
        "problem": "Find the volume of the solid shown, giving your answer correct to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = 3 * np.cos(theta)\n y = 1.5 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 3], [0, 0], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(1, 0.1, '22.36m', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]",
        "answer": "$46827.83$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 455,
        "problem": "Find the volume of the following solid figure.\n\nRound your answer to three decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 7\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='#C6E6EB', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='#8ED7DE', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(0.3, -radius/2, '6', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$452.389$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 456,
        "problem": "Find the volume of the following solid figure.\n\nRound your answer to three decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 6\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='#C6E6EB', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='#8ED7DE', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(-3.3, -radius/2, '$25.36cm$', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$34159.095$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 457,
        "problem": "Consider the following solid figure with a height of 35 cm. Find the surface area of the solid figure.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n radius = 20\n height = 70\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = radius * np.cos(theta)\n y = radius/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(5,8))\n \n ax.plot(x, y + height, 'k')\n \n ax.plot(x[:50], y[:50], 'k--')\n ax.plot(x[50:], y[50:], 'k')\n \n ax.plot([radius, radius], [height, 0], 'k')\n ax.plot([-radius, -radius], [height, 0], 'k')\n \n ax.annotate(\n     '', xy=(radius + 4, height), xytext=(radius + 4, 0),\n     arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([radius+3, radius+5], [height, height], color=\"black\")\n ax.plot([radius+3, radius+5], [0, 0], color=\"black\")\n ax.text(radius + 6, height/2, f'{35}', va='center', fontsize=12)\n \n ax.annotate(\n     '', xy=(0, height+0.5), xytext=(-radius-0.5, height+0.5),\n     arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([0, 0], [height-0.2, height+1.2], color=\"black\")\n ax.plot([-radius, -radius], [height-0.2, height+1.2], color=\"black\")\n ax.text(-radius/2, height + 1, f'{10}', ha='center', fontsize=12)\n \n ax.set_xlim(-radius*1.5, radius*1.5)\n ax.set_ylim(-radius, height + radius)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$2827.43$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 458,
        "problem": "The following solid figure has a base radius of 21 m.\n\nFind the surface area.\n\nGive your answer to the nearest two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n # Cylinder dimensions\n radius = 17\n length = 36\n \n fig, ax = plt.subplots(figsize=(8, 5))\n \n # Draw the side edges of the cylinder\n ax.plot([0, length], [radius, radius], 'k')\n ax.plot([0, length], [-radius, -radius], 'k')\n \n # Draw the front ellipse (solid)\n theta = np.linspace(0, 2 * np.pi, 100)\n x_front = length + radius/2 * np.cos(theta)\n y_front = radius * np.sin(theta)\n ax.plot(x_front, y_front, 'k')\n \n # Draw the back ellipse (dashed)\n x_back = 0 + radius/2 * np.cos(theta)\n y_back = radius * np.sin(theta)\n ax.plot(x_back, y_back, 'k--')\n \n # Draw the radius on the front face\n ax.plot([length, length], [0, radius], 'k')\n ax.plot(length, 0, 'bo', markersize=3) # Center point\n \n # Add dimension lines and labels\n # Length\n ax.annotate('', xy=(0, -radius-5), xytext=(length, -radius-5),\n             arrowprops=dict(arrowstyle='<->', lw=1.5))\n ax.text(length/2, -radius-8, '49', ha='center', va='top', fontsize=12)\n \n # Radius\n ax.annotate('', xy=(length, 0), xytext=(length, radius),\n             arrowprops=dict(arrowstyle='<->', lw=1.5))\n ax.text(length+3, radius/2, '21', va='center', fontsize=12)\n \n # Set aspect, limits and remove axes\n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-radius*0.7, length+radius*1.2)\n ax.set_ylim(-radius*1.3, radius*1.3)\n \n plt.show() \n[/python]",
        "answer": "$9236.28$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 459,
        "problem": "Find the surface area of the given solid figure. The radius is 98. All measurements in the diagram are in mm.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n # Cylinder parameters\n radius = 80\n height = 82\n \n # Plot setup\n fig, ax = plt.subplots(figsize=(7,5))\n ax.set_xlim(-120, 160)\n ax.set_ylim(-70, 90)\n \n # Angles for ellipse\n theta = np.linspace(0, 2*np.pi, 100)\n \n # Top ellipse (at y = height/2)\n x_top = radius * np.cos(theta)\n y_top = 0.25 * radius * np.sin(theta) + height/2\n \n # Bottom ellipse (at y = -height/2)\n x_bot = radius * np.cos(theta)\n y_bot = 0.25 * radius * np.sin(theta) - height/2\n \n # Draw top and bottom ellipses\n ax.plot(x_top, y_top, 'k')\n ax.plot(x_bot, y_bot, 'k', linestyle='dashed')\n \n # Draw sides\n ax.plot([-radius, -radius], [-height/2, height/2], 'k')\n ax.plot([radius, radius], [-height/2, height/2], 'k')\n \n # Draw radius annotation\n ax.plot([0, radius], [height/2, height/2], 'b', marker='o')\n ax.text(radius/2, height/2+7, '98', ha='center', va='bottom', fontsize=12)\n \n # Draw height annotation with double arrow\n ax.annotate(\n     '', xy=(radius+15, height/2), xytext=(radius+15, -height/2),\n     arrowprops=dict(arrowstyle='<->', linewidth=2))\n ax.text(radius+22, 0, '80', va='center', fontsize=12)\n \n # Remove axes\n ax.axis('off')\n \n plt.show() \n[/python]",
        "answer": "$109603.88$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 460,
        "problem": "Find the height $h$ mm of this closed solid figure if its surface area (S) is 27288(mm)^2.\n\nRound your answer to the nearest whole number.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n radius = 20\n height = 70\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = radius * np.cos(theta)\n y = radius/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(5,8))\n \n ax.plot(x, y + height, 'k')\n \n ax.plot(x[:50], y[:50], 'k--')\n ax.plot(x[50:], y[50:], 'k')\n \n ax.plot([radius, radius], [height, 0], 'k')\n ax.plot([-radius, -radius], [height, 0], 'k')\n \n ax.annotate(\n     '', xy=(radius + 4, height), xytext=(radius + 4, 0),\n     arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([radius+3, radius+5], [height, height], color=\"black\")\n ax.plot([radius+3, radius+5], [0, 0], color=\"black\")\n ax.text(radius + 6, height/2, '$hmm$', va='center', fontsize=12)\n \n ax.annotate(\n     '', xy=(0, height+0.5), xytext=(-radius-0.5, height+0.5),\n     arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([0, 0], [height-0.2, height+1.2], color=\"black\")\n ax.plot([-radius, -radius], [height-0.2, height+1.2], color=\"black\")\n ax.text(-radius/2, height + 1, '$43mm$', ha='center', fontsize=12)\n \n ax.set_xlim(-radius*1.5, radius*1.5)\n ax.set_ylim(-radius, height + radius)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$58$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "length"
    },
    {
        "index": 461,
