id stringlengths 1 4 | question stringlengths 31 709 | options listlengths 6 6 | image_path stringlengths 12 15 | image imagewidth (px) 56 2.52k | answer stringclasses 5
values | solution stringclasses 259
values | level int64 1 5 | subject stringclasses 16
values | Answer(Option) stringclasses 5
values |
|---|---|---|---|---|---|---|---|---|---|
2241 | A square piece of paper has side length $1$ 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 o... | [
"$2$",
"$1+\\frac{2}{3}\\sqrt{3}$",
"$\\frac{13}{6}$",
"$1+\\frac{3}{4}\\sqrt{3}$",
"$\\frac{7}{3}$",
"I don't know"
] | images/2241.jpg | A | null | 4 | transformation geometry | A | |
2243 | Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move... | [
"(6,1,1)",
"(6,2,1)",
"(6,2,2)",
"(6,3,1)",
"(6,3,2)",
"I don't know"
] | images/2243.jpg | B | null | 4 | combinatorics | B | |
2245 | Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of th... | [
"$14$",
"$10\\sqrt{2}$",
"$16$",
"$12\\sqrt{2}$",
"$18$",
"I don't know"
] | images/2245.jpg | E | null | 4 | metric geometry - area | E | |
2246 | A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of 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?
<image1> | [
"$6$",
"$7$",
"$5+2\\sqrt{2}$",
"$8$",
"$9$",
"I don't know"
] | images/2246.jpg | B | null | 2 | solid geometry | B | |
2249 | In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?
<image1> | [
"$3\\sqrt{5}$",
"$10$",
"$6\\sqrt{5}$",
"$20$",
"$25$",
"I don't know"
] | images/2249.jpg | D | null | 4 | metric geometry - area | D | |
2250 | The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
<image1> | [
"$15\\frac{1}{8}$",
"$15\\frac{3}{8}$",
"$15\\frac{1}{2}$",
"$15\\frac{5}{8}$",
"$15\\frac{7}{8}$",
"I don't know"
] | images/2250.jpg | D | null | 4 | metric geometry - area | D | |
2252 | A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
<image1> | [
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$2-\\sqrt{3}$",
"$\\sqrt{3}-\\sqrt{2}$",
"$\\sqrt{2}-1$",
"I don't know"
] | images/2252.jpg | C | null | 4 | metric geometry - length | C | |
2254 | 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 radius of $C_4$?
<image1> | [
"$\\frac{1}{14}$",
"$\\frac{1}{12}$",
"$\\frac{1}{10}$",
"$\\frac{3}{28}$",
"$\\frac{1}{9}$",
"I don't know"
] | images/2254.jpg | D | null | 4 | metric geometry - length | D | |
2255 | Six regular hexagonal blocks of side length 1 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 i... | [
"$\\frac{13 \\sqrt{3}}{3}$",
"$\\frac{216 \\sqrt{3}}{49}$",
"$\\frac{9 \\sqrt{3}}{2}$",
"$\\frac{14 \\sqrt{3}}{3}$",
"$\\frac{243 \\sqrt{3}}{49}$",
"I don't know"
] | images/2255.jpg | C | null | 4 | metric geometry - area | C | |
2256 | Square $ABCD$ is rotated $20^\circ$ clockwise about its center to obtain square $EFGH$, as shown below. What is the degree measure of $\angle EAB$?
<image1> | [
"$20^\\circ$",
"$30^\\circ$",
"$32^\\circ$",
"$35^\\circ$",
"$45^\\circ$",
"I don't know"
] | images/2256.jpg | B | null | 4 | transformation geometry | B | |
2257 | 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 quadrilateral $ ADEC$ is:
<image1> | [
"$75$",
"$58\\frac{1}{2}$",
"$48$",
"$37\\frac{1}{2}$",
"$\\text{none of these}$",
"I don't know"
] | images/2257.jpg | B | null | 5 | metric geometry - area | B | |
2258 | In the figure, $ \overline{CD}, \overline{AE}$ and $ \overline{BF}$ are one-third of their respective sides. It follows that $ \overline{AN_2}: \overline{N_2N_1}: \overline{N_1D} = 3: 3: 1$, and similarly for lines $ BE$ and $ CF$. Then the area of triangle $ N_1N_2N_3$ is:
<image1> | [
"$\\frac{1}{10} \\triangle ABC$",
"$\\frac{1}{9} \\triangle ABC$",
"$\\frac{1}{7} \\triangle ABC$",
"$\\frac{1}{6} \\triangle ABC$",
"$\\text{none of these}$",
"I don't know"
] | images/2258.jpg | C | null | 5 | metric geometry - area | C | |
2259 | In the right triangle shown the sum of the distances $ BM$ and $ MA$ is equal to the sum of the distances $ BC$ and $ CA$. If $ MB = x$, $ CB = h$, and $ CA = d$, then $ x$ equals:
<image1> | [
"$\\frac{hd}{2h + d}$",
"$d - h$",
"$\\frac{1}{2}d$",
"$h + d - \\sqrt{2d}$",
"$\\sqrt{h^2 + d^2} - h$",
"I don't know"
] | images/2259.jpg | A | null | 5 | metric geometry - length | A | |
2260 | Given triangle $ PQR$ with $ \overline{RS}$ bisecting $ \angle R$, $ PQ$ extended to $ D$ and $ \angle n$ a right angle, then:
<image1> | [
"$\\angle m = \\frac{1}{2}(\\angle p - \\angle q)$",
"$\\angle m = \\frac{1}{2}(\\angle p + \\angle q)$",
