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 |
|---|---|---|---|---|---|---|---|---|---|
2358 | In $ \triangle ABC$, we have $ \angle C = 3 \angle A$, $ a = 27$, and $ c = 48$. What is $ b$?
<image1> | [
"$33$",
"$35$",
"$37$",
"$39$",
"$\\text{not uniquely determined}$",
"I don't know"
] | images/2358.jpg | B | null | 5 | metric geometry - length | B | |
2359 | $\triangle ABC$ is a right angle at $C$ and $\angle A = 20^\circ$. If $BD$ is the bisector of $\angle ABC$, then $\angle BDC =$
<image1> | [
"$40^\\circ$",
"$45^\\circ$",
"$50^\\circ$",
"$55^\\circ$",
"$60^\\circ$",
"I don't know"
] | images/2359.jpg | D | null | 4 | metric geometry - angle | D | |
2360 | 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?
<image1> | [
"$28\\text{ inches}$",
"$29\\text{ inches}$",
"$30\\text{ inches}$",
"$31\\text{ inches}$",
"$32\\text{ inches}$",
"I don't know"
] | images/2360.jpg | C | null | 5 | algebra | C | |
2363 | In the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$.
<image1>
A necessary and sufficient condition for the equality of the two shaded areas, given $0 < \theta < \frac{\pi}{2}$, is | [
"$\\tan \\theta = \\theta$",
"$\\tan \\theta = 2\\theta$",
"$\\tan \\theta = 4\\theta$",
"$\\tan 2\\theta = \\theta \\$",
"$\\tan \\frac{\\theta}{2} = \\theta$",
"I don't know"
] | images/2363.jpg | B | null | 5 | metric geometry - area | B | |
2364 | In the adjoining figure, $AB$ is a diameter of the circle, $CD$ is a chord parallel to $AB$, and $AC$ intersects $BD$ at $E$, with $\angle AED = \alpha$. The ratio of the area of $\triangle CDE$ to that of $\triangle ABE$ is
<image1> | [
"$\\cos \\alpha$",
"$\\sin \\alpha$",
"$\\cos^2\\alpha$",
"$\\sin^2\\alpha$",
"$1 - \\sin \\alpha$",
"I don't know"
] | images/2364.jpg | C | null | 5 | metric geometry - area | C | |
2365 | $ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then $AO + AQ + AR$ equals
<image1> | [
"$3$",
"$1 + \\sqrt{5}$",
"$4$",
"$2 + \\sqrt{5}$",
"$5$",
"I don't know"
] | images/2365.jpg | C | null | 5 | metric geometry - length | C | |
2366 | A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle $ABC$ of side length $3$. The perimeter of the remaining quadrilateral is
<image1> | [
"$6$",
"$6\\frac{1}{2}$",
"$7$",
"$7\\frac{1}{2}$",
"$8$",
"I don't know"
] | images/2366.jpg | E | null | 5 | metric geometry - length | E | |
2367 | In the $\triangle ABC$ shown, $D$ is some interior point, and $x, y, z, w$ are the measures of angles in degrees. Solve for $x$ in terms of $y, z$ and $w$.
<image1> | [
"$w-y-z$",
"$w-2y-2z$",
"$180-w-y-z \\$",
"$2w-y-z$",
"$180-w+y+z$",
"I don't know"
] | images/2367.jpg | A | null | 4 | metric geometry - angle | A | |
2368 | In the figure the sum of the distances $AD$ and $BD$ is
<image1> | [
"$\\text{between 10 and 11}$",
"$12$",
"$\\text{between 15 and 16}$",
"$\\text{between 16 and 17}$",
"$17$",
"I don't know"
] | images/2368.jpg | C | null | 5 | metric geometry - length | C | |
2369 | $ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Then $\sin \theta=$
<image1> | [
"$\\frac{\\sqrt{5}}{5}$",
"$\\frac{3}{5}$",
"$\\frac{\\sqrt{10}}{5}$",
"$\\frac{4}{5}$",
"$\\text{none of these}$",
"I don't know"
] | images/2369.jpg | B | null | 4 | metric geometry - angle | B | |
2371 | 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 $\frac{AD}{AB}$ is
<image1> | [
"$\\frac{1}{\\sqrt{2}}$",
"$\\frac{2}{2+\\sqrt{2}}$",
"$\\frac{1}{\\sqrt{3}}$",
"$\\frac{1}{\\sqrt[3]{6}}$",
"$\\frac{1}{\\sqrt[4]{12}}$",
"I don't know"
] | images/2371.jpg | E | null | 5 | metric geometry - length | E | |
2374 | On each horizontal line in the figure below, the five large dots indicate the populations of cities $A$, $B$, $C$, $D$ and $E$ in the year indicated. Which city had the greatest percentage increase in population from 1970 to 1980?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/2374.jpg | C | null | 3 | statistics | C | |
2375 | $ABC$ 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 ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ is
<image1> | [
"$\\frac{1}{36}$",
"$\\frac{1}{6}$",
"$\\frac{1}{4}$",
"$\\frac{\\sqrt{3}}{4}$",
"$\\frac{9+8\\sqrt{3}}{36}$",
"I don't know"
] | images/2375.jpg | C | null | 5 | metric geometry - area | C | |
2376 | In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY$ / $YZ$, in this rect... | [
"$4$",
"$1+2\\sqrt{3}$",
"$2\\sqrt{5}$",
"$\\frac{8+4\\sqrt{3}}{3}$",
"$5$",
"I don't know"
] | images/2376.jpg | E | null | 5 | metric geometry - length | E | |
2377 | In the figure, $AB \perp BC$, $BC \perp CD$, and $BC$ is tangent to the circle with center $O$ and diameter $AD$. In which one of the following cases is the area of $ABCD$ an integer?