        "problem": "A solid figure has a surface area of 54105(mm)^2.\n\nWhat must the height $h$ mm of the solid figure be?\n\nRound your answer to the nearest whole number.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n radius = 20\n height = 70\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = radius * np.cos(theta)\n y = radius/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(5,8))\n \n ax.plot(x, y + height, 'k')\n \n ax.plot(x[:50], y[:50], 'k--')\n ax.plot(x[50:], y[50:], 'k')\n \n ax.plot([radius, radius], [height, 0], 'k')\n ax.plot([-radius, -radius], [height, 0], 'k')\n \n ax.annotate(\n     '', xy=(radius + 4, height), xytext=(radius + 4, 0),\n     arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([radius+3, radius+5], [height, height], color=\"black\")\n ax.plot([radius+3, radius+5], [0, 0], color=\"black\")\n ax.text(radius + 6, height/2, '$hmm$', va='center', fontsize=12)\n \n ax.annotate(\n     '', xy=(0, height+0.5), xytext=(-radius-0.5, height+0.5),\n     arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([0, 0], [height-0.2, height+1.2], color=\"black\")\n ax.plot([-radius, -radius], [height-0.2, height+1.2], color=\"black\")\n ax.text(-radius/2, height + 1, '$79mm$', ha='center', fontsize=12)\n \n ax.set_xlim(-radius*1.5, radius*1.5)\n ax.set_ylim(-radius, height + radius)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$30$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "length"
    },
    {
        "index": 462,
        "problem": "Find the surface area of the solid figure shown. The radius is 6 cm.\n\nGive your answer to the nearest two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n radius = 7\n height = 12\n \n fig, ax = plt.subplots(figsize=(6,10))\n \n theta = np.linspace(0, 2*np.pi, 100)\n x_top = radius * np.cos(theta)\n y_top = radius/2 * np.sin(theta) + height\n \n x_bottom = radius * np.cos(theta)\n y_bottom = radius/2 * np.sin(theta)\n \n ax.plot([radius, radius], [height, 0], color='k')\n ax.plot([-radius, -radius], [height, 0], color='k')\n \n ax.plot(x_top, y_top, color='k')\n ax.plot(x_bottom[50:], y_bottom[50:], color='k')\n \n theta_hidden = np.linspace(np.pi, 2*np.pi, 100)\n x_hidden = radius * np.cos(theta_hidden)\n y_hidden = radius/2 * np.sin(theta_hidden) + height\n ax.plot(x_hidden, y_hidden, color='k', alpha=0.3, linestyle='dashed')\n \n ax.plot([0, radius], [height, height], color='k')\n ax.text(radius/2, height+0.5, '6 cm', fontsize=12, style='italic', ha='center')\n \n ax.text(radius+1.5, height/2, '10 cm', fontsize=12, style='italic', ha='center')\n \n ax.set_xlim(-10, 12)\n ax.set_ylim(-6, 16)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$603.19$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 463,
        "problem": "Find the surface area of the solid figure shown.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#19bfa7')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = 3 * np.cos(theta)\n y = 1.5 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#19bfa7')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 3], [0, 0], linestyle='--', linewidth=3, color=\"blue\")\n \n ax.text(1, 0.1, '11cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]",
        "answer": "$1520.53$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 464,
        "problem": "Find the surface area of the solid figure shown.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n y = 3 * np.cos(theta)\n x = 1.1 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 0], [3, -3], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(0.2, 0.1, '9 cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]",
        "answer": "$254.47$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 465,
        "problem": "Find the surface area of the following solid figure.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n # Set up the plot\n fig, ax = plt.subplots(figsize=(6,6))\n radius = 30\n \n # Draw the main circle (sphere)\n circle = plt.Circle((0, 0), radius, fill=False, color='black')\n ax.add_artist(circle)\n \n # Draw the dashed radius (equator perspective)\n theta = np.linspace(0, 2*np.pi, 200)\n x_ellipse = radius * np.cos(theta)\n y_ellipse = 0.4 * radius * np.sin(theta)   # Slightly \"flatten\" to suggest equator\n \n ax.plot(x_ellipse, y_ellipse, 'k--')\n \n # Draw the radius line\n ax.plot([0, radius], [0, 0], 'k')\n \n # Add the radius text\n plt.text(radius/2-5, 1, '56.17 $mm$', fontsize=14, fontstyle='italic')\n \n # Formatting\n ax.set_aspect('equal')\n ax.set_xlim(-radius*1.2, radius*1.2)\n ax.set_ylim(-radius*1.2, radius*1.2)\n ax.axis('off')\n \n plt.show() \n[/python]",
        "answer": "$39647.77$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 466,
        "problem": "What is the area of the circular base of the solid figure?\n\nRound your answer to three decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 7\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='olive', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='olive', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(0.3, -radius/2, '6', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$113.097$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 467,
        "problem": "Find the total surface area.\n\nRound your answer to three decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 7\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='olive', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='olive', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(0.3, -radius/2, '8', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$603.186$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 468,
        "problem": "Find the surface area of the following solid figure.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 7\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='olive', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='olive', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(-2, -radius/2, '$74.74cm$', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$52647.45$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 469,
        "problem": "One side of the left solid figure is 33. The height of middle solid figure is 11.\n\nFind the surface area of the solid.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n import matplotlib.patches as patches\n \n \n fig, ax = plt.subplots(figsize=(8,4))\n \n width, height = 15, 20\n \n # \u68af\u5f62\u524d\u5e95\u4e3a14\uff0c\u4e0a\u5e9510\uff0c\u9ad83\uff0c\u5de6\u504f\u79fb2\uff0c\u53f3\u504f\u79fb2\n # \u524d\u4fa7\u68af\u5f62\n A = np.array([0, 0])\n B = np.array([width, 0])\n D = A + np.array([0, height])\n C = B + np.array([0, height])\n \n # \u68f1\u67f1\u539a\u5ea6\n dx, dy = -5, 6\n dxy = np.array([dx, dy])\n # \u540e\u9762\u56db\u4e2a\u70b9\n A1 = A + dxy\n B1 = B + dxy\n C1 = C + dxy\n D1 = D + dxy\n \n x0 = np.array([25, 0])\n A2 = A + x0\n B2 = B + x0\n C2 = C + x0\n D2 = D + x0\n \n A3, B3, C3, D3 = A1 + x0,  B1 + x0, C1 + x0, D1 + x0\n \n for X, Y in [(A, B), (B, C), (C, D), (D, A)]:\n     ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n for X, Y in [(C1, D1), (D1, A1), (D, D1), (C, C1), (A, A1), (A2, A3)]:\n     ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n for X, Y in [(A2, B2), (B2, C2), (C2, D2), (D2, A2)]:\n     ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n for X, Y in [(C3, D3), (D3, A3), (D2, D3), (C2, C3)]:\n     ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n for X, Y in [(A1, B1), (B1, C1), (A3, B3), (B3, C3), (B, B1), (B2, B3)]:\n     ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k--')\n \n # Big cylinder base\n main1 = patches.Arc(((B1[0] + B[0])/2, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2), 3, 13, theta1=0, theta2=360, color='black', linewidth=2, linestyle='--')\n #ax.plot([0, 0], [0, 0], 'ko')\n ax.add_patch(main1)\n \n main2 = patches.Arc(((A2[0] + A3[0])/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2), 3, 13, theta1=270, theta2=90, color='black', linewidth=2)\n main3 = patches.Arc(((A2[0] + A3[0])/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2), 3, 13, theta1=90, theta2=270, color='black', linewidth=2, linestyle='--')\n #ax.plot([0, 0], [0, 0], 'ko')\n ax.add_patch(main2)\n ax.add_patch(main3)\n \n ax.plot([(B1[0] + B[0])/2, (A2[0] + A3[0])/2], [((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 + 6.5, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 + 6.5], 'k')\n ax.plot([(B1[0] + B[0])/2, (A2[0] + A3[0])/2], [((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 - 6.5, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 - 6.5], 'k')\n \n \n ax.plot([B[0], A2[0]], [-2, -2], 'k')\n ax.plot([B[0], B[0]], [-1, -3], 'k')\n ax.plot([A2[0], A2[0]], [-1, -3], 'k')\n ax.plot([(A2[0] + A3[0])/2, (A2[0] + A3[0])/2], [((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2+6.5], 'k')\n \n ax.text(B2[0] + 1, C[1]/2, r\"$20$\", fontsize=12, style='italic')\n ax.text((B2[0] + A2[0])/2, -2, r\"$25$\", fontsize=12, style='italic')\n \n ax.text((A2[0] + A3[0])/2-1.2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2+0.5, r\"$6$\", fontsize=10, style='italic')\n \n ax.set_aspect('equal')\n ax.axis('off')\n plt.xlim(-8, 60)\n plt.ylim(-5, 30)\n plt.tight_layout()\n plt.show() \n[/python]",