"$\\angle d = \\frac{1}{2} (\\angle q + \\angle p)$",
"$\\angle d = \\frac{1}{2}\\angle m$",
"$\\text{none of these is correct}$",
"I don't know"
] | images/2260.jpg | B | null | 4 | metric geometry - angle | B | |
2261 | In the diagram, if points $ A$, $ B$ and $ C$ are points of tangency, then $ x$ equals:
<image1> | [
"$\\frac{3}{16}\"$",
"$\\frac{1}{8}\"$",
"$\\frac{1}{32}\"$",
"$\\frac{3}{32}\"$",
"$\\frac{1}{16}\"$",
"I don't know"
] | images/2261.jpg | E | null | 5 | metric geometry - length | E | |
2262 | In the figure, $ PA$ is tangent to semicircle $ SAR$; $ PB$ is tangent to semicircle $ RBT$; $ SRT$ is a straight line; the arcs are indicated in the figure. Angle $ APB$ is measured by:
<image1> | [
"$\\frac{1}{2}(a - b)$",
"$\\frac{1}{2}(a + b)$",
"$(c - a) - (d - b)$",
"$a - b$",
"$a + b$",
"I don't know"
] | images/2262.jpg | E | null | 4 | metric geometry - angle | E | |
2263 | In circle $ O$ chord $ AB$ is produced so that $ BC$ equals a radius of the circle. $ CO$ is drawn and extended to $ D$. $ AO$ is drawn. Which of the following expresses the relationship between $ x$ and $ y$?
<image1> | [
"$x=3y \\\\$",
"$x=2y \\\\$",
"$x=60^\\circ \\\\$",
"$\\text{there is no special relationship between }x\\text{ and }y \\\\$",
"$x=2y \\text{ or }x=3y\\text{, depending upon the length of }AB$",
"I don't know"
] | images/2263.jpg | A | null | 4 | metric geometry - angle | A | |
2264 | In the figure $ \overline{AB} = \overline{AC}$, angle $ BAD = 30^{\circ}$, and $ \overline{AE} = \overline{AD}$.
<image1>Then angle $ CDE$ equals: | [
"$7\\frac{1}{2}^{\\circ}$",
"$10^{\\circ}$",
"$12\\frac{1}{2}^{\\circ}$",
"$15^{\\circ}$",
"$20^{\\circ}$",
"I don't know"
] | images/2264.jpg | D | null | 4 | metric geometry - angle | D | |
2265 | Circle $ O$ has diameters $ AB$ and $ CD$ perpendicular to each other. $ AM$ is any chord intersecting $ CD$ at $ P$. Then $ AP\cdot AM$ is equal to:
<image1> | [
"$AO\\cdot OB$",
"$AO\\cdot AB$",
"$CP\\cdot CD$",
"$CP\\cdot PD$",
"$CO\\cdot OP$",
"I don't know"
] | images/2265.jpg | B | null | 5 | metric geometry - length | B | |
2266 | In right triangle $ ABC$, $ BC = 5$, $ AC = 12$, and $ AM = x$; $ \overline{MN} \perp \overline{AC}$, $ \overline{NP} \perp \overline{BC}$; $ N$ is on $ AB$. If $ y = MN + NP$, one-half the perimeter of rectangle $ MCPN$, then:
<image1> | [
"$y = \\frac{1}{2}(5 + 12)$",
"$y = \\frac{5x}{12} + \\frac{12}{5}$",
"$y = \\frac{144 - 7x}{12}$",
"$y = 12 \\,\\,$",
"$y = \\frac{5x}{12} + 6$",
"I don't know"
] | images/2266.jpg | C | null | 5 | metric geometry - length | C | |
2267 | In triangle $ ABC$, $ AC = CD$ and $ \angle CAB - \angle ABC = 30^\circ$. Then $ \angle BAD$ is:
<image1> | [
"$30^\\circ$",
"$20^\\circ$",
"$22\\frac{1}{2}^\\circ$",
"$10^\\circ$",
"$15^\\circ$",
"I don't know"
] | images/2267.jpg | E | null | 4 | metric geometry - angle | E | |
2268 | In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}... | [
"$r\\sqrt{2}$",
"$r$",
"$\\text{not a side of an inscribed regular polygon}$",
"$\\frac{r\\sqrt{3}}{2}$",
"$r\\sqrt{3}$",
"I don't know"
] | images/2268.jpg | A | null | 5 | metric geometry - length | A | |
2269 | Let $ ABC$ be an equilateral triangle inscribed in circle $ O$. $ M$ is a point on arc $ BC$. Lines $ \overline{AM}$, $ \overline{BM}$, and $ \overline{CM}$ are drawn. Then $ AM$ is:
<image1> | [
"$\\text{equal to }{BM + CM}$",
"$\\text{less than }{BM + CM}$",
"$\\text{greater than }{BM + CM}$",
"$\\text{equal, less than, or greater than }{BM + CM}\\text{, depending upon the position of }{ {M} }$",
"$\\text{none of these}$",
"I don't know"
] | images/2269.jpg | A | null | 5 | metric geometry - length | A | |
2270 | The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:
<image1> | [
"4: 3",
"3: 2",
"4: 1",
"3: 1",
"6: 1",
"I don't know"
] | images/2270.jpg | C | null | 5 | metric geometry - length | C | |
2271 | In the accompanying figure $ \overline{CE}$ and $ \overline{DE}$ are equal chords of a circle with center $ O$. Arc $ AB$ is a quarter-circle. Then the ratio of the area of triangle $ CED$ to the area of triangle $ AOB$ is:
<image1> | [
"$\\sqrt{2} : 1$",
"$\\sqrt{3} : 1$",
"$4 : 1$",
"$3 : 1$",
"$2 : 1$",
"I don't know"
] | images/2271.jpg | E | null | 5 | metric geometry - area | E | |
2272 | In a general triangle $ ADE$ (as shown) lines $ \overline{EB}$ and $ \overline{EC}$ are drawn. Which of the following angle relations is true?