<image1> | [
"AB=3, CD=1",
"AB=5, CD=2",
"AB=7, CD=3",
"AB=9, CD=4",
"AB=11, CD=5",
"I don't know"
] | images/2377.jpg | D | null | 5 | metric geometry - area | D | |
2381 | In $\triangle ABC, \angle A = 100^\circ, \angle B = 50^\circ, \angle C = 30^\circ, \overline{AH}$ is an altitude, and $\overline{BM}$ is a median. Then $\angle MHC =$
<image1> | [
"$15^\\circ$",
"$22.5^\\circ$",
"$30^\\circ$",
"$40^\\circ$",
"$45^\\circ$",
"I don't know"
] | images/2381.jpg | C | null | 4 | metric geometry - angle | C | |
2382 | Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is
<image1> | [
"$\\sin \\alpha$",
"$\\frac{1}{\\sin \\alpha}$",
"$\\frac{1}{1 - \\cos \\alpha}$",
"$\\frac{1}{\\sin^2 \\alpha}$",
"$\\frac{1}{(1 - \\cos \\alpha)^2}$",
"I don't know"
] | images/2382.jpg | B | null | 5 | metric geometry - area | B | |
2383 | In $\triangle ABC$, $AB = 5, BC = 7, AC = 9$ and $D$ is on $\overline{AC}$ with $BD = 5$. Find the ratio of $AD: DC$.
<image1> | [
"4:3",
"7:5",
"11:6",
"13:5",
"19:8",
"I don't know"
] | images/2383.jpg | E | null | 5 | metric geometry - length | E | |
2385 | A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At w... | [
"(35,44)",
"(36,45)",
"(37,45)",
"(44,35)",
"(45,36)",
"I don't know"
] | images/2385.jpg | D | null | 5 | algebra | D | |
2387 | An acute isosceles triangle, $ABC$ is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC=\angle ACB=2\angle D$ and $x$ is the radian measure of $\angle A$, then $x=$
<image1> | [
"$\\frac{3}{7}\\pi$",
"$\\frac{4}{9}\\pi$",
"$\\frac{5}{11}\\pi$",
"$\\frac{6}{13}\\pi$",
"$\\frac{7}{15}\\pi$",
"I don't know"
] | images/2387.jpg | A | null | 4 | metric geometry - angle | A | |
2389 | Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The figure shows the average announced by each person (not the original number the person picked). The number picked by ... | [
"$1$",
"$5$",
"$6$",
"$10$",
"$\\text{not uniquely determined from the given information}$",
"I don't know"
] | images/2389.jpg | A | null | 5 | algebra | A | |
2390 | In the arrow-shaped polygon [see figure], the angles at vertices $A$, $C$, $D$, $E$ and $F$ are right angles, $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. The area of the polygon is closest to
<image1> | [
"288",
"291",
"294",
"297",
"300",
"I don't know"
] | images/2390.jpg | E | null | 5 | metric geometry - area | E | |
2392 | Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle. If $PA = AB = 4$, then the area of the smaller circle is
<image1> | [
"$1.44\\pi$",
"$2\\pi$",
"$2.56\\pi$",
"$\\sqrt{8}\\pi$",
"$4\\pi$",
"I don't know"
] | images/2392.jpg | B | null | 5 | metric geometry - area | B | |
2393 | If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is
<image1> | [
"$\\frac{1}{3}$",
"$\\frac{2}{5}$",
"$\\frac{7}{15}$",
"$\\frac{8}{15}$",
"$\\frac{3}{5}$",
"I don't know"
] | images/2393.jpg | C | null | 5 | metric geometry - area | C | |
2394 | Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is
<image1> | [
"$\\frac{8}{5}$",
"$\\frac{7}{20}\\sqrt{21}$",
"$\\frac{1+\\sqrt{5}}{2}$",
"$\\frac{13}{8}$",
"$\\sqrt{3}$",
"I don't know"
] | images/2394.jpg | B | null | 5 | metric geometry - length | B | |
2396 | Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. 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 region... | [
"$10$",
"$12$",
"$15$",
"$10\\sqrt{3}$",
"$12\\sqrt{3}$",
"I don't know"
] | images/2396.jpg | E | null | 5 | metric geometry - area | E | |
2399 | Semicircle $\stackrel{\frown}{AB}$ has center $C$ and radius $1$. Point $D$ is on $\stackrel{\frown}{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\stackrel{\frown}{AE}$ and $\stackrel{\frown}{BF}$ have $B$ and $A$ as thei... | [
"$(2 - \\sqrt{2})\\pi$",
"$2\\pi - \\pi\\sqrt{2} - 1$",
"$\\left(1 - \\frac{\\sqrt{2}}{2}\\right)\\pi$",
"$\\frac{5\\pi}{2} - \\pi\\sqrt{2} - 1$",
"$(3 - 2\\sqrt{2})\\pi$",
"I don't know"
] | images/2399.jpg | B | null | 5 | metric geometry - area | B | |
2400 | 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 $\angle BED=$
<image1> | [
"$50^{\\circ}$",
"$55^{\\circ}$",
"$60^{\\circ}$",
"$65^{\\circ}$",
"$70^{\\circ}$",
"I don't know"
] | images/2400.jpg | D | null | 4 | metric geometry - angle | D | |
2401 | The convex pentagon $ABCDE$ has $\angle A=\angle B=120^{\circ}$, $EA=AB=BC=2$ and $CD=DE=4$. What is the area of $ABCDE$?