        "answer": "$8128.50$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 470,
        "problem": "Find the surface area of the composite figure shown. The radius is 4 cm.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\n import numpy as np\n import matplotlib.pyplot as plt\n \n radius = 4 \n height = 9 \n \n fig = plt.figure(figsize=(6, 12))\n ax = fig.add_subplot(111)\n \n theta = np.linspace(0, np.pi, 100)\n x_hemisphere = radius * np.cos(theta)\n y_hemisphere = -radius * np.sin(theta)\n \n \n x_cone = np.array([-radius, 0, radius])\n y_cone = np.array([0, height, 0])\n \n ax.plot(x_hemisphere, y_hemisphere, color='deepskyblue', linewidth=4)\n ax.fill_between(x_hemisphere, y_hemisphere, 0, color='deepskyblue', alpha=0.6)\n \n \n theta = np.linspace(0, 2*np.pi, 200)\n x_circle = radius * np.cos(theta)\n y_circle = radius/ 5 * np.sin(theta)\n ax.plot(x_circle, y_circle, color=\"deepskyblue\", linewidth=4)\n ax.fill_between(x_circle, y_circle, 0, color=\"gray\")\n ax.plot([-radius, 0], [0, height], color='gray')\n ax.plot([radius, 0], [0, height], color='gray')\n \n ax.plot([0, 0], [0, height], 'k--', lw=1)\n ax.text(0.1, height/1.7, r'$10cm$', fontsize=15)\n \n ax.plot([0, radius], [0, 0], 'black', linestyle='dashed')\n \n ax.plot([0.5, 0.5], [0, 0.5], color='black')\n ax.plot([0, 0.5], [0.5, 0.5], color='black')\n \n ax.axis('off')\n \n ax.set_aspect('equal')\n ax.set_xlim(-7, 7)\n ax.set_ylim(-6, 13) \n[/python]",
        "answer": "$235.87$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 471,
        "problem": "The height of the bottom solid figure is 33.1 mm.\n\nHence find the total surface area of the shape, correct to two decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n # Parameters\n diameter = 9 \n radius = diameter / 2\n height = 18  \n \n fig, ax = plt.subplots(figsize=(4, 6))\n \n theta = np.linspace(np.pi, 2 * np.pi, 100)\n x = radius * np.cos(theta)\n y = radius * 0.25 * np.sin(theta) - height \n \n ax.plot(x, y, color='teal', linewidth=2)\n \n top_y = 0\n top_ellipse = radius * np.cos(theta)\n top_ellipse_y = radius * 0.25 * np.sin(theta) + top_y\n \n ax.plot(top_ellipse, top_ellipse_y, color='teal', linewidth=2, alpha=1, solid_capstyle='round')\n \n theta = np.linspace(0, np.pi, 100)\n ax.plot(radius * np.cos(theta), \n         radius * 0.25 * np.sin(theta) - height, \n         '--', color='teal', alpha=0.5, linewidth=2) \n \n ax.plot(radius * np.cos(theta), \n         radius * 0.25 * np.sin(theta), \n         '--', color='teal', alpha=0.5, linewidth=2) \n \n ax.plot([radius, radius], [top_y, -height], color='teal', linewidth=2)\n ax.plot([-radius, -radius], [top_y, -height], color='teal', linewidth=2)\n \n theta2 = np.linspace(0, np.pi, 100)\n x2 = radius * np.cos(theta2)\n y2 = radius * np.sin(theta2)\n \n ax.plot(x2, y2, color='teal', linewidth=2)\n \n ax.annotate(\n     '', xy=(0, -height-2), xytext=(radius, -height-2),\n     arrowprops=dict(arrowstyle='<->', color='#43536D', linewidth=2)\n )\n ax.text(radius/2, -height-3, f\"10.5 mm\", ha='center', va='top', fontsize=14, color=\"#43536D\")\n \n ax.plot([radius+1.2, radius+1.2], [-height, 0], color='#43536D', linewidth=2)\n ax.plot([radius+0.8, radius+1.6], [0, 0], color='#43536D', linewidth=2)\n ax.plot([radius+0.8, radius+1.6], [-height, -height], color='#43536D', linewidth=2)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]",
        "answer": "$3222.80$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 472,
        "problem": "A solid figure has one side of length 14cm as shown.\n\nIf the size of \\angle DFH is \\theta \u00b0, find theta to two decimal places.",
        "geo_code": "[python]\n import numpy as np\n import matplotlib.pyplot as plt\n from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n \n length = 9 \n width = 4\n height = 4\n \n C = np.array([0, width, height])\n B = np.array([length, width, height])\n A = np.array([length, 0, height])\n D = np.array([0, 0, height])\n F = np.array([length, width, 0])\n E = np.array([length, 0, 0])\n H = np.array([0, 0, 0])\n G = np.array([0, width, 0])\n \n verts = [A, B, C, D, E, F, G, H]\n \n fig = plt.figure(figsize=(8,6))\n ax = fig.add_subplot(111, projection='3d')\n \n for point, name in zip(verts, ['A','B','C','D','E','F','G','H']):\n     ax.scatter(*point, s=40)\n     ax.text(*(point + np.array([0.2,0.2,0.2])), name, fontsize=12, color='k')\n \n edges = [\n     (C, B), (B, A), (A, D), (D, C),\n     (E, F), (E, H), (G, H), (G, F),\n     (C, G), (B, F), (A, E), (D, H) \n ]\n for edge in edges:\n     ax.plot(*zip(*edge), color='k')\n \n ax.plot(*zip(*[D, F]), 'r--')\n ax.plot(*zip(*[H, F]), 'g--')\n ax.plot(*zip(*[E, F]), 'b--')\n \n arc_radius = 0.5\n angle = np.arctan(width/length)\n angle_values = np.linspace(0, angle, 30)\n arc_x = length - arc_radius * np.cos(angle_values)\n arc_y = width - arc_radius * np.sin(angle_values)\n arc_z = np.zeros_like(arc_x)\n ax.plot(arc_x, arc_y, arc_z, 'b')\n \n ax.text(length-0.3, width-1, 0, r'$\\theta$', color='b', fontsize=16)\n \n ax.set_xlim([-.5, length+1])\n ax.set_ylim([-.5, width+1])\n ax.set_zlim([-.5, height+1])\n \n ax.text(0, width, height/2, '3 cm', color='k', fontsize=12)\n ax.text(-0.5, width/3, height+0.3, '3 cm', color='k', fontsize=12)\n ax.text(length/2, 1.5, -0.5, 'z cm', color='k', fontsize=12)\n \n ax.set_box_aspect([2,1,1])\n ax.axis('off')\n plt.tight_layout() \n[/python]",
        "answer": "$11.83$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "angle"
    },
    {
        "index": 473,
        "problem": "All edges of the following solid are 7 cm long.Find z, the size of \\angle AGH, correct to 2 decimal places.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n \n fig, ax = plt.subplots(figsize=(4, 3))\n \n \n O = [0, 0]\n A = [-5.1, 4.1]\n B = [5.1, 4.1]\n C = [5.1, -4.1]\n D = [-5.1, -4.1]\n E = [2, 6]\n F = [12.2, 6]\n G = [12.2, -2.2]\n H = [2, -2.2]\n \n ax.plot([A[0], B[0]], [A[1], B[1]], color='black')\n ax.plot([B[0], C[0]], [B[1], C[1]], color='black')\n ax.plot([C[0], D[0]], [C[1], D[1]], color='black')\n ax.plot([D[0], A[0]], [D[1], A[1]], color='black')\n \n ax.plot([E[0], F[0]], [E[1], F[1]], color='black')\n ax.plot([F[0], G[0]], [F[1], G[1]], color='black')\n ax.plot([G[0], C[0]], [G[1], C[1]], color='black')\n \n ax.plot([A[0], E[0]], [A[1], E[1]], color='black')\n ax.plot([B[0], F[0]], [B[1], F[1]], color='black')\n \n for X, Y in [(H, E), (H, D), (H, G)]:\n     ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n ax.text(A[0]-1, A[1], r\"$D$\", fontsize=12, style='italic')\n ax.text(B[0]-0.5, B[1]+0.5, r\"$C$\", fontsize=12, style='italic')\n ax.text(C[0], C[1]-1, r\"$G$\", fontsize=12, style='italic')\n ax.text(D[0], D[1]-1, r\"$H$\", fontsize=12, style='italic')\n ax.text(E[0], E[1]+0.5, r\"$A$\", fontsize=12, style='italic')\n ax.text(F[0], F[1]+0.5, r\"$B$\", fontsize=12, style='italic')\n ax.text(G[0], G[1]-1, r\"$F$\", fontsize=12, style='italic')\n ax.text(H[0]-1, H[1], r\"$E$\", fontsize=12, style='italic')\n \n ax.axis('off')\n plt.tight_layout()\n plt.show() \n[/python]",
