Split (2)

test · 1.53k rows

id stringlengths 1 4	question stringlengths 31 709	options listlengths 6 6	image_path stringlengths 12 15	answer stringclasses 5 values	solution stringclasses 259 values	level int64 1 5	subject stringclasses 16 values	Answer(Option) stringclasses 5 values
2720	A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each 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 $ab$ ? <image1>	[ "$\\frac{1}{5}$", "$\\frac{2}{5}$", "$\\frac{1}{2}$", "$1$", "$4$", "I don't know" ]	images/2720.jpg	C	null	2	metric geometry - length	C
2725	Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ ad $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? <image1>	[ "$\\frac{1}{4}$", "$\\frac{7}{24}$", "$\\frac{1}{3}$", "$\\frac{3}{8}$", "$\\frac{5}{12}$", "I don't know" ]	images/2725.jpg	C	null	4	combinatorial geometry	C
2726	A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ 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? <image1>	[ "$238\\pi$", "$240\\pi$", "$260\\pi$", "$280\\pi$", "$500\\pi$", "I don't know" ]	images/2726.jpg	A	null	2	metric geometry - length	A
2731	A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch? Note: $1$ mile= $5280$ feet <image1>	[ "$\\frac{\\pi}{11}$", "$\\frac{\\pi}{10}$", "$\\frac{\\pi}{5}$", "$\\frac{2\\pi}{5}$", "$\\frac{2\\pi}{3}$", "I don't know" ]	images/2731.jpg	B	null	2	metric geometry - length	B
2732	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? <image1>	[ "$\\frac{11}{32}$", "$\\frac{3}{8}$", "$\\frac{13}{32}$", "$\\frac{7}{16}$", "$\\frac{15}{32}$", "I don't know" ]	images/2732.jpg	D	null	3	metric geometry - area	D
2735	A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle? <image1>	[ "$\\frac{1}{6}$", "$\\frac{1}{5}$", "$\\frac{1}{4}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$", "I don't know" ]	images/2735.jpg	A	null	3	analytic geometry	A
2736	In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$? <image1>	[ "$6\\sqrt{2}$", "$9$", "$12$", "$9\\sqrt{2}$", "$32$", "I don't know" ]	images/2736.jpg	C	null	3	metric geometry - area	C
2737	One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space? <image1>	[ "$9$", "$12\\frac{1}{2}$", "$15$", "$15\\frac{1}{2}$", "$17$", "I don't know" ]	images/2737.jpg	C	null	3	metric geometry - area	C
2739	Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA$. The area of the "bat wings" is <image1>	[ "$2$", "$2 \\frac{1}{2}$", "$3$", "$3 \\frac{1}{2}$", "$5$", "I don't know" ]	images/2739.jpg	C	null	3	metric geometry - area	C
2740	A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle? <image1>	[ "$4 \\sqrt{3}$", "$\\frac{120}{17}$", "$10$", "$\\frac{17\\sqrt{2}}{2}$", "$\\frac{17\\sqrt{3}}{2}$", "I don't know" ]	images/2740.jpg	B	null	2	metric geometry - length	B
2743	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 $\triangle ABD$? <image1>	[ "$\\frac{3}{4}$", "$\\frac{3}{2}$", "$2$", "$\\frac{12}{5}$", "$\\frac{5}{2}$", "I don't know" ]	images/2743.jpg	D	null	3	metric geometry - area	D
2745	In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle? <image1>	[ "$\\frac{7}{6}$", "$\\frac{13}{5}$", "$\\frac{59}{18}$", "$\\frac{10}{3}$", "$\\frac{60}{13}$", "I don't know" ]	images/2745.jpg	D	null	2	metric geometry - length	D
2746	In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length 2, and $m\angle TUS = 60^\circ$. Arcs $\overarc{TR}$ and $\overarc{SR}$ are each one-sixth of a circle with radius 2. What is the area of the region shown? <image1>	[ "$3\\sqrt{3}-\\pi$", "$4\\sqrt{3}-\\frac{4\\pi}{3}$", "$2\\sqrt{3}$", "$4\\sqrt{3}-\\frac{2\\pi}{3}$", "$4+\\frac{4\\pi}{3}$", "I don't know" ]	images/2746.jpg	B	null	3	metric geometry - area	B
2749	In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units? <image1>	[ "$\\frac{1}{4}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$", "$1$", "$\\frac{\\pi}{2}$", "I don't know" ]	images/2749.jpg	D	null	3	metric geometry - area	D
2751	In $\triangle ABC,$ a point $E$ is on $\overline{AB}$ with $AE=1$ and $EB=2$. Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}$. What is the ratio of the area of $CDEF$ to the area of $\triangle ABC?$ <image1>	[ "$\\frac{4}{9}$", "$\\frac{1}{2}$", "$\\frac{5}{9}$", "$\\frac{3}{5}$", "$\\frac{2}{3}$", "I don't know" ]	images/2751.jpg	A	null	3	metric geometry - area	A