<image1> | [
"$10$",
"$7\\sqrt{3}$",
"$15$",
"$9\\sqrt{3}$",
"$12\\sqrt{5}$",
"I don't know"
] | images/2401.jpg | B | null | 5 | metric geometry - area | B | |
2402 | Amy painted a dart board 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'c... | [
"$2\\sqrt{3}-2$",
"$\\frac{3}{2}$",
"$\\frac{\\sqrt{5}+1}{2}$",
"$\\sqrt{3}$",
"$2$",
"I don't know"
] | images/2402.jpg | A | null | 5 | metric geometry - area | A | |
2404 | Points $A, B, C$ and $D$ are on a circle of diameter $1$, and $X$ is on diameter $\overline{AD}$. If $BX=CX$ and $3 \angle BAC=\angle BXC=36^{\circ}$, then $AX=$
<image1> | [
"$\\cos 6^{\\circ}\\cos 12^{\\circ} \\sec 18^{\\circ}$",
"$\\cos 6^{\\circ}\\sin 12^{\\circ} \\csc 18^{\\circ}$",
"$\\cos 6^{\\circ}\\sin 12^{\\circ} \\sec 18^{\\circ} \\$",
"$\\sin 6^{\\circ}\\sin 12^{\\circ} \\csc 18^{\\circ}$",
"$\\sin 6^{\\circ} \\sin 12^{\\circ} \\sec 18^{\\circ}$",
"I don't know"
] | images/2404.jpg | B | null | 5 | metric geometry - length | B | |
2405 | Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not... | [
"$\\text{exactly 2 such triangles} \\$",
"$\\text{exactly 3 such triangles} \\$",
"$\\text{exactly 7 such triangles} \\$",
"$\\text{exactly 15 such triangles} \\$",
"$\\text{more than 15 such triangles}$",
"I don't know"
] | images/2405.jpg | E | null | 5 | combinatorics | E | |
2410 | In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$
<image1> | [
"$30^{\\circ}$",
"$36^{\\circ}$",
"$48^{\\circ}$",
"$60^{\\circ}$",
"$72^{\\circ}$",
"I don't know"
] | images/2410.jpg | B | null | 4 | metric geometry - angle | B | |
2412 | 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 $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is
<image1> | [
"$\\frac{2}{7}$",
"$\\frac{1}{3}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"$\\frac{7}{9}$",
"I don't know"
] | images/2412.jpg | E | null | 4 | analytic geometry | E | |
2414 | 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 $AB/BC=$
<image1> | [
"$\\frac{1}{2}\\csc{\\frac{1}{4}}$",
"$2\\cos{\\frac{1}{2}}$",
"$4\\sin{\\frac{1}{2}}$",
"$\\csc{\\frac{1}{2}}$",
"$2\\sec{\\frac{1}{2}}$",
"I don't know"
] | images/2414.jpg | A | null | 5 | metric geometry - length | A | |
2415 | The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x$?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/2415.jpg | C | null | 3 | solid geometry | C | |
2416 | In $\triangle ABC$, $\angle C = 90^\circ, AC = 6$ and $BC = 8$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED = 90^\circ$. If $DE = 4$, then $BD =$
<image1> | [
"$5$",
"$\\frac{16}{3}$",
"$\\frac{20}{3}$",
"$\\frac{15}{2}$",
"$8$",
"I don't know"
] | images/2416.jpg | C | null | 5 | metric geometry - length | C | |
2419 | Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is
<image1> | [
"$\\frac{1}{6}$",
"$\\frac{1}{4}$",
"$\\frac{1}{3}$",
"$\\frac{2}{5}$",
"$\\frac{1}{2}$",
"I don't know"
] | images/2419.jpg | C | null | 5 | metric geometry - area | C | |
2420 | In the figure, $\overline{AB}$ and $\overline{CD}$ are diameters of the circle with center $O$, $\overline{AB} \perp \overline{CD}$, and chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is
<image1> | [
"$23\\pi$",
"$\\frac{47}{2}\\pi$",
"$24\\pi$",
"$\\frac{49}{2}\\pi$",
"$25\\pi$",
"I don't know"
] | images/2420.jpg | C | null | 5 | metric geometry - area | C | |
2423 | Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See fig... | [
"1",
"m/n",
"n/m",
"2m/n",
"2n/m",
"I don't know"
] | images/2423.jpg | B | null | 5 | metric geometry - area | B | |
2425 | The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon?