        "answer": "$54.74$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "angle"
    },
    {
        "index": 474,
        "problem": "Find the length of AF, where $AB=2, BC=3, CD=6$.",
        "geo_code": "[python]\n import matplotlib.pyplot as plt\n from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n import numpy as np\n \n A = np.array([0, 0, 0])\n B = np.array([1, 0, 0])\n C = np.array([1, 1, 0])\n D = np.array([0, 1, 0])\n E = np.array([1, 1, 1.5])\n F = np.array([0, 1, 1.5])\n \n fig = plt.figure()\n ax = fig.add_subplot(111, projection='3d')\n \n verts = [A, B, C, D, E, F]\n \n edges = [\n     [A, B], [B, C],\n     [E, F], [A, E],\n     [A, F], [B, E], [C, E]\n ]\n for edge in edges:\n     ax.plot(*zip(*edge), color='black', linestyle='-', linewidth=1)\n \n edges = [[A,D], [C, D], [F, D]]\n for edge in edges:\n     ax.plot(*zip(*edge), color='black', linestyle='--', linewidth=1)\n \n ax.plot(*zip(A, E), color='red', linewidth=2)\n \n labels = ['A', 'B', 'D', 'C', 'E', 'F']\n label_indices = [0, 1, 3, 2, 4, 5]\n for label, idx in zip(labels, label_indices):\n     ax.text(*(verts[idx] + 0.05), label, fontsize=12)\n \n ax.set_box_aspect([1,1,1.5])\n ax.view_init(20, -30)\n ax.axis('off') \n[/python]",
        "answer": "$7$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "length"
    },
    {
        "index": 475,
        "problem": "The solid is 3 cm thick. Calculate the volume of the solid, correct to one decimal places.",
        "geo_code": "[python]\n\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(4,4))\n\ntheta1 = 24\ntheta2 = 66\nr = 9\n\nangle = np.deg2rad(np.linspace(theta1, theta2, 200))\nx_arc = r * np.cos(angle)\ny_arc = r * np.sin(angle)\n\nax.plot(x_arc, y_arc, color=\"black\")\n\nax.plot([0, x_arc[0]], [0, y_arc[0]], color=\"black\")\nax.plot([0, x_arc[-1]], [0, y_arc[-1]], color=\"black\")\nax.plot([0, 0], [0, -0.3], color=\"black\")\nax.plot([x_arc[0], x_arc[0]], [y_arc[0], y_arc[0]-0.3], color=\"black\")\nax.plot([0, x_arc[0]], [-0.3, y_arc[0]-0.3], color=\"black\")\n\n\nax.text(1.2, 1.3, r\"$42^\\circ$\", fontsize=16)\n\narc_mid = 0\nx_label = (r+0.2)*np.cos(np.deg2rad(arc_mid))/2\ny_label = (r+0.2)*np.sin(np.deg2rad(arc_mid)) +1\nax.text(x_label, y_label-0.4, \"9\", fontsize=16, ha='center')\n\nax.set_aspect('equal')\nax.set_xlim(-1.5, r+1)\nax.set_ylim(-1.5, r+2)\nax.axis('off')\n\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$89.1$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 476,
        "problem": "Find the volume of the figure shown, correct to two decimal places. The height of both solid figures is 2 cm.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nmain = patches.Arc((0, 0), 18, 4, theta1=180, theta2=360, color='black', linewidth=2)\nax.add_patch(main)\n\ntop_main = patches.Arc((0, 3), 18, 4, theta1=-204, theta2=24, color='black', linewidth=2)\nax.add_patch(top_main)\n\nsmall = patches.Arc((0, 3), 8, 2, theta1=180, theta2=360, color='black', linewidth=2)\nax.add_patch(small)\n\nax.plot([-4, -4], [3, 7], color='black', linewidth=2)\nax.plot([4, 4], [3, 7], color='black', linewidth=2)\n\n\nax.plot([-9, -9], [0, 3], color='black', linewidth=2)\nax.plot([9, 9], [0, 3], color='black', linewidth=2)\n\ntop_small = patches.Ellipse((0, 7), 8, 2, fill=False, color='black', linewidth=2)\nax.add_patch(top_small)\n\nax.plot([0, 9], [0, 0], color='black')\nax.plot([0, 0], [0, 0], 'ko')\nax.text(4.5, 0.1, \"$\\\\bf{9\\\\ cm}$\", fontsize=14, style='italic')\n\nax.plot([0, 4], [7, 7], color='black')\nax.plot([0, 0], [7, 7], 'ko')\nax.text(2, 7.3, \"$\\\\bf{4\\\\ cm}$\", fontsize=14, style='italic')\n\nax.set_aspect('equal')\nax.set_xlim(-12, 15)\nax.set_ylim(-4, 12)\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$609.47$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 477,
        "problem": "Find the volume of the figure shown. The middle solid figure has a diameter of 3 cm and height of 10 cm.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\nfrom matplotlib.patches import Arrow\n\nfig, ax = plt.subplots(figsize=(6, 6))\n\nellipse_width = 8.1\nellipse_height = 1\nrect_height = 2.1\nsmall_cylinder_width = 3.1\nsmall_ellipse_height = 0.8\nsmall_cylinder_height = 6\n\ntop_center = (0, small_cylinder_height/2 + rect_height)\ntop_ellipse = patches.Arc(\n    top_center, ellipse_width, ellipse_height*2, \n    angle=0, theta1=0, theta2=360, linewidth=1.5, color='black'\n)\nax.plot([0, ellipse_width / 2], [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height], color='black')\nax.text(ellipse_width / 4, small_cylinder_height/2 + rect_height + 0.1, \"$\\\\bf{8\\\\ cm}$\", fontsize=8)\nax.add_patch(top_ellipse)\n\nax.add_line(plt.Line2D(\n    [ellipse_width/2, ellipse_width/2], \n    [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height - rect_height],\n    color='black'\n))\nax.add_line(plt.Line2D(\n    [-ellipse_width/2, -ellipse_width/2], \n    [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height - rect_height],\n    color='black'\n))\nax.plot([ellipse_width/2 + 1, ellipse_width/2 + 1], [small_cylinder_height/2 + rect_height, -small_cylinder_height/2 - rect_height], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [small_cylinder_height/2, small_cylinder_height/2], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [-small_cylinder_height/2 - rect_height, -small_cylinder_height/2 - rect_height], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [-small_cylinder_height/2, -small_cylinder_height/2], color='black')\nax.text(ellipse_width/2 + 1.2, (small_cylinder_height + rect_height)/2, \"$\\\\bf{2\\\\ cm}$\", fontsize=8)\nax.text(ellipse_width/2 + 1.2, -(small_cylinder_height + rect_height)/2, \"$\\\\bf{2\\\\ cm}$\", fontsize=8)\n\narrow1 = Arrow(\n    x=0, y=-rect_height/2,\n    dx=small_cylinder_width/2, dy=0,\n    width=0.2,\n    color='black'\n)\narrow2 = Arrow(\n    x=0, y=-rect_height/2,\n    dx=-small_cylinder_width/2, dy=0,\n    width=0.2,\n    color='black'\n)\nax.add_patch(arrow1)\nax.add_patch(arrow2)\nbottom_center = (0, -small_cylinder_height/2 - rect_height)\nbottom_ellipse = patches.Arc(\n    bottom_center, ellipse_width, ellipse_height*2, \n    angle=0, theta1=180, theta2=360, linewidth=1.5, color='black'\n)\nax.add_patch(bottom_ellipse)\n\nax.add_line(plt.Line2D(\n    [ellipse_width/2, ellipse_width/2], \n    [-small_cylinder_height/2 - rect_height, -small_cylinder_height/2],\n    color='black'\n))\nax.add_line(plt.Line2D(\n    [-ellipse_width/2, -ellipse_width/2], \n    [-small_cylinder_height/2 - rect_height, -small_cylinder_height/2],\n    color='black'\n))\n\ntop_inner_ellipse = patches.Arc(\n    (0, small_cylinder_height/2), ellipse_width, ellipse_height*2,\n    angle=0, theta1=180, theta2=360, linewidth=1.2, color='black'\n)\nax.plot([0, ellipse_width / 2], [-small_cylinder_height/2 - rect_height - 1.3, -small_cylinder_height/2 - rect_height - 1.3], color='black')\nax.text(ellipse_width / 4, -small_cylinder_height/2 - rect_height - 1.7, \"$\\\\bf{8\\\\ cm}$\", fontsize=8)\nax.plot([0, 0], [-small_cylinder_height/2 - rect_height - 1.2, -small_cylinder_height/2 - rect_height - 1.4], color='black')\nax.plot([ellipse_width / 2, ellipse_width / 2], [-small_cylinder_height/2 - rect_height - 1.2, -small_cylinder_height/2 - rect_height - 1.4], color='black')\nax.add_patch(top_ellipse)\nax.add_patch(top_inner_ellipse)\n\nbottom_inner_ellipse = patches.Arc(\n    (0, -small_cylinder_height/2), ellipse_width, ellipse_height*2, theta1=-211, theta2=31, linewidth=1.2, color='black'\n)\nax.add_patch(bottom_inner_ellipse)\n\nax.add_line(plt.Line2D(\n    [small_cylinder_width/2, small_cylinder_width/2], \n    [-small_cylinder_height/2, small_cylinder_height/2 - 1],\n    color='black'\n))\nax.add_line(plt.Line2D(\n    [-small_cylinder_width/2, -small_cylinder_width/2], \n    [-small_cylinder_height/2, small_cylinder_height/2 - 1],\n    color='black'\n))\n\nmid_top = (0, small_cylinder_height/2)\nmid_bottom = (0, -small_cylinder_height/2)\n\nmid_ellipse_bottom =  patches.Arc(\n    mid_bottom, small_cylinder_width, small_ellipse_height, theta1=180, theta2=360, color='black'\n)\nax.add_patch(mid_ellipse_bottom)\n\nax.set_xlim(-6, 6)\nax.set_ylim(-10, 7)\nax.set_aspect('equal')\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$874.93$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 478,