<image1> | [
"$x + z = a + b$",
"$y + z = a + b$",
"$m + x = w + n \\\\$",
"$x + z + n = w + c + m$",
"$x + y + n = a + b + m$",
"I don't know"
] | images/2272.jpg | E | null | 4 | metric geometry - angle | E | |
2273 | $ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR + PS$ is equal to:
<image1> | [
"PQ",
"AE",
"PT + AT",
"AF",
"EF",
"I don't know"
] | images/2273.jpg | D | null | 5 | metric geometry - length | D | |
2274 | In this diagram a scheme is indicated for associating all the points of segment $ \overline{AB}$ with those of segment $ \overline{A'B'}$, and reciprocally. To described this association scheme analytically, let $ x$ be the distance from a point $ P$ on $ \overline{AB}$ to $ D$ and let $ y$ be the distance from the ass... | [
"$13a$",
"$17a - 51$",
"$17 - 3a$",
"$\\frac{17 - 3a}{4}$",
"$12a - 34$",
"I don't know"
] | images/2274.jpg | C | null | 5 | metric geometry - length | C | |
2275 | In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then:
<image1> | [
"$AP^2 = PB \\times AB$",
"$AP \\times DO = PB \\times AD$",
"$AB^2 = AD \\times DE$",
"$AB \\times AD = OB \\times AO$",
"$\\text{none of these}$",
"I don't know"
] | images/2275.jpg | A | null | 5 | metric geometry - length | A | |
2276 | In this diagram $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which is inscribed equilateral triangle $DEF$. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. Then:
<image1> | [
"$b=\\frac{a+c}{2}$",
"$b=\\frac{a-c}{2}$",
"$a=\\frac{b-c}{2}$",
"$a=\\frac{b+c}{2}$",
"$\\text{none of these}$",
"I don't know"
] | images/2276.jpg | D | null | 4 | metric geometry - angle | D | |
2277 | Chord $EF$ is the perpendicular bisector of chord $BC$, intersecting it in $M$. Between $B$ and $M$ point $U$ is taken, and $EU$ extended meets the circle in $A$. Then, for any selection of $U$, as described, triangle $EUM$ is similar to triangle:
<image1> | [
"EFA",
"EFC",
"ABM",
"ABU",
"FMC",
"I don't know"
] | images/2277.jpg | A | null | 5 | metric geometry - area | A | |
2280 | In triangle $ABC$ lines $CE$ and $AD$ are drawn so that
$\frac{CD}{DB}=\frac{3}{1}$ and $\frac{AE}{EB}=\frac{3}{2}$. Let $r=\frac{CP}{PE}$
where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:
<image1> | [
"$3$",
"$\\frac{3}{2}$",
"$4$",
"$5$",
"$\\frac{5}{2}$",
"I don't know"
] | images/2280.jpg | D | null | 5 | metric geometry - length | D | |
2281 | In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\frac{1}{2}$ inches. The length of $RS$, in inches, is:
<image1> | [
"$\\text{undetermined}$",
"$4$",
"$5\\frac{1}{2}$",
"$6$",
"$6\\frac{1}{4}$",
"I don't know"
] | images/2281.jpg | E | null | 5 | metric geometry - length | E | |
2282 | $P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches.
<image1>
Then $PB$, in inches, equals: | [
"$2\\sqrt{3}$",
"$3\\sqrt{2}$",
"$3\\sqrt{3}$",
"$4\\sqrt{2}$",
"$2$",
"I don't know"
] | images/2282.jpg | B | null | 5 | metric geometry - length | B | |
2283 | In this figure the radius of the circle is equal to the altitude of the equilateral triangle $ABC$. The circle is made to roll along the side $AB$, remaining tangent to it at a variable point $T$ and intersecting lines $AC$ and $BC$ in variable points $M$ and $N$, respectively. Let $n$ be the number of degrees in arc $... | [
"$\\text{varies from }30^{\\circ}\\text{ to }90^{\\circ}$",
"$\\text{varies from }30^{\\circ}\\text{ to }60^{\\circ}$",
"$\\text{varies from }60^{\\circ}\\text{ to }90^{\\circ}$",
"$\\text{remains constant at }30^{\\circ}$",
"$\\text{remains constant at }60^{\\circ}$",
"I don't know"
] | images/2283.jpg | E | null | 4 | metric geometry - angle | E | |
2284 | The magnitudes of the sides of triangle $ABC$ are $a$, $b$, and $c$, as shown, with $c\le b\le a$. Through interior point $P$ and the vertices $A$, $B$, $C$, lines are drawn meeting the opposite sides in $A'$, $B'$, $C'$, respectively.