<image1> | [
"$\\displaystyle \\frac{1}{2}$",
"$\\displaystyle \\frac{\\sqrt{3}}{3}$",
"$\\displaystyle \\frac{2}{3}$",
"$\\displaystyle \\frac{3}{4}$",
"$\\displaystyle \\frac{\\sqrt{3}}{2}$",
"I don't know"
] | images/2425.jpg | D | null | 5 | metric geometry - area | D | |
2426 | Triangles $ABC$ and $ABD$ are isosceles with $AB =AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then $\angle C + \angle D$ is
<image1> | [
"$115^\\circ$",
"$120^\\circ$",
"$130^\\circ$",
"$135^\\circ$",
"$\\text{not uniquely determined}$",
"I don't know"
] | images/2426.jpg | D | null | 4 | metric geometry - angle | D | |
2429 | In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is
<image1> | [
"$2$",
"$3 - \\frac{\\sqrt{3}}{2}$",
"$\\frac{11}{5}$",
"$\\sqrt{5}$",
"$\\frac{9}{4}$",
"I don't know"
] | images/2429.jpg | C | null | 5 | metric geometry - area | C | |
2432 | In the figure, polygons $ A$, $ E$, and $ F$ are isosceles right triangles; $ B$, $ C$, and $ D$ are squares with sides of length $ 1$; and $ G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces.
<image1>
The volume of this polyhedron is | [
"1/2",
"2/3",
"3/4",
"5/6",
"4/3",
"I don't know"
] | images/2432.jpg | D | null | 3 | solid geometry | D | |
2434 | <image1>
Each of the sides of the five congruent rectangles is labeled with an integer, as shown above. These five rectangles are placed, without rotating or reflecting, in positions $I$ through $V$ so that the labels on coincident sides are equal.
<image2>
Which of the rectangles is in position $I$? | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/2434.jpg | E | null | 5 | combinatorial geometry | E | |
2435 | A square $ ABCD$ with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points $ E,F,G$ where $ E$ is the midpoint of $ BC$, $ F,G$ are on $ AB$ and $ CD$, respectively, and they're positioned that $ AF < FB, DG < GC$ and $ F$ is ... | [
"$\\frac{3}{5}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"$\\frac{5}{6}$",
"$\\frac{7}{8}$",
"I don't know"
] | images/2435.jpg | D | null | 5 | metric geometry - length | D | |
2437 | The figure shown is the union of a circle and two semicircles of diameters of $ a$ and $ b$, all of whose centers are collinear.
<image1>
The ratio of the area of the shaded region to that of the unshaded region is | [
"$\\sqrt{\\frac{a}{b}}$",
"$\\frac{a}{b}$",
"$\\frac{a^2}{b^2}$",
"$\\frac{a + b}{2b}$",
"$\\frac{a^2 + 2ab}{b^2 + 2ab}$",
"I don't know"
] | images/2437.jpg | B | null | 5 | metric geometry - area | B | |
2440 | A circle centered at $ O$ has radius $ 1$ and contains the point $ A$. Segment $ AB$ is tangent to the circle at $ A$ and $ \angle{AOB} = \theta$. If point $ C$ lies on $ \overline{OA}$ and $ \overline{BC}$ bisects $ \angle{ABO}$, then $ OC =$
<image1> | [
"$\\sec^2\\theta - \\tan\\theta$",
"$\\frac{1}{2}$",
"$\\frac{\\cos^2\\theta}{1 + \\sin\\theta}$",
"$\\frac{1}{1 + \\sin\\theta}$",
"$\\frac{\\sin\\theta}{\\cos^2\\theta}$",
"I don't know"
] | images/2440.jpg | D | null | 5 | metric geometry - length | D | |
2441 | The graph below shows a portion of the curve defined by the quartic polynomial $ P(x) = x^4 + ax^3 + bx^2 + cx + d$. Which of the following is the smallest?
<image1> | [
"$P( - 1)$",
"$\\text{The product of the zeros of }P$",
"$\\text{The product of the non - real zeros of }P$",
"$\\text{The sum of the coefficients of }P$",
"$\\text{The sum of the real zeros of }P$",
"I don't know"
] | images/2441.jpg | C | null | 4 | analytic geometry | C | |
2444 | A point $ P$ is selected at random from the interior of the pentagon with vertices $ A = (0,2)$, $B = (4,0)$, $C = (2 \pi + 1, 0)$, $D = (2 \pi + 1,4)$, and $ E = (0,4)$. What is the probability that $ \angle APB$ is obtuse?