        "problem": "The radius of hole is 2cm. Find the volume of the solid, correct to two decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nO = [0, 0]\nA = [-5, 4]\nB = [5, 4]\nC = [5, -4]\nD = [-5, -4]\nE = [2, 10]\nF = [12, 10]\nG = [12, 2]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.text(B[0] + 0.5, 0, \"$\\\\bf{8\\\\ cm}$\", fontsize=11)\nax.plot([C[0], D[0]], [C[1], D[1]], color='black')\nax.plot([D[0], A[0]], [D[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\nax.plot([F[0], G[0]], [F[1], G[1]], color='black')\nax.plot([G[0], C[0]], [G[1], C[1]], color='black')\nax.text((C[0] + G[0]) / 2 + 0.5, (C[1] + G[1]) / 2 - 0.5, \"$\\\\bf{14\\\\ cm}$\", fontsize=11)\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([B[0], F[0]], [B[1], F[1]], color='black')\n\n\ntop_small = patches.Circle((0, 0), 2, fill=False, color='black', linewidth=2)\nax.plot([0, 2], [0, 0], color='black')\nax.add_patch(top_small)\n\nax.text(0 - 1.3, -A[1] - 1, \"$\\\\bf{10\\\\ cm}$\", fontsize=11)\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$944.07$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 479,
        "problem": "Find the total surface area of the solid with a side length equal to 6 cm.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nO = [0, 0]\nA = [-5.1, 4.1]\nB = [5.1, 4.1]\nC = [5.1, -4.1]\nD = [-5.1, -4.1]\nE = [2, 6]\nF = [12.2, 6]\nG = [12.2, -2.2]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], D[0]], [C[1], D[1]], color='black')\nax.plot([D[0], A[0]], [D[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\nax.plot([F[0], G[0]], [F[1], G[1]], color='black')\nax.plot([G[0], C[0]], [G[1], C[1]], color='black')\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([B[0], F[0]], [B[1], F[1]], color='black')\n\nax.plot([0, 0], [-3.8, -4.4], color='black')\n\nax.plot([0, 0], [3.8, 4.4], color='black')\nax.plot([0, 0], [-3.8, -4.4], color='black')\nax.plot([7.1, 7.1], [5.7, 6.3], color='black')\n\nax.plot([-4.8, -5.4], [0, 0], color='black')\nax.plot([11.9, 12.4], [2, 2], color='black')\nax.plot([4.8, 5.4], [0, 0], color='black')\nax.plot([-1.6, -1.3], [5.40, 4.80], color='black')\nax.plot([8.6, 8.8], [-2.8, -3.4], color='black')\nax.text(0 - 1.3, -A[1] - 1, \"$\\\\bf{6\\\\ cm}$\", fontsize=11)\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$216$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 480,
        "problem": "Find the surface area of the solid figure with length equal to 12 m.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nO = [0, 0]\nA = [-5.1, 3.1]\nB = [5.1, 3.1]\nC = [5.1, -3.1]\nD = [-5.1, -3.1]\nE = [5, 10]\nF = [15.2, 10]\nG = [15.2, 3.8]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], D[0]], [C[1], D[1]], color='black')\nax.plot([D[0], A[0]], [D[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\nax.plot([F[0], G[0]], [F[1], G[1]], color='black')\nax.plot([G[0], C[0]], [G[1], C[1]], color='black')\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([B[0], F[0]], [B[1], F[1]], color='black')\n\nax.text(0 - 1.3, -A[1] - 1, \"$\\\\bf{6\\\\ m}$\", fontsize=11)\nax.text(A[0] - 4, 0, \"$\\\\bf{4\\\\ m}$\", fontsize=11)\n\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$288$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 481,
        "problem": "Find the surface area of the solid figure shown. The height is 21 cm.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nA = [0, 4.8]\nB = [-2.5, 0]\nC = [2.5, 0]\nE = [10, 10]\nF = [12.5, 5.2]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], A[0]], [C[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\n\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.text(-1, -1, \"$\\\\bf{12\\\\ cm}$\", fontsize=11)\nax.text(1.7, 2.4, \"$\\\\bf{10\\\\ cm}$\", fontsize=11)\nax.plot([B[0] - 0.5, B[0] - 0.5], [A[1], 0], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [A[1], A[1]], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [0, 0], color='black')\n\nax.plot([0, 0.5, 0.5], [0.5, 0.5, 0], color='black')\n\n\nax.plot([0, 0], [A[1], 0], color='black', linestyle='--')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\nax.plot([B[0]/2 - 0.4, B[0]/2 + 0.4], [2.6, 2.3], color='black')\nax.plot([C[0]/2 - 0.4, C[0]/2 + 0.4], [2.3, 2.6], color='black')\n\nax.text(B[0] - 3, 2.4, \"$\\\\bf{8\\\\ cm}$\", fontsize=11)\n\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$768$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 482,
        "problem": "Find the surface area of the solid figure shown. The height is 20 cm.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(5,7))\n\nA = np.array([0, 0])\nB = np.array([10, 0])\nC = np.array([3.6, -4.8])\n\nA0 = C * 0.9\nB0 = A0 + 0.1* (B - C)\nC0 = C + 0.1* (B - C)\n\n\ndy = 12\nA1 = A + np.array([0, dy])\nB1 = B + np.array([0, dy])\nC1 = C + np.array([0, dy])\n\nA10 = A0 + np.array([0, dy])\nB10 = B0 + np.array([0, dy])\nC10 = C0 + np.array([0, dy])\n\nfor p1, p2 in [(B,C), (C,A),\n               (A1,B1), (B1,C1), (C1,A1),\n               (A,A1), (B,B1), (C,C1),\n               (A0, B0), (B0, C0), (A10, B10), (B10, C10)]:\n    ax.plot([p1[0], p2[0]],[p1[1], p2[1]],'k', lw=1.5)\n\nax.plot([A[0], B[0]], [A[1], B[1]], 'k--')\nax.text(1, -2.5, r'$6\\,cm$', fontsize=14, ha='center', va='top', style='italic')\nax.text(4, dy+0.45, r'$10\\,cm$', fontsize=14, ha='left', va='center', style='italic')\nax.text(9, -2.65, r'$8\\,cm$', fontsize=14, ha='right', va='center', style='italic')\n\nax.axis('equal')\nax.axis('off')\nplt.show()[/python]",
        "answer": "$528$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 483,
        "problem": "Find the surface area of the solid prism. The bottom face has edges of 18cm and 8cm.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(8,4))\n\nA = np.array([0, 0])\nB = np.array([14, 0])\nC = np.array([12, 3])\nD = np.array([2, 3])\n\nw = 5\nA1 = A + np.array([w, w])\nB1 = B + np.array([w, w])\nC1 = C + np.array([w, w])\nD1 = D + np.array([w, w])\n\nfront = np.array([A,B,C,D,A])\nax.plot(front[:,0], front[:,1], 'k', lw=3)\nback = np.array([B1,C1,D1])\nax.plot(back[:,0], back[:,1], 'k', lw=3)\nax.plot([B[0],B1[0]], [B[1],B1[1]], 'k', lw=3)\nax.plot([C[0],C1[0]], [C[1],C1[1]], 'k', lw=3)\nax.plot([D[0],D1[0]], [D[1],D1[1]], 'k', lw=3)\n\nax.plot([7,7],[0,3],'k--', lw=2)\nax.text(7.3,1.1,'3 cm',fontsize=13, va='center')\n\nax.text(7, 3.5, '10 cm', fontsize=13, ha='center', va='bottom')\nax.text(0, 2.2, '5 cm', fontsize=13, ha='right', va='center')\nax.text(18, 7, '5 cm', fontsize=13, ha='left', va='center')\n\nax.set_aspect('equal')\nax.axis('off')\nax.set_xlim(-2,20)\nax.set_ylim(-1,10)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$338$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 484,