2753	From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? <image1>	[ "$\\frac{2}{7}$", "$\\frac{5}{42}$", "$\\frac{11}{14}$", "$\\frac{5}{7}$", "$\\frac{6}{7}$", "I don't know" ]	images/2753.jpg	D	null	2	combinatorics	D
2754	In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of edges $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ <image1>	[ "$\\frac{5}{4}$", "$\\frac{4}{3}$", "$\\frac{3}{2}$", "$\\frac{25}{16}$", "$\\frac{9}{4}$", "I don't know" ]	images/2754.jpg	C	null	4	solid geometry	C
2757	A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $t$ from start to finish?$\phantom{h}$ <image1>	[ "A", "B", "C", "D", "E", "I don't know" ]	images/2757.jpg	B	null	2	statistics	B
2758	There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square? <image1>	[ "$\\frac{1}{5}$", "$\\frac{1}{4}$", "$\\frac{2}{5}$", "$\\frac{9}{20}$", "$\\frac{1}{2}$", "I don't know" ]	images/2758.jpg	C	null	2	combinatorics	C
2759	The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made? <image1>	[ "$\\text{The mean increases by 1 and the median does not change.}$", "$\\text{The mean increases by 1 and the median increases by 1.}$", "$\\text{The mean increases by 1 and the median increases by 5.}$", "$\\text{The mean increases by 5 and the median increases by 1.}$", "$\\text{The mean increases by 5 an...	images/2759.jpg	B	null	2	statistics	B
2760	The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face? <image1>	[ "$\\text{red}$", "$\\text{white}$", "$\\text{green}$", "$\\text{brown}$", "$\\text{purple}\n$", "I don't know" ]	images/2760.jpg	A	null	4	solid geometry	A
2765	There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities? <image1>	[ "65{,}000", "75{,}000", "85{,}000", "95{,}000", "105{,}000", "I don't know" ]	images/2765.jpg	D	null	2	statistics	D
2769	When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below. <image1> For example, starting with an input of $N = 7$, the machine will output $3 \cdot 7 + 1 = 22$. Then if the output is repeatedly inserted into the machine five more times, the final output is $26$. $$ 7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$$When the same 6-step process is applied to a different starting value of $N$, the final output is $1$. What is the sum of all such integers $N$? $$ N \to \_\_ \to \_\_ \to \_\_ \to \_\_ \to \_\_ \to 1$$	[ "73", "74", "75", "82", "83", "I don't know" ]	images/2769.jpg	E	null	5	arithmetic	E
2770	A large square region is paved with $n^2$ gray square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n = 3$. When $n = 24$, the $576$ gray tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n$? <image1>	[ "$\\frac{6}{25}$", "$\\frac{1}{4}$", "$\\frac{9}{25}$", "$\\frac{7}{16}$", "$\\frac{9}{16}$", "I don't know" ]	images/2770.jpg	A	null	3	metric geometry - area	A
2773	The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image? <image1> <image2>	[ "A", "B", "C", "D", "E", "I don't know" ]	images/2773.jpg	E	null	3	analytic geometry	E
2774	One sunny day, Ling decided to take a hike in the mountains. She left her house at $8 \, \textsc{am}$, drove at a constant speed of $45$ miles per hour, and arrived at the hiking trail at $10 \, \textsc{am}$. After hiking for $3$ hours, Ling drove home at a constant speed of $60$ miles per hour. Which of the following graphs best illustrates the distance between Ling's car and her house over the course of her trip? <image1>	[ "A", "B", "C", "D", "E", "I don't know" ]	images/2774.jpg	E	null	2	statistics	E
2775	The arrows on the two spinners shown below are spun. Let the number $N$ equal 10 times the number on Spinner $A$, added to the number on Spinner $B$. What is the probability that $N$ is a perfect square number? <image1>	[ "$\\frac{1}{16}$", "$\\frac{1}{8}$", "$\\frac{1}{4}$", "$\\frac{3}{8}$", "$\\frac{1}{2} $", "I don't know" ]	images/2775.jpg	B	null	2	combinatorics	B
2782	A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures? <image1> <image2>	[ "A", "B", "C", "D", "E", "I don't know" ]	images/2782.jpg	E	null	3	transformation geometry	E
2787	The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded? <image1>	[ "$\\frac{1}{4}$", "$\\frac{11}{36}$", "$\