<image1>
Let $s=AA'+BB'+CC'$. Then, for all positions of point $P$, $s$ is less than... | [
"2a+b",
"2a+c",
"2b+c",
"a+2b",
"a+b+c",
"I don't know"
] | images/2284.jpg | A | null | 5 | metric geometry - length | A | |
2285 | An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots,k,\cdots,n$, $n\geq 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbered $k$ a... | [
"180",
"360",
"180(n+2)",
"180(n-2)",
"180(n-4)",
"I don't know"
] | images/2285.jpg | E | null | 4 | metric geometry - angle | E | |
2286 | Triangle $ABC$ is inscribed in a circle with center $O'$. A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$.
<image1>
Then, we must have: | [
"$CD=BD=O'D$",
"$AO=CO=OD$",
"$CD=CO=BD \\\\$",
"$CD=OD=BD$",
"$O'B=O'C=OD $",
"I don't know"
] | images/2286.jpg | D | null | 5 | metric geometry - length | D | |
2287 | <image1>
In this figure $AB$ is a diameter of a circle, centered at $O$, with radius $a$. A chord $AD$ is drawn and extended to meet the tangent to the circle at $B$ in point $C$. Point $E$ is taken on $AC$ so that $AE=DC$. Denoting the distances of $E$ from the tangent through $A$ and from the diameter $AB$ by $x$... | [
"$y^2=\\frac{x^3}{2a-x}$",
"$y^2=\\frac{x^3}{2a+x}$",
"$y^4=\\frac{x^2}{2-x} \\\\$",
"$x^2=\\frac{y^2}{2a-x}$",
"$x^2=\\frac{y^2}{2a+x}$",
"I don't know"
] | images/2287.jpg | A | null | 5 | metric geometry - length | A | |
2288 | <image1>
In this diagram semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, so that they are mutually tangent. If $\overline{CD} \bot \overline{AB}$, then the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius is: | [
"$1:2$",
"$1:3$",
"$\\sqrt{3}:7$",
"$1:4$",
"$\\sqrt{2}:6$",
"I don't know"
] | images/2288.jpg | D | null | 5 | metric geometry - area | D | |
2289 | In this diagram, not drawn to scale, figures $\text{I}$ and $\text{III}$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $\text{II}$ is a square region with area $32$ sq. in. Let the length of segment $AD$ be decreased by $12\frac{1}{2} \%$ of itself, while... | [
"$12\\frac{1}{2}$",
"$25$",
"$50$",
"$75$",
"$87\\frac{1}{2}$",
"I don't know"
] | images/2289.jpg | D | null | 5 | metric geometry - area | D | |
2290 | In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD, O, G, H, J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF$. Then, as $CD$ and $EF$ are tr... | [
"$0$",
"$1$",
"$\\sqrt{2}$",
"$\\frac{1}{\\sqrt{2}}+\\frac{1}{2}$",
"$\\frac{1}{\\sqrt{2}}+1$",
"I don't know"
] | images/2290.jpg | D | null | 5 | metric geometry - area | D | |
2291 | <image1>
A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at a point $5$ inches from the center of $M$ is: | [
"$1$",
"$15$",
"$15\\frac{1}{3}$",
"$15\\frac{1}{2}$",
"$15\\frac{3}{4}$",
"I don't know"
] | images/2291.jpg | B | null | 5 | metric geometry - length | B | |
2292 | Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:
<image1>
<image2>
<image3>
<image4>
<image5> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/2292.jpg | D | null | 4 | analytic geometry | D | |
2293 | In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively.
<image1>
Then the measure of $AB$ is equal to | [
"$\\frac{1}{2}a+2b$",
"$\\frac{3}{2}b+\\frac{3}{4}a$",
"$2a-b$",
"$4b-\\frac{1}{2}a$",
"$a+b$",
"I don't know"
] | images/2293.jpg | E | null | 4 | metric geometry - angle | E | |
2294 | Points $A,B,Q,D,$ and $C$ lie on the circle shown and the measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$ respectively.