        "problem": "Find the surface area of this solid. The thickness of the solid is 3cm.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\n\ndx, dy = -3, 3\n\nA = np.array([0, 0])\nB = np.array([w1, 0])\nC = np.array([w1-w2, h1])\nD = np.array([0, h1])\n\nE = np.array([w1, h2])\nF = np.array([w1-w2, h2])\n\nA_ = A + np.array([dx, dy])\nB_ = B + np.array([dx, dy])\nC_ = C + np.array([dx, dy])\nD_ = D + np.array([dx, dy])\nF_ = F + np.array([dx, dy])\nE_ = E + np.array([dx, dy])\n\nax = plt.gca()\nfor p1, p2 in [(A,B),(B,E),(E,F),(F,C), (C, D), (D, A)]:\n    ax.plot([p1[0], p2[0]], [p1[1], p2[1]], color='k')\n\nfor p1, p2 in [(A,A_),(C,C_),(D,D_), (E,E_), (F,F_), (A_, D_), (D_, C_), (F_, C_), (E_, F_)]:\n    ax.plot([p1[0],p2[0]], [p1[1],p2[1]], color='k')\n\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='k')\n\nax.text(-4, h1/2+dy, '8cm', va='center', ha='right', fontsize=13, fontstyle='italic')\nax.text(w1/2, -1.4, '17cm', va='top', ha='center', fontsize=13, fontstyle='italic')\nax.text(w1 + 1, h2/2, '16cm', va='center', ha='left', fontsize=13, fontstyle='italic')\nax.text(w1-w2/2+dx, h2+dy+0.6, '4cm', va='bottom', ha='center', fontsize=13, fontstyle='italic')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-6, 28)\nplt.ylim(-4, 22)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$534$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 485,
        "problem": "Find the surface area of the following solid prism. The right rectangle surface has one edge of 10cm.\n\nGive your answer to the nearest one decimal place.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(7,7))\n\nheight = 10\ndepth = 3\nwidth = 13\n\ndx, dy = 5, 3.5\n\nA = np.array([0,0])\nB = np.array([width,0])\nC = np.array([width, depth])\nD = np.array([0,height])\n\nA1 = A + np.array([dx,dy])\nB1 = B + np.array([dx,dy])\nC1 = C + np.array([dx,dy])\nD1 = D + np.array([dx,dy])\n\n\nax.plot([A[0],B[0]],[A[1],B[1]],'k')\nax.plot([B[0],C[0]],[B[1],C[1]],'k')\nax.plot([C[0],D[0]],[C[1],D[1]],'k')\nax.plot([D[0],A[0]],[D[1],A[1]],'k')\nax.plot([A1[0],B1[0]],[A1[1],B1[1]],'k--')\nax.plot([B1[0],C1[0]],[B1[1],C1[1]],'k')\nax.plot([B1[0],B[0]],[B1[1],B[1]],'k')\nax.plot([C1[0],D1[0]],[C1[1],D1[1]],'k')\nax.plot([D1[0],A1[0]],[D1[1],A1[1]],'k--')\nax.plot([D[0],D1[0]],[D[1],D1[1]],'k')\nax.plot([C[0],C1[0]], [C[1],C1[1]], 'k')\n\nax.plot([A[0], A1[0]], [A[1], A1[1]], 'k--')\n\nax.text(width/2-0.8, -1.1, r'$13\\,cm$', fontsize=15, style='italic')\nax.text(-1.0, height/2, r'$10\\,cm$', fontsize=15, style='italic', va='center', ha='right')\nax.text(width+dx+0.5, dy+depth/2, r'$3\\,cm$', fontsize=15, style='italic', va='center', ha='left')\nax.text((A1[0]+B1[0])/2+0.5, (C1[1]+D1[1])/2+1, r'$y\\,cm$', fontsize=15, style='italic', ha='center')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-4, width+dx+depth+2)\nplt.ylim(-3, max(height,dy+height)+depth+2)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$577.0$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 486,
        "problem": "Find the total surface area of the solid figure shown.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nA = [0, 4.8]\nB = [-2.5, 0]\nC = [2.5, 0]\nE = [10, 10]\nF = [12.5, 5.2]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], A[0]], [C[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\n\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.text(-1, -1, \"$\\\\bf{12\\\\ cm}$\", fontsize=11)\nax.text(1.7, 2.4, \"$\\\\bf{10\\\\ cm}$\", fontsize=11)\nax.plot([B[0] - 0.5, B[0] - 0.5], [A[1], 0], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [A[1], A[1]], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [0, 0], color='black')\n\nax.plot([0, 0.5, 0.5], [0.5, 0.5, 0], color='black')\n\n\nax.plot([0, 0], [A[1], 0], color='black', linestyle='--')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\nax.plot([B[0]/2 - 0.4, B[0]/2 + 0.4], [2.6, 2.3], color='black')\nax.plot([C[0]/2 - 0.4, C[0]/2 + 0.4], [2.3, 2.6], color='black')\n\nax.text(B[0] - 3, 2.4, \"$\\\\bf{x\\\\ cm}$\", fontsize=11)\n\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$608$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 487,
        "problem": "Find the surface area of the figure shown. The height is 3 cm.\n\nGive your answer to the nearest two decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(7,7))\n\nheight = 10\nwidth = 15\nwidth1 = 8\n\ndx, dy = 5, 3.5\n\nA = np.array([0,0])\nB = np.array([width,0])\nC = np.array([width1, height])\nD = np.array([0, height])\n\nA1 = A + np.array([dx,dy])\nB1 = B + np.array([dx,dy])\nC1 = C + np.array([dx,dy])\nD1 = D + np.array([dx,dy])\n\n\nax.plot([A[0],B[0]],[A[1],B[1]],'k')\nax.plot([B[0],C[0]],[B[1],C[1]],'k')\nax.plot([C[0],D[0]],[C[1],D[1]],'k')\nax.plot([D[0],A[0]],[D[1],A[1]],'k')\nax.plot([B1[0],C1[0]],[B1[1],C1[1]],'k')\nax.plot([B1[0],B[0]],[B1[1],B[1]],'k')\nax.plot([C1[0],D1[0]],[C1[1],D1[1]],'k')\nax.plot([D[0],D1[0]],[D[1],D1[1]],'k')\nax.plot([C[0],C1[0]], [C[1],C1[1]], 'k')\n\n\nax.text(width/2-0.8, -1.1, r'$12\\,cm$', fontsize=15, style='italic')\nax.text(-1.0, height/2, r'$5\\,cm$', fontsize=15, style='italic', va='center', ha='right')\nax.text((A[0]+C[0])/2+0.5, C[1]+0.5, r'$9\\,cm$', fontsize=15, style='italic', ha='center')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-4, width+dx+depth+2)\nplt.ylim(-3, max(height,dy+height)+depth+2)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$200.49$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 488,
        "problem": "Find the surface area of the figure shown.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(8,6))\n\nbar_width = 16\nbar_depth = 3\nbar_height = 3\nvert_width = 4\nvert_height = 7\nvert_depth = 1\n\ndx, dy = 2, 1.2\n\nA = np.array([0, 0])\nB = np.array([bar_width, 0])\nC = np.array([bar_width, bar_height])\nD = np.array([0, bar_height])\n\nA1 = A + np.array([dx, dy])\nB1 = B + np.array([dx, dy])\nC1 = C + np.array([dx, dy])\nD1 = D + np.array([dx, dy])\n\nleft_margin = (bar_width - vert_width) / 2\nright_margin = left_margin + vert_width\n\nE = np.array([left_margin, 0])\nF = E + np.array([vert_width, 0])\nG = F + np.array([0, -vert_height])\nH = E + np.array([0, -vert_height])\n\nE1 = E + np.array([dx, dy])\nF1 = F + np.array([dx, dy])\nG1 = G + np.array([dx, dy])\nH1 = H + np.array([dx, dy])\n\nfor X, Y in [(A,B), (B,C), (C,D), (D,A), (B,B1), (C,C1), (D,D1), (B1,C1), (C1,D1)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(F,G),(G,H),(H,E),(G,G1)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nax.plot([F1[0],G1[0]], [F1[1]-1.3,G1[1]], 'k') \n\n\nax.text(bar_width/2, 2.5, r\"$12\\,cm$\", ha='center', va='top', fontsize=14, style='italic')\nax.text(-0.7, bar_height/2, r\"$1\\,cm$\", ha='right', va='center', fontsize=14, style='italic')\nax.text(bar_width+dx-1, bar_height/2-dy-0.5, r\"$1\\,cm$\", ha='left', va='center', fontsize=14, style='italic')\nax.text((left_margin+vert_width/2 + 4), -vert_height-0.5, r\"$1\\,cm$\",  ha='center', va='bottom', fontsize=14, style='italic')\n\nax.text((left_margin+vert_width/2), -vert_height-1, r\"$4\\,cm$\",  ha='center', va='bottom', fontsize=14, style='italic')\nax.text(left_margin-0.2, -vert_height/2, r\"$7\\,cm$\", ha='right', va='center', fontsize=14, style='italic')\n\nax.plot([bar_width/6, bar_width/6], [0.35, -0.35], 'k')\nax.plot([bar_width*5/6, bar_width*5/6], [0.35, -0.35], 'k')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-2, 20)\nplt.ylim(-15, 10)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$120$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 489,