<image1> | [
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$\\frac{5}{16}$",
"$\\frac{3}{8}$",
"$\\frac{1}{2}$",
"I don't know"
] | images/2444.jpg | C | null | 4 | metric geometry - angle | C | |
2445 | A circle centered at $ A$ with a radius of 1 and a circle centered at $ B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is
<image1> | [
"$\\frac{1}{3}$",
"$\\frac{2}{5}$",
"$\\frac{5}{12}$",
"$\\frac{4}{9}$",
"$\\frac{1}{2}$",
"I don't know"
] | images/2445.jpg | D | null | 5 | metric geometry - length | D | |
2446 | In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF = FG = GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$.
<image1> | [
"$\\frac{5}{2}$",
"$\\frac{35}{12}$",
"$3$",
"$\\frac{7}{2}$",
"$\\frac{35}{8}$",
"I don't know"
] | images/2446.jpg | C | null | 5 | metric geometry - area | C | |
2447 | In $ \triangle ABC$, $ \angle ABC = 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD = CD$ and $ \angle DAB = 15^\circ$. Find $ \angle ACB$.
<image1> | [
"$54^\\circ$",
"$60^\\circ$",
"$72^\\circ$",
"$75^\\circ$",
"$90^\\circ$",
"I don't know"
] | images/2447.jpg | D | null | 4 | metric geometry - angle | D | |
2449 | 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 $ \triangle ABC$, and extend $ \overline{BP}$ to meet $ \overline{AC}$ at $ D$. What is the probability that $ BD > 5\sqrt{2}$?
<image1> | [
"$\\frac{2 - \\sqrt{2}}{2}$",
"$\\frac{1}{3}$",
"$\\frac{3 - \\sqrt{3}}{3}$",
"$\\frac{1}{2}$",
"$\\frac{5 - \\sqrt{5}}{5}$",
"I don't know"
] | images/2449.jpg | C | null | 5 | metric geometry - length | C | |
2450 | The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
<image1>
<image2>
<image3>
<image4>
<image5> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/2450.jpg | B | null | 4 | analytic geometry | B | |
2451 | Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$.
<image1> | [
"$32$",
"$16 + 16\\sqrt{3}$",
"$48$",
"$32 + 16\\sqrt{3}$",
"$64$",
"I don't know"
] | images/2451.jpg | D | null | 5 | metric geometry - area | D | |
2452 | Square $ ABCD$ has sides of length $ 4$, and $ M$ is the midpoint of $ \overline{CD}$. A circle with radius $ 2$ and center $ M$ intersects a circle with raidus $ 4$ and center $ A$ at points $ P$ and $ D$. What is the distance from $ P$ to $ \overline{AD}$?
<image1> | [
"$3$",
"$\\frac{16}{5}$",
"$\\frac{13}{4}$",
"$2\\sqrt{3}$",
"$\\frac{7}{2}$",
"I don't know"
] | images/2452.jpg | B | null | 5 | metric geometry - length | B | |
2454 | Part of the graph of $ f(x) = x^3 + bx^2 + cx + d$ is shown. What is $ b$?
<image1> | [
"$-\\!4$",
"$-\\!2$",
"$0$",
"$2$",
"$4$",
"I don't know"
] | images/2454.jpg | B | null | 4 | analytic geometry | B | |
2456 | The graph of the line $ y = mx + b$ is shown. Which of the following is true?
<image1> | [
"mb < - 1",
"- 1 < mb < 0",
"mb = 0",
"0 < mb < 1",
"mb > 1",
"I don't know"
] | images/2456.jpg | B | null | 4 | analytic geometry | B | |
2458 | Square $ ABCD$ has side length $ 2$. A semicircle with diameter $ \overline{AB}$ is constructed inside the square, and the tangent to the semicricle from $ C$ intersects side $ \overline{AD}$ at $ E$. What is the length of $ \overline{CE}$?
<image1> | [
"$\\frac{2 + \\sqrt{5}}{2}$",
"$\\sqrt{5}$",
"$\\sqrt{6}$",
"$\\frac{5}{2}$",
"$5 - \\sqrt{5}$",
"I don't know"
] | images/2458.jpg | D | null | 5 | metric geometry - length | D | |
2459 | Circles $ A$, $ B$ and $ C$ are externally tangent to each other and internally tangent to circle $ D$. Circles $ B$ and $ C$ are congruent. Circle $ A$ has radius $ 1$ and passes through the center of $ D$. What is the radius of circle $ B$?