<image1>
The sum of the measures of angles $P$ and $Q$ is | [
"$80^\\circ$",
"$62^\\circ$",
"$40^\\circ$",
"$46^\\circ$",
"$\\text{None of these}$",
"I don't know"
] | images/2294.jpg | C | null | 4 | metric geometry - angle | C | |
2295 | <image1>
Pascal's triangle is an array of positive integers(See figure), in which the first row is $1$, the second row is two $1$'s, each row begins and ends with $1$, and the $k^\text{th}$ number in any row when it is not $1$, is the sum of the $k^\text{th}$ and $(k-1)^\text{th}$ numbers in the immediately preceding ... | [
"$\\frac{n^2-n}{2n-1}$",
"$\\frac{n^2-n}{4n-2}$",
"$\\frac{n^2-2n}{2n-1}$",
"$\\frac{n^2-3n+2}{4n-2}$",
"$\\text{None of these}$",
"I don't know"
] | images/2295.jpg | D | null | 5 | combinatorics | D | |
2296 | <image1>
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 midpoints of $BF$. The point $E$ divides side $BC$ in the ratio | [
"1:4",
"1:3",
"2:5",
"4:11",
"3:8",
"I don't know"
] | images/2296.jpg | B | null | 5 | metric geometry - length | B | |
2297 | <image1>
Quadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length | [
"$\\frac{7}{2}$",
"$\\frac{5\\sqrt{2}}{2}$",
"$\\sqrt{11}$",
"$\\sqrt{13}$",
"$2\\sqrt{3}$",
"I don't know"
] | images/2297.jpg | A | null | 5 | metric geometry - length | A | |
2298 | <image1>
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 ratio of segment $PM$ to $MQ$ is | [
"5:12",
"5:13",
"5:19",
"1:4",
"5:21",
"I don't know"
] | images/2298.jpg | C | null | 5 | metric geometry - length | C | |
2300 | <image1>
The radius of the smallest circle containing the symmetric figure composed of the $3$ unit squares shown above is | [
"$\\sqrt{2}$",
"$\\sqrt{1.25}$",
"$1.25$",
"$\\frac{5\\sqrt{17}}{16}$",
"$\\text{None of these}$",
"I don't know"
] | images/2300.jpg | D | null | 5 | metric geometry - length | D | |
2301 | <image1>
In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to | [
"3x+2",
"3x+1",
"2x+3",
"2x+2",
"2x+1",
"I don't know"
] | images/2301.jpg | E | null | 5 | metric geometry - length | E | |
2302 | <image1>
A rectangular piece of paper $6$ inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease $L$ in terms of angle $\theta$ is | [
"$3\\sec ^2\\theta\\csc\\theta$",
"$6\\sin\\theta\\sec\\theta$",
"$3\\sec\\theta\\csc\\theta$",
"$6\\sec\\theta\\csc ^2\\theta$",
"$\\text{None of these}$",
"I don't know"
] | images/2302.jpg | A | null | 5 | transformation geometry | A | |
2303 | <image1>
Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is | [
"$4\\sqrt{5}$",
"$\\sqrt{65}$",
"$2\\sqrt{17}$",
"$3\\sqrt{7}$",
"$6\\sqrt{2}$",
"I don't know"
] | images/2303.jpg | B | null | 5 | metric geometry - length | B | |
2304 | <image1>
Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The ... | [
"$20\\pi/3$",
"$32\\pi/3$",
"$12\\pi$",
"$40\\pi/3$",
"$15\\pi$",
"I don't know"
] | images/2304.jpg | D | null | 5 | metric geometry - length | D | |
2305 | A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below.
<image1>
The inequalities
\[ |x| + |y| \leq \sqrt{2(x^2 + y^2)} \leq 2\mbox{Max}(|x|, |y|)\]
are represented... | [
"$I$",
"$II$",
"$III$",
"$IV$",
"$\\mbox{none of these}*An inequality of the form f(x, y) \\leq g(x, y), for all x and y is represented geometrically by a figure showing the containment \\{\\mbox{The set of points }(x, y)\\mbox{ such that }g(x, y) \\leq a\\} \\subset\\ \\{\\mbox{The set of points }(x, y)... | images/2305.jpg | B | null | 4 | analytic geometry | B | |
2306 | In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.
<image1>
Of the three equations
\[ \textbf{I.}\ d-s=1, \qquad \textbf{II.}\ ds=1, \qquad \textbf{III.}... | [
"$\\textbf{I} \\text{only}$",
"$\\textbf{II} \\text{only}$",
"$\\textbf{III} \\text{only}$",
"$\\textbf{I} \\text{and} \\textbf{II} \\text{only}$",
"$\\textbf{I, II} \\text{and} \\textbf{III}$",
"I don't know"
] | images/2306.jpg | E | null | 5 | metric geometry - length | E | |
2307 | In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is
<image1> | [
"$2\\sqrt{3}-3$",
"$1-\\frac{\\sqrt{3}}{3}$",
"$\\frac{\\sqrt{3}}{4}$",
"$\\frac{\\sqrt{2}}{3}$",
"$4-2\\sqrt{3}$",
"I don't know"
] | images/2307.jpg | A | null | 5 | metric geometry - area | A | |
2308 | 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 point of tangency. If $ TP=4$ and $ T'Q=9$ then $ r$ is
<image1> | [
"$25/6$",
"$6$",
"$25/4 \\$",
"$\\text{a number other than }25/6, 6, 25/4 \\$",
"$\\text{not determinable from the given information}$",
"I don't know"
] | images/2308.jpg | B | null | 5 | metric geometry - length | B | |
2309 | 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 $k$, then the area ... | [
"$k$",
"$\\frac{6k}{5}$",
"$\\frac{9k}{8}$",
"$\\frac{5k}{4}$",
"$2k$",
"I don't know"
] | images/2309.jpg | C | null | 5 | metric geometry - area | C | |
2310 | 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 $ BC$?