        "problem": "Find the surface area of the solid figure. The height is 6 cm.\n\nRound your answer to one decimal place.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(5,4))\n\na = 10\nh = a * np.sqrt(3) / 2\n\nO = np.array([0, 0])\nE = np.array([0, 10])\nA = np.array([-a/2 - h/2, -h/2])\nB = np.array([a - h/2, -h/2])\n\ndx, dy = h/2, h\nA1 = A + np.array([dx, dy])\nB1 = B + np.array([dx, dy])\n\n\nfor X, Y in [(A, A1), (B,B1), (A,B), (A,E), (A1,E), (B,E), (B1,E)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\n\nax.plot([E[0],0],[E[1],0], 'k--')\nax.plot([A1[0],B1[0]],[A1[1],B1[1]], 'k--')\n\n\nax.text((B[0]+B1[0])/2+0.5, 0, r\"$10\\,cm$\", fontsize=12, style='italic')\nax.plot([(A[0]+B[0])/2, (A[0]+B[0])/2], [A[1]-0.3, A[1]+0.3], 'k')\nax.plot([(A1[0]+B1[0])/2, (A1[0]+B1[0])/2], [A1[1]-0.3, A1[1]+0.3], 'k')\nax.plot([(A[0]+A1[0])/2-0.25, (A[0]+A1[0])/2+0.25], [0.20, -0.20], 'k', lw=2)\nax.plot([(B[0]+B1[0])/2-0.25, (B[0]+B1[0])/2+0.25], [0.20, -0.20], 'k', lw=2)\n\nax.plot([0, 1, 1], [1, 1, 0], color='black')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-a, a)\nplt.ylim(-a, h+2)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$256.2$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 490,
        "problem": "Find the surface area of the solid shown. The radius is 3cm.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(4,3))\n\n\nellipse = patches.Ellipse((0, 0), 16, 6, fill=False, color='black', linewidth=2)\nax.add_patch(ellipse)\nax.plot([-8, 0, 8], [0, 14, 0], color='black')\nax.plot([0, 8], [0, 0], color='black')\n\nax.text(4.5, 7.5, r\"$10\\,cm$\", fontsize=12, style='italic')\n\nax.set_aspect('equal')\nax.set_xlim(-12, 12)\nax.set_ylim(-4, 18)\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$122.52$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 491,
        "problem": "The solid figure has a diameter of 4cm. Find the surface area of the solid formed, correct to two decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(5,5))\n\n\nwidth = 20\nheight = 4\nellipse1 = patches.Ellipse((0, 0), width, height, fill=False, color='black', linewidth=2)\nellipse2 = patches.Ellipse((0, 10), width/2, height/2, fill=False, color='black', linewidth=2)\nax.add_patch(ellipse1)\nax.add_patch(ellipse2)\nax.plot([-width/2, -width/4, width/4, width/2, -width/2], [0, 10, 10, 0, 0], color='black')\nax.plot([0, 8], [0, 0], color='black')\nax.plot([-width/4, 0, width/4], [10, 20, 10], 'k--', color='black')\nax.plot([1, width/2 + 1.5], [21, 0.5], color='black')\nax.plot([0.5, 1.5], [20.75, 21.5], color='black')\nax.plot([width/4 + 0.75,  width/4 + 1.75], [10.5, 11.25], color='black')\nax.plot([width/2 + 1, width/2 + 2], [0, 0.5], color='black')\n\nax.plot([0, 0], [10, 10], 'ko')\nax.plot([0, 0], [0, 0], 'ko')\n\nax.text(width/3 + 2.5, 5, r\"$8\\,cm$\", fontsize=12, style='italic')\nax.text(width/4 - 0.5, 15, r\"$8\\,cm$\", fontsize=12, style='italic')\nax.text(-1, 7.5, r\"$2\\,cm$\", fontsize=11, style='italic')\n\nax.set_aspect('equal')\nax.set_xlim(-width, width)\nax.set_ylim(-4, 30)\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$91.11$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 492,
        "problem": "Find the surface area of the solid figure. The perpendicular height is 8mm.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(5,4))\n\na = 11\nh = a * np.sqrt(3) / 2\n\nO = np.array([0, 0])\nE = np.array([0, 20])\nA = np.array([-a/2 - h/2, -h/2])\nB = np.array([a - h/2, -h/2])\n\ndx, dy = h/2, h\nA1 = A + np.array([dx, dy])\nB1 = B + np.array([dx, dy])\n\n\nfor X, Y in [(B,B1), (A,B), (A,E), (B,E), (B1,E)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\n\nax.plot([E[0],0],[E[1],0], 'k--')\nax.plot([A1[0],B1[0]],[A1[1],B1[1]], 'k--')\nax.plot([A[0], A1[0]], [A[1], A1[1]], 'k--')\nax.plot([A1[0], E[0]], [A1[1], E[1]], 'k--')\n\n\nax.text((B[0]+B1[0])/2+1, 0, r\"$10\\,mm$\", fontsize=11, style='italic')\nax.text(-dx + 1, A[1]-1.5, r\"$10\\,mm$\", fontsize=11, style='italic')\nax.text(0-4, E[1]/2-2, r\"$8\\,mm$\", fontsize=11, style='italic')\n\nax.plot([0, 1, 1], [1, 1, 0], color='black')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-a, a)\nplt.ylim(-a, 20)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$288.6$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 493,
        "problem": "Find the surface area of the solid figure given. The slant height is 13.7cm.\n\nGive your answer correct to 2 decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(5,4))\n\n\nellipse = patches.Ellipse((0, 0), 7, 20, fill=False, color='black', linewidth=2)\nax.add_patch(ellipse)\nE = np.array([-25, 0])\nax.plot([0, E[0], 0], [10, 0, -10], color='black', linewidth=2)\nax.plot([0, 0], [0, -10], 'k--', linewidth=2)\n\nax.text(-2, 1, r\"$9.1\\,cm$\", fontsize=10, style='italic')\nax.text(E[0]/2-3, 6.5, r\"$13.7\\,cm$\", fontsize=10, style='italic')\nax.plot([0, 0], [0, 0], 'ko')\n\nax.set_aspect('equal')\nax.set_xlim(-28, 5)\nax.set_ylim(-16, 18)\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$651.82$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 494,
        "problem": "Find the surface area of the solid figure shown.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\nh = 8\n\nA = np.array([0, 4.8])\nB = np.array([-2.5, 0])\nC = np.array([2.5, 0])\nE = np.array([10, 10])\nF = np.array([12.5, 5.2])\nB1 = B + np.array([0, -h])\nC1 = C + np.array([0, -h])\nF1 = F + np.array([0, -h])\n\n\nfor X, Y in [(B,B1), (C,C1), (B1,C1), (C1,F1), (F,F1)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k', color='black')\n\n\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], A[0]], [C[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\n\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.plot([B[0] - 0.5, B[0] - 0.5], [A[1], 0], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [A[1], A[1]], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [0, 0], color='black')\n\nax.plot([0, 0.5, 0.5], [0.5, 0.5, 0], color='black')\n\n\nax.plot([0, 0], [A[1], 0], color='black', linestyle='--')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\nax.plot([B[0]/2 - 0.4, B[0]/2 + 0.4], [2.6, 2.3], color='black')\nax.plot([C[0]/2 - 0.4, C[0]/2 + 0.4], [2.3, 2.6], color='black')\n\nax.text(B[0] - 3, 2.4, \"$\\\\bf{3\\\\ m}$\", fontsize=11)\nax.text(B[0] - 3, -h/2, \"$\\\\bf{6\\\\ m}$\", fontsize=11)\nax.text(-1, -h-1.5, \"$\\\\bf{5\\\\ m}$\", fontsize=11)\nax.text((C1[0] + F1[0])/2, (C1[1] + F1[1])/2-1, \"$\\\\bf{10\\\\ m}$\", fontsize=11)\n\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$323.10$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 495,
        "problem": "Find the surface area of the figure shown, rounded to two decimal places. The height is 12 cm.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nr = 9\nmain = patches.Arc(O, r, r, theta1=90, theta2=360, color='black', linewidth=2)\nax.add_patch(main)\n\ndx, dy = 12, 6\ndxy = np.array([dx, dy])\nO = np.array([0, 0])\nO1 = O + np.array([dx, dy])\nbackend1 = patches.Arc(O1, r, r, theta1=90, theta2=120, color='black', linewidth=2)\nbackend2 = patches.Arc(O1, r, r, theta1=120, theta2=300, color='black', linewidth=2, linestyle='--')\nbackend3 = patches.Arc(O1, r, r, theta1=300, theta2=360, color='black', linewidth=2)\nax.add_patch(backend1)\nax.add_patch(backend2)\nax.add_patch(backend3)\n\nA = [0, r/2]\nB = [r/2, 0]\nC = [-r/4, r*1.7/4]\nD = [r/4, -r*1.7/4]\n\nA1 = A + dxy\nB1 = B + dxy\nC1 = C + dxy\nD1 = D + dxy\n\nfor X, Y in [(O,O1), (A, O), (B, O), (A1, O1), (B1, O1), (A, A1), (B, B1), (C, C1), (D, D1)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\n\nax.text(1, -1, \"$\\\\bf{5\\\\ cm}$\", fontsize=10, style='italic')\n\nax.set_aspect('equal')\nax.set_xlim(-12, 20)\nax.set_ylim(-6, 12)\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$520.55$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 496,