<image1> | [
"$\\frac{2}{3}$",
"$\\frac{\\sqrt{3}}{2}$",
"$\\frac{7}{8}$",
"$\\frac{8}{9}$",
"$\\frac{1 + \\sqrt{3}}{3}$",
"I don't know"
] | images/2459.jpg | D | null | 5 | metric geometry - length | D | |
2460 | In $ \triangle ABC$ , $ AB = 13$, $ AC = 5$, and $ BC = 12$. Points $ M$ and $ N$ lie on $ \overline{AC}$ and $ \overline{BC}$, respectively, with $ CM = CN = 4$. Points $ J$ and $ K$ are on $ \overline{AB}$ so that $ \overline{MJ}$ and $ \overline{NK}$ are perpendicular to $ \overline{AB}$. What is the area of pentago... | [
"$15$",
"$\\frac{81}{5}$",
"$\\frac{205}{12}$",
"$\\frac{240}{13}$",
"$20$",
"I don't know"
] | images/2460.jpg | D | null | 5 | metric geometry - area | D | |
2461 | In $ \triangle ABC$, $ AB = BC$, and $ BD$ is an altitude. Point $ E$ is on the extension of $ \overline{AC}$ such that $ BE = 10$. The values of $ \tan CBE$, $ \tan DBE$, and $ \tan ABE$ form a geometric progression, and the values of $ \cot DBE$, $ \cot CBE$, $ \cot DBC$ form an arithmetic progression. What is the ar... | [
"$16$",
"$\\frac{50}{3}$",
"$10\\sqrt{3}$",
"$8\\sqrt{5}$",
"$18$",
"I don't know"
] | images/2461.jpg | B | null | 5 | metric geometry - area | B | |
2463 | A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$?
<image1> | [
"$\\frac{1}{12}$",
"$\\frac{1}{9}$",
"$\\frac{1}{8}$",
"$\\frac{1}{6}$",
"$\\frac{1}{4}$",
"I don't know"
] | images/2463.jpg | A | null | 3 | solid geometry | A | |
2464 | The vertices of a $ 3 - 4 - 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
<image1> | [
"$12\\pi$",
"$\\frac{25\\pi}{2}$",
"$13\\pi$",
"$\\frac{27\\pi}{2}$",
"$14\\pi$",
"I don't know"
] | images/2464.jpg | E | null | 5 | metric geometry - area | E | |
2465 | Square $ ABCD$ has side length $ s$, a circle centered at $ E$ has radius $ r$, and $ r$ and $ s$ are both rational. The circle passes through $ D$, and $ D$ lies on $ \overline{BE}$. Point $ F$ lies on the circle, on the same side of $ \overline{BE}$ as $ A$. Segment $ AF$ is tangent to the circle, and $ AF = \sqrt{9 ... | [
"$\\frac{1}{2}$",
"$\\frac{5}{9}$",
"$\\frac{3}{5}$",
"$\\frac{5}{3}$",
"$\\frac{9}{5}$",
"I don't know"
] | images/2465.jpg | B | null | 5 | metric geometry - length | B | |
2466 | Circles with centers $ (2,4)$ and $ (14,9)$ have radii 4 and 9, respectively. The equation of a common external tangent to the circles can be written in the form $ y = mx + b$ with $ m > 0$. What is $ b$?
<image1> | [
"$\\frac{908}{199}$",
"$\\frac{909}{119}$",
"$\\frac{130}{17}$",
"$\\frac{911}{119}$",
"$\\frac{912}{119}$",
"I don't know"
] | images/2466.jpg | E | null | 4 | analytic geometry | E | |
2469 | A ship sails $ 10$ miles in a straight line from $ A$ to $ B$, turns through an angle between $ 45^{\circ}$ and $ 60^{\circ}$, and then sails another $ 20$ miles to $ C$. Let $ AC$ be measured in miles. Which of the following intervals contains $ AC^2$?
<image1> | [
"[400,500]",
"[500,600]",
"[600,700]",
"[700,800]",
"[800,900]",
"I don't know"
] | images/2469.jpg | D | null | 5 | metric geometry - length | D | |
2473 | There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?
<image1> | [
"$(C, A, E, D, B)$",
"$(C, A, D, E, B)$",
"$(C, D, E, A, B) \\ [1ex]$",
"$(C, E, A, D, B)$",
"$(C, E, D, A, B)$",
"I don't know"
] | images/2473.jpg | E | null | 1 | descriptive geometry | E | |
2474 | A unit cube has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3'$, and $P_4'$. Vertices $P_2, P_3$, and $P_4$ are adjacent to $P_1$, and for $1\leq i\leq 4$, vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'$, and ... | [
"$\\frac{3\\sqrt{2}}{4}$",
"$\\frac{7\\sqrt{6}}{16}$",
"$\\frac{\\sqrt{5}}{2}$",
"$\\frac{2\\sqrt{3}}{3}$",
"$\\frac{\\sqrt{6}}{2}$",
"I don't know"
] | images/2474.jpg | A | null | 3 | solid geometry | A | |
2475 | 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 side-length of the square?