<image1> | [
"$2\\sqrt{26}$",
"$2\\sqrt{31}$",
"$9$",
"$4+2\\sqrt{13}$",
"$\\text{not enough information given to solve the problem}$",
"I don't know"
] | images/2310.jpg | B | null | 5 | metric geometry - length | B | |
2311 | In the adjoining figure $AB$ and $BC$ are adjacent sides of square $ABCD$; $M$ is the midpoint of $AB$; $N$ is the midpoint of $BC$; and $AN$ and $CM$ intersect at $O$. The ratio of the area of $AOCD$ to the area of $ABCD$ is
<image1> | [
"$\\frac{5}{6}$",
"$\\frac{3}{4}$",
"$\\frac{2}{3}$",
"$\\frac{\\sqrt{3}}{2}$",
"$\\frac{(\\sqrt{3}-1)}{2}$",
"I don't know"
] | images/2311.jpg | C | null | 5 | metric geometry - area | C | |
2312 | 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 $\frac{EG}{GF}$ equals
<image1> | [
"$\\frac{3}{2}$",
"$\\frac{4}{3}$",
"$\\frac{5}{4}$",
"$\\frac{6}{5} \\$",
"$\\text{not enough information to solve the problem}$",
"I don't know"
] | images/2312.jpg | A | null | 5 | metric geometry - length | A | |
2313 | <image1>
In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC=DC=3$, $OD=2$, and $AB=6$, then the radius of the circle is | [
"$3+\\sqrt{3}$",
"$15/\\pi$",
"$9/2$",
"$2\\sqrt{6}$",
"$\\sqrt{22}$",
"I don't know"
] | images/2313.jpg | E | null | 5 | metric geometry - length | E | |
2314 | <image1>
In the adjoining figure, circle $\mathit{K}$ has diameter $\mathit{AB}$; cirlce $\mathit{L}$ is tangent to circle $\mathit{K}$ and to $\mathit{AB}$ at the center of circle $\mathit{K}$; and circle $\mathit{M}$ tangent to circle $\mathit{K}$, to circle $\mathit{L}$ and $\mathit{AB}$. The ratio of the area of c... | [
"$12$",
"$14$",
"$16$",
"$18$",
"$\\text{not an integer}$",
"I don't know"
] | images/2314.jpg | C | null | 5 | metric geometry - area | C | |
2315 | <image1>
In the adjoining figure, every point of circle $\mathit{O'}$ is exterior to circle $\mathit{O}$. Let $\mathit{P}$ and $\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is | [
"$\\text{the average of the lengths of the internal and external common tangents}$",
"$\\text{equal to the length of an external common tangent if and only if circles }\\mathit{O}\\text{ and }\\mathit{O'}\\text{ have equal radii}$",
"$\\text{always equal to the length of an external common tangent}$",
"$\\tex... | images/2315.jpg | C | null | 5 | metric geometry - length | C | |
2316 | <image1>
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals | [
"$30^\\circ$",
"$40^\\circ$",
"$50^\\circ$",
"$65^\\circ$",
"$\\text{none of these}$",
"I don't know"
] | images/2316.jpg | C | null | 4 | metric geometry - angle | C | |
2317 | <image1>
In the adjoining figure $\measuredangle E=40^\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\measuredangle ACD$. | [
"$10^\\circ$",
"$15^\\circ$",
"$20^\\circ$",
"$\\left(\\frac{45}{2}\\right)^\\circ$",
"$30^\\circ$",
"I don't know"
] | images/2317.jpg | B | null | 4 | metric geometry - angle | B | |
2318 | <image1>
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 three, then the perimeter of the triangle is | [
"$36+9\\sqrt{2}$",
"$36+6\\sqrt{3}$",
"$36+9\\sqrt{3}$",
"$18+18\\sqrt{3}$",
"$45$",
"I don't know"
] | images/2318.jpg | D | null | 5 | metric geometry - length | D | |
2319 | <image1>
If $a,b,$ and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then | [
"$d=a+b$",
"$d^2=a^2+b^2$",
"$d^2=a^2+ab+b^2$",
"$b=\\frac{a+d}{2}$",
"$b^2=ad$",
"I don't know"
] | images/2319.jpg | A | null | 5 | metric geometry - length | A | |
2321 | <image1>
Vertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is | [
"$1$",
"$\\frac{\\sqrt{2}}{2}$",
"$\\frac{\\sqrt{3}}{2}$",
"$4-2\\sqrt{3}$",
"$\\frac{1}{2}+\\frac{\\sqrt{3}}{4}$",
"I don't know"
] | images/2321.jpg | C | null | 5 | metric geometry - area | C | |
2322 | <image1>
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 length of segment $QR$ is | [
"$4.75$",
"$4.8$",
"$5$",
"$4\\sqrt{2}$",
"$3\\sqrt{3}$",
"I don't know"
] | images/2322.jpg | B | null | 5 | metric geometry - length | B | |
2323 | <image1>
If $\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\measuredangle A_{44}A_{45}A_{43}$ equals | [
"$30^\\circ$",
"$45^\\circ$",
"$60^\\circ$",
"$90^\\circ$",
"$120^\\circ$",
"I don't know"
] | images/2323.jpg | E | null | 4 | metric geometry - angle | E | |
2326 | <image1>
In the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $AB$ equals length $OD$, and the m... | [
"$10^\\circ$",
"$15^\\circ$",
"$20^\\circ$",
"$25^\\circ$",
"$30^\\circ$",
"I don't know"
] | images/2326.jpg | B | null | 4 | metric geometry - angle | B | |
2327 | <image1>
Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle w... | [
"$\\textbf{I. }\\text{only}$",
"$\\textbf{II. }\\text{only}$",
"$\\textbf{I. }\\text{and }\\textbf{II. }\\text{only}$",
"$\\textbf{II. }\\text{and }\\textbf{III. }\\text{only}$",
"$\\textbf{I. },\\textbf{II. },\\text{and }\\textbf{III. }$",
"I don't know"
] | images/2327.jpg | C | null | 5 | transformation geometry | C | |
2328 | The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.