        "problem": "The middle solid has radius of 6 and height of 11.\nFind the surface area of the solid.\n\nRound your answer to two decimal places.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(8,4))\n\nwidth, height = 15, 20\n\nA = np.array([0, 0])\nB = np.array([width, 0])\nD = A + np.array([0, height])\nC = B + np.array([0, height])\n\ndx, dy = -5, 6\ndxy = np.array([dx, dy])\nA1 = A + dxy\nB1 = B + dxy\nC1 = C + dxy\nD1 = D + dxy\n\nx0 = np.array([25, 0])\nA2 = A + x0\nB2 = B + x0\nC2 = C + x0\nD2 = D + x0\n\nA3, B3, C3, D3 = A1 + x0,  B1 + x0, C1 + x0, D1 + x0\n\nfor X, Y in [(A, B), (B, C), (C, D), (D, A)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(C1, D1), (D1, A1), (D, D1), (C, C1), (A, A1), (A2, A3)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(A2, B2), (B2, C2), (C2, D2), (D2, A2)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(C3, D3), (D3, A3), (D2, D3), (C2, C3)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(A1, B1), (B1, C1), (A3, B3), (B3, C3), (B, B1), (B2, B3)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k--')\n\nmain1 = patches.Arc(((B1[0] + B[0])/2, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2), 3, 13, theta1=0, theta2=360, color='black', linewidth=2, linestyle='--')\nax.add_patch(main1)\n\nmain2 = patches.Arc(((A2[0] + A3[0])/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2), 3, 13, theta1=270, theta2=90, color='black', linewidth=2)\nmain3 = patches.Arc(((A2[0] + A3[0])/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2), 3, 13, theta1=90, theta2=270, color='black', linewidth=2, linestyle='--')\nax.add_patch(main2)\nax.add_patch(main3)\n\nax.plot([(B1[0] + B[0])/2, (A2[0] + A3[0])/2], [((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 + 6.5, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 + 6.5], 'k')\nax.plot([(B1[0] + B[0])/2, (A2[0] + A3[0])/2], [((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 - 6.5, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 - 6.5], 'k')\n\nax.plot([B[0], A2[0]], [-2, -2], 'k')\nax.plot([B[0], B[0]], [-1, -3], 'k')\nax.plot([A2[0], A2[0]], [-1, -3], 'k')\nax.plot([(A2[0] + A3[0])/2, (A2[0] + A3[0])/2], [((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2+6.5], 'k')\n\nax.text(B2[0] + 1, C[1]/2, r\"$20\\,cm$\", fontsize=12, style='italic')\nax.text((B2[0] + A2[0])/2, -2, r\"$25\\,cm$\", fontsize=12, style='italic')\nax.text((B2[0] + B3[0])/2, (C3[1] + C2[1])/2, r\"$33\\,cm$\", fontsize=12, style='italic')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-8, 60)\nplt.ylim(-5, 30)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$8128.50$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "area"
    },
    {
        "index": 497,
        "problem": "Given the following solid. The length of the base is 10cm.\nFind the volume of the solid.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(5,4))\n\na = 10\nh = a * np.sqrt(3) / 2\n\nO = np.array([0, 0])\nE = np.array([0, 10])\nA = np.array([-a/2 - h/2, -h/2])\nB = np.array([a - h/2, -h/2])\n\ndx, dy = h/2, h\nA1 = A + np.array([dx, dy])\nB1 = B + np.array([dx, dy])\n\n\nfor X, Y in [(A, A1), (B,B1), (A,B), (A,E), (A1,E), (B,E), (B1,E)]:\n    ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\n\nax.plot([E[0],0],[E[1],0], 'k--')\nax.plot([A1[0],B1[0]],[A1[1],B1[1]], 'k--')\n\n\nax.text((B[0]+B1[0])/2+0.5, 0, r\"$10\\,cm$\", fontsize=12, style='italic')\nax.text(-3, 2, r\"$6\\,cm$\", fontsize=12, style='italic')\nax.plot([(A[0]+B[0])/2, (A[0]+B[0])/2], [A[1]-0.3, A[1]+0.3], 'k')\nax.plot([(A1[0]+B1[0])/2, (A1[0]+B1[0])/2], [A1[1]-0.3, A1[1]+0.3], 'k')\nax.plot([(A[0]+A1[0])/2-0.25, (A[0]+A1[0])/2+0.25], [0.20, -0.20], 'k', lw=2)\nax.plot([(B[0]+B1[0])/2-0.25, (B[0]+B1[0])/2+0.25], [0.20, -0.20], 'k', lw=2)\n\nax.plot([0, 1, 1], [1, 1, 0], color='black')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-a, a)\nplt.ylim(-a, h+2)\nplt.tight_layout()\nplt.show()[/python]",
        "answer": "$200$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 498,
        "problem": "Consider the two similar solids shown. The larger solid has a radius of 12 cm.\n\n\nFind the volume of solid A, in simplest exact form.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nr = 6\nR = 10\nOA = [0, 0]\nOB = [10, 0]\nmainA1 = patches.Arc(OA, r, r, theta1=00, theta2=360, color='black', linewidth=2)\nmainA2 = patches.Arc(OA, r, 2, theta1=00, theta2=360, color='black', linewidth=2)\nax.add_patch(mainA1)\nax.add_patch(mainA2)\nax.plot([0, r/2], [0, 0], color='black')\n\nmainB1 = patches.Arc(OB, R, R, theta1=0, theta2=360, color='black', linewidth=2)\nmainB2 = patches.Arc(OB, r, R, theta1=0, theta2=360, color='black', linewidth=2)\nax.add_patch(mainB1)\nax.add_patch(mainB2)\nax.plot([OB[0], OB[0]], [0, R/2], color='black')\n\n\nax.text(-1, -0.1, r\"$3$\", fontsize=10, style='italic')\nax.text(0, r/2+1, r\"$A$\", fontsize=12, style='italic')\nax.text(OB[0], R/2+1, r\"$B$\", fontsize=12, style='italic')\n\n\nax.set_aspect('equal')\nax.set_xlim(-5, 20)\nax.set_ylim(-6, 6)\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$36$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "volume"
    },
    {
        "index": 499,
        "problem": "Consider the two solids shown. The larger solid has a radius of 12 cm.\nWhat is the ratio of the volume of solid B to solid A?",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nr = 6\nR = 10\nOA = [0, 0]\nOB = [10, 0]\nmainA1 = patches.Arc(OA, r, r, theta1=00, theta2=360, color='black', linewidth=2)\nmainA2 = patches.Arc(OA, r, 2, theta1=00, theta2=360, color='black', linewidth=2)\nax.add_patch(mainA1)\nax.add_patch(mainA2)\nax.plot([0, r/2], [0, 0], color='black')\n\nmainB1 = patches.Arc(OB, R, R, theta1=0, theta2=360, color='black', linewidth=2)\nmainB2 = patches.Arc(OB, r, R, theta1=0, theta2=360, color='black', linewidth=2)\nax.add_patch(mainB1)\nax.add_patch(mainB2)\nax.plot([OB[0], OB[0]], [0, R/2], color='black')\n\n\nax.text(-1, -0.1, r\"$3$\", fontsize=10, style='italic')\nax.text(0, r/2+1, r\"$A$\", fontsize=12, style='italic')\nax.text(OB[0], R/2+1, r\"$B$\", fontsize=12, style='italic')\n\n\nax.set_aspect('equal')\nax.set_xlim(-5, 20)\nax.set_ylim(-6, 6)\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$64$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "ratio"
    },
    {
        "index": 500,
        "problem": "Two similar solids as shown in the figure, the larger solid has volumes of 5760\\mathrm{cm}^3.\n\n\nFind the fully simplified ratio of the volume of the larger solid to the volume of the smaller solid.",
        "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nr = 5\nR = 10\nH, h = 15, 10\nOA = [0, 0]\nOA1 = [0, H]\nOB = [15, 0]\nOB1 = [15, h]\nmainA1 = patches.Arc((0, 0), R, r, theta1=0, theta2=180, color='black', linewidth=2, linestyle='--')\nmainA2 = patches.Arc((0, 0), R, r, theta1=180, theta2=360, color='black', linewidth=2)\nmainA3 = patches.Arc((0, H), R, r, theta1=0, theta2=360, color='black', linewidth=2)\nmainB1 = patches.Arc((15, 0), r, 3, theta1=0, theta2=180, color='black', linewidth=2, linestyle=\"--\")\nmainB2 = patches.Arc((15, 0), r, 3, theta1=180, theta2=360, color='black', linewidth=2)\nmainB3 = patches.Arc((15, h), r, 3, theta1=0, theta2=360, color='black', linewidth=2)\nax.add_patch(mainA1)\nax.add_patch(mainA2)\nax.add_patch(mainA3)\nax.add_patch(mainB1)\nax.add_patch(mainB2)\nax.add_patch(mainB3)\nax.plot([-R/2, -R/2], [0, H], color='black')\nax.plot([R/2, R/2], [0, H], color='black')\n\nax.plot([15-r/2, 15-r/2], [0, h], color='black')\nax.plot([15+r/2, 15+r/2], [0, h], color='black')\n\n\nax.text(12, -4, r\"$Volume\\,= \\,90\\,cm^3$\", fontsize=12, style='italic')\nax.text(R/2+1, H/2, r\"$c\\,\\,cm$\", fontsize=12, style='italic')\nax.text(15+r/2+1, h/2, r\"$20\\,\\,cm$\", fontsize=12, style='italic')\n\n\nax.set_aspect('equal')\nax.set_xlim(-8, 20)\nax.set_ylim(-6, 20)\nax.axis('off')\n\nplt.show()[/python]",
        "answer": "$100$",
        "category": "Global Abstract Integration",
        "source": "Mathverse",
        "problem_type": "ratio"
    }
]