<image1> | [
"$29\\sqrt{3}$",
"$\\frac{21}{2}\\sqrt{2}+\\frac{41}{2}\\sqrt{3}$",
"$20\\sqrt{3}+16$",
"$20\\sqrt{2}+13\\sqrt{3}$",
"$21\\sqrt{6}$",
"I don't know"
] | images/2475.jpg | A | null | 5 | metric geometry - length | A | |
2476 | Let $S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}$, and $(x,y) \neq (0,0) \}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle ABC$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)= \tan (\angle CBA)$. What ... | [
"$1$",
"$\\frac{625}{144}$",
"$\\frac{125}{24}$",
"$6$",
"$\\frac{625}{24}$",
"I don't know"
] | images/2476.jpg | B | null | 5 | combinatorial geometry | B | |
2477 | Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$... | [
"$1$",
"$\\frac{3}{2}$",
"$\\frac{21}{13}$",
"$\\frac{13}{8}$",
"$\\frac{5}{3}$",
"I don't know"
] | images/2477.jpg | C | null | 5 | metric geometry - length | C | |
2479 | A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?
<image1> | [
"$2+2\\sqrt{7}$",
"$3+2\\sqrt{5}$",
"$4+2\\sqrt{7}$",
"$4\\sqrt{5}$",
"$4\\sqrt{7}$",
"I don't know"
] | images/2479.jpg | A | null | 3 | solid geometry | A | |
2481 | In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
<image1> | [
"$\\frac{1}{2}(\\sqrt{6}-2)$",
"$\\frac{1}{4}$",
"$2-\\sqrt{3}$",
"$\\frac{\\sqrt{3}}{6}$",
"$1-\\frac{\\sqrt{2}}{2}$",
"I don't know"
] | images/2481.jpg | C | null | 5 | metric geometry - length | C | |
2482 | 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 $k\geq 1$, the circles in $\textstyle\bigcup_{j=0}^{k-1} L_j$ are ordered according to their points of tang... | [
"$\\frac{286}{35}$",
"$\\frac{583}{70}$",
"$\\frac{715}{73}$",
"$\\frac{143}{14}$",
"$\\frac{1573}{146}$",
"I don't know"
] | images/2482.jpg | D | null | 5 | algebra | D | |
2484 | 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 ratio $AF$ : $FD$?
<image1> | [
"3:2",
"5:3",
"2:1",
"7:3",
"5:2",
"I don't know"
] | images/2484.jpg | C | null | 5 | metric geometry - length | C | |
2485 | 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 $PQ$?... | [
"$1$",
"$\\frac{5}{8}\\sqrt{3}$",
"$\\frac{4}{5}\\sqrt{2}$",
"$\\frac{8}{15}\\sqrt{5}$",
"$\\frac{6}{5}$",
"I don't know"
] | images/2485.jpg | D | null | 5 | metric geometry - length | D | |
2486 | In the figure below, semicircles with centers at $A$ and $B$ and with radii $2$ and $1$, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $\overline{JK}$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A ci... | [
"$\\frac{3}{4}$",
"$\\frac{6}{7}$",
"$\\frac{1}{2}\\sqrt{3}$",
"$\\frac{5}{8}\\sqrt{2}$",
"$\\frac{11}{12}$",
"I don't know"
] | images/2486.jpg | B | null | 5 | metric geometry - length | B | |
2487 | Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\triangle{ABP}$, $\triangle{BCP}$, $\triangle{CDP}$, and $\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?
<image1> | [
"$100\\sqrt{2}$",
"$100\\sqrt{3}$",
"$200$",
"$200\\sqrt{2}$",
"$200\\sqrt{3}$",
"I don't know"
] | images/2487.jpg | C | null | 5 | metric geometry - area | C | |
2490 | Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\sin(2\angle BAD)$?
<image1> | [
"$\\frac{1}{3}$",
"$\\frac{\\sqrt{2}}{2}$",
"$\\frac{3}{4}$",
"$\\frac{7}{9}$",
"$\\frac{\\sqrt{3}}{2}$",
"I don't know"
] | images/2490.jpg | D | null | 4 | metric geometry - angle | D | |
2491 | The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$?$
<image1> | [
"$17$",
"$15 + 2\\sqrt{2}$",
"$13 + 4\\sqrt{2}$",
"$11 + 6\\sqrt{2}$",
"$21$",
"I don't know"
] | images/2491.jpg | C | null | 5 | metric geometry - length | C | |
2493 | 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?