<image1> | [
"$\\frac{1}{2}$",
"$\\frac{3}{4}$",
"$\\frac{\\sqrt{2}}{2}$",
"$\\frac{\\sqrt{3}}{2}$",
"$\\frac{\\sqrt{3}}{3}$",
"I don't know"
] | images/2328.jpg | C | null | 3 | solid geometry | C | |
2329 | Circles with centers $A ,~ B,$ and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$.
<image1>
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... | [
"$3r-2$",
"$r^2$",
"$r+\\sqrt{3(r-1)}$",
"$1+\\sqrt{3(r^2-1)}$",
"$\\text{none of these}$",
"I don't know"
] | images/2329.jpg | D | null | 5 | metric geometry - length | D | |
2330 | <image1>
In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ 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 $\trian... | [
"$\\frac{1}{4}\\cos 10^\\circ$",
"$\\frac{\\sqrt{3}}{8}$",
"$\\frac{1}{4}\\cos 40^\\circ$",
"$\\frac{1}{4}\\cos 50^\\circ$",
"$\\frac{1}{8}$",
"I don't know"
] | images/2330.jpg | B | null | 5 | metric geometry - area | B | |
2331 | In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares.
<image1>
The measure of $\angle GDA$ is | [
"$90^\\circ$",
"$105^\\circ$",
"$120^\\circ$",
"$135^\\circ$",
"$150^\\circ$",
"I don't know"
] | images/2331.jpg | C | null | 4 | metric geometry - angle | C | |
2332 | If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, then the length of $PQ$ divided by the length of $AQ$ is
<image1> | [
"$\\frac{\\sqrt{3}}{2}$",
"$\\frac{\\sqrt{3}}{3}$",
"$\\frac{\\sqrt{2}}{2}$",
"$\\frac{1}{2}$",
"$\\frac{2}{3}$",
"I don't know"
] | images/2332.jpg | B | null | 5 | metric geometry - length | B | |
2334 | 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 ratio of the area of triangle $BDF$ to the area of quadrilateral $FDCE$ is
<image1> | [
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$\\frac{1}{3}$",
"$\\frac{2}{5}$",
"$\\text{none of these}$",
"I don't know"
] | images/2334.jpg | A | null | 5 | metric geometry - area | A | |
2335 | In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals
<image1> | [
"$2$",
"$\\frac{5}{2}$",
"$\\frac{5}{2} - \\sin \\theta$",
"$\\frac{5}{2} - \\frac{1}{2} \\sin \\theta$",
"$\\frac{5}{2} - \\frac{1}{2} \\sin \\left(\\frac{1}{2} \\theta\\right)$",
"I don't know"
] | images/2335.jpg | B | null | 5 | metric geometry - length | B | |
2336 | <image1>
Equilateral $ \triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $ T$ and tangent to sides $ AB$ and $ AC$ at points $ P$ and $ Q$. If side $ BC$ has length $ 12$, then segment $ PQ$ has length | [
"$6$",
"$6\\sqrt{3}$",
"$8$",
"$8\\sqrt{3}$",
"$9$",
"I don't know"
] | images/2336.jpg | C | null | 5 | metric geometry - length | C | |
2337 | <image1>
In triangle $ ABC$ in the adjoining figure, $ AD$ and $ AE$ trisect $ \angle BAC$. The lengths of $ BD$, $ DE$ and $ EC$ are $ 2$, $ 3$, and $ 6$, respectively. The length of the shortest side of $ \triangle ABC$ is | [
"$2\\sqrt{10}$",
"$11$",
"$6\\sqrt{6}$",
"$6$",
"$\\text{not uniquely determined by the given information}$",
"I don't know"
] | images/2337.jpg | A | null | 5 | metric geometry - length | A | |
2338 | In the adjoining figure triangle $ ABC$ is inscribed in a circle. Point $ D$ lies on $ \stackrel{\frown}{AC}$ with $ \stackrel{\frown}{DC} = 30^\circ$, and point $ G$ lies on $ \stackrel{\frown}{BA}$ with $ \stackrel{\frown}{BG}\, > \, \stackrel{\frown}{GA}$. Side $ AB$ and side $ AC$ each have length equal to the leng... | [
"$\\frac{2 - \\sqrt{3}}{3}$",
"$\\frac{2\\sqrt{3} - 3}{3}$",
"$7\\sqrt{3} - 12$",
"$3\\sqrt{3} - 5$",
"$\\frac{9 - 5\\sqrt{3}}{3}$",
"I don't know"
] | images/2338.jpg | C | null | 5 | metric geometry - area | C | |
2340 | In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circle $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ h... | [
"$20$",
"$15\\sqrt{2}$",
"$24$",
"$25$",
"$\\text{none of these}$",
"I don't know"
] | images/2340.jpg | C | null | 5 | metric geometry - length | C | |
2341 | In the adjoining figure of a rectangular solid, $\angle DHG=45^\circ$ and $\angle FHB=60^\circ$. Find the cosine of $\angle BHD$.
<image1> | [
"$\\frac{\\sqrt{3}}{6}$",
"$\\frac{\\sqrt{2}}{6}$",
"$\\frac{\\sqrt{6}}{3}$",
"$\\frac{\\sqrt{6}}{4}$",
"$\\frac{\\sqrt{6}-\\sqrt{2}}{4}$",
"I don't know"
] | images/2341.jpg | D | null | 3 | solid geometry | D | |
2342 | 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 $BC=s$. The length of $BN$ is
<image1> | [
"$s\\sqrt{2}$",
"$\\frac{3}{2}s\\sqrt{2}$",
"$2s\\sqrt{2}$",
"$\\frac{1}{2}s\\sqrt{5}$",
"$\\frac{1}{2}s\\sqrt{6}$",
"I don't know"
] | images/2342.jpg | E | null | 5 | metric geometry - length | E | |
2343 | In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If $AG=2$, $GF=13$, $FC=1$, and $HJ=7$, then $DE$ equals
<image1> | [
"$2\\sqrt{22}$",
"$7\\sqrt{3}$",
"$9$",
"$10$",
"$13$",
"I don't know"
] | images/2343.jpg | A | null | 5 | metric geometry - length | A | |
2344 | The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with prob... | [
"$\\frac{11}{32}$",
"$\\frac{1}{2}$",
"$\\frac{4}{7}$",
"$\\frac{21}{32}$",
"$\\frac{3}{4}$",
"I don't know"
] | images/2344.jpg | D | null | 5 | combinatorics | D | |
2345 | In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, and sides $BC$ and $ED$. Each side has length of 1. Also, $\measuredangle FAB = \measuredangle BCD = 60^\circ$. The area of the figure is
<image1> | [
"$\\frac{\\sqrt{3}}{2}$",
"$1$",
"$\\frac{3}{2}$",
"$\\sqrt{3}$",
"$2$",
"I don't know"
] | images/2345.jpg | D | null | 5 | metric geometry - area | D | |
2347 | Triangle $\triangle ABC$ in the figure has area $10$. 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
<image1> | [
"$4$",
"$5$",
"$6$",
"$\\frac{5}{3}\\sqrt{10}$",
"$\\text{not uniquely determined}$",
"I don't know"
] | images/2347.jpg | C | null | 5 | metric geometry - area | C | |
2348 | Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$. The point $P$ is on $CN$ and $\angle CAP = \angle CBP = 10^{\circ}$. If $\stackrel{\frown}{MA} = 40^{\circ}$, then $\stackrel{\frown}{BN}$ equals
<image1> | [
"$10^{\\circ}$",
"$15^{\\circ}$",
"$20^{\\circ}$",
"$25^{\\circ}$",
"$30^{\\circ}$",
"I don't know"
] | images/2348.jpg | C | null | 4 | metric geometry - angle | C | |
2349 | A rectangle intersects a circle as shown: $AB=4$, $BC=5$, and $DE=3$. Then $EF$ equals:
<image1> | [
"$6$",
"$7$",
"$\\frac{20}{3}$",
"$8$",
"$9$",
"I don't know"
] | images/2349.jpg | B | null | 5 | metric geometry - length | B | |
2350 | A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\triangle ABC$ is:
<image1> | [
"$120$",
"$144$",
"$150$",
"$216$",
"$144\\sqrt{5}$",
"I don't know"
] | images/2350.jpg | C | null | 5 | metric geometry - area | C | |
2351 | In the obtuse triangle $ABC$, $AM = MB, MD \perp BC, EC \perp BC$. If the area of $\triangle ABC$ is 24, then the area of $\triangle BED$ is
<image1> | [
"$9$",
"$12$",
"$15$",
"$18$",
"$\\text{not uniquely determined}$",
"I don't know"
] | images/2351.jpg | B | null | 5 | metric geometry - area | B | |
2352 | In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $ 60^{\circ}$. What is the perimeter of the monster in cm?
<image1> | [
"$\\pi + 2$",
"$2\\pi$",
"$\\frac{5}{3} \\pi$",
"$\\frac{5}{6} \\pi + 2$",
"$\\frac{5}{3} \\pi + 2$",
"I don't know"
] | images/2352.jpg | E | null | 5 | metric geometry - length | E | |
2353 | In right $ \triangle ABC$ with legs $ 5$ and $ 12$, arcs of circles are drawn, one with center $ A$ and radius $ 12$, the other with center $ B$ and radius $ 5$. They intersect the hypotenuse at $ M$ and $ N$. Then, $ MN$ has length:
<image1> | [
"$2$",
"$\\frac{13}{5}$",
"$3$",
"$4$",
"$\\frac{24}{5}$",
"I don't know"
] | images/2353.jpg | D | null | 5 | metric geometry - length | D | |
2354 | The odd positive integers $1,3,5,7,\cdots,$ are arranged into in five columns continuing with the pattern shown on the right. Counting from the left, the column in which $ 1985$ appears in is the
<image1> | [
"$\\text{ first}$",
"$\\text{ second}$",
"$\\text{ third}$",
"$\\text{ fourth}$",
"$\\text{ fifth}$",
"I don't know"
] | images/2354.jpg | B | null | 5 | algebra | B | |
2357 | In a circle with center $ O$, $ AD$ is a diameter, $ ABC$ is a chord, $ BO = 5$, and $ \angle ABO = \stackrel{\frown}{CD} = 60^{\circ}$. Then the length of $ BC$ is:
<image1> | [
"$3$",
"$3 + \\sqrt{3}$",
"$5 - \\frac{\\sqrt{3}}{2}$",
"$5$",
"$\\text{none of the above}$",
"I don't know"
] | images/2357.jpg | D | null | 5 | metric geometry - length | D |
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