<image1> | [
"4",
"$4\\sqrt{3}$",
"12",
"18",
"$12\\sqrt{3}$",
"I don't know"
] | images/2493.jpg | E | null | 5 | metric geometry - length | E | |
2494 | 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 $ABCD?$
<image1> | [
"$15$",
"$5\\sqrt{11}$",
"$3\\sqrt{35}$",
"$18$",
"$7\\sqrt{7}$",
"I don't know"
] | images/2494.jpg | C | null | 5 | metric geometry - area | C | |
2495 | 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 sh... | [
"$3$",
"$2+\\sqrt{2}$",
"$1+\\sqrt{6}$",
"$2\\sqrt{3}$",
"$3\\sqrt{2}$",
"I don't know"
] | images/2495.jpg | B | null | 5 | metric geometry - length | B | |
2497 | Five rectangles, $A$, $B$, $C$, $D$, and $E$, are arranged in a square as shown below. These rectangles have dimensions $1\times6$, $2\times4$, $5\times6$, $2\times7$, and $2\times3$, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/2497.jpg | B | null | 5 | combinatorial geometry | B | |
2501 | 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 angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ w... | [
"$\\arccos\\frac{5}{6}$",
"$\\arccos\\frac{4}{5}$",
"$\\arccos\\frac{3}{10}$",
"$\\arcsin\\frac{4}{5}$",
"$\\arcsin\\frac{5}{6}$",
"I don't know"
] | images/2501.jpg | A | null | 4 | metric geometry - angle | A | |
2503 | A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\sqrt{3}$ inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at t... | [
"$6 + 3\\pi$",
"$6 + 6\\pi$",
"$6\\sqrt{3}$",
"$6\\sqrt{5}$",
"$6\\sqrt{3} + \\pi$",
"I don't know"
] | images/2503.jpg | E | null | 3 | solid geometry | E | |
2505 | <image1>
The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory? | [
"$\\frac{1}{2}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"$\\frac{4}{5}$",
"$\\frac{9}{10}$",
"I don't know"
] | images/2505.jpg | C | null | 2 | statistics | C | |
2507 | <image1>
A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $ \text{X}$ is: | [
"$\\text{Z}$",
"$\\text{U}$",
"$\\text{V}$",
"$\\text{W}$",
"$\\text{Y}$",
"I don't know"
] | images/2507.jpg | E | null | 4 | solid geometry | E | |
2509 | <image1>
Five cards are lying on a table as shown. Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over? | [
"$3$",
"$4$",
"$6$",
"$\\text{P}$",
"$\\text{Q}$",
"I don't know"
] | images/2509.jpg | A | null | 4 | logic | A | |
2511 | <image1>
The table displays the grade distribution of the $ 30$ students in a mathematics class on the last two tests. For example, exactly one student received a "D" on Test 1 and a "C" on Test 2. What percent of the students received the same grade on both tests? | [
"$12 \\%$",
"$25 \\%$",
"$33 \\frac{1}{3} \\%$",
"$40 \\%$",
"$50 \\%$",
"I don't know"
] | images/2511.jpg | D | null | 2 | statistics | D | |
2512 | <image1>
Given that all angles shown are marked, the perimeter of the polygon shown is | [
"$14$",
"$20$",
"$28$",
"$48$",
"$\\text{cannot be determined from the information given}$",
"I don't know"
] | images/2512.jpg | C | null | 2 | metric geometry - length | C | |
2515 | <image1>
The large circle has diameter $ \overline{AC}$. The two small circles have their centers on $ \overline{AC}$ and just touch at $ O$, the center of the large circle. If each small circle has radius $ 1$, what is the value of the ratio of the area of the shaded region to the area of one of the small circles? | [
"$\\text{between }\\frac{1}{2} \\text{ and }1$",
"$1$",
"$\\text{between 1 and }\\frac{3}{2}$",
"$\\text{between }\\frac{3}{2} \\text{ and }2 \\\\$",
"$\\text{cannot be determined from the information given}$",
"I don't know"
] | images/2515.jpg | B | null | 3 | metric geometry - area | B | |
2516 | The area of the rectangular region is
<image1> | [
"$\\text{.088 m}^2$",
"$\\text{.62 m}^2$",
"$\\text{.88 m}^2$",
"$\\text{1.24 m}^2$",
"$\\text{4.22 m}^2$",
"I don't know"
] | images/2516.jpg | A | null | 3 | metric geometry - area | A | |
2518 | What fraction of the large $12$ by $18$ rectangular region is shaded?
<image1> | [
"$\\frac{1}{108}$",
"$\\frac{1}{18}$",
"$\\frac{1}{12}$",
"$\\frac{2}{9}$",
"$\\frac{1}{3}$",
"I don't know"
] | images/2518.jpg | C | null | 3 | metric geometry - area | C | |
2519 | $\text{ABCD}$ 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 between
<image1> | [
"$4\\text{ and }5$",
"$5\\text{ and }6$",
"$6\\text{ and }7$",
"$7\\text{ and }8$",
"$8\\text{ and }9$",
"I don't know"
] | images/2519.jpg | D | null | 3 | metric geometry - area | D | |
2520 | The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of
<image1> | [
"10.05",
"10.15",
"10.25",
"10.3",
"10.6",
"I don't know"
] | images/2520.jpg | B | null | 2 | metric geometry - length | B | |
2522 | If $ \angle\text{CBD} $ is a right angle, then this protractor indicates that the measure of $ \angle\text{ABC} $ is approximately
<image1> | [
"$20^\\circ$",
"$40^\\circ$",
"$50^\\circ$",
"$70^\\circ$",
"$120^\\circ$",
"I don't know"
] | images/2522.jpg | B | null | 5 | metric geometry - angle | B | |
2525 | The shaded region formed by the two intersecting perpendicular rectangles, in square units, is
<image1> | [
"$23$",
"$38$",
"$44$",
"$46$",
"$\\text{unable to be determined from the information given}$",
"I don't know"
] | images/2525.jpg | C | null | 3 | metric geometry - area | C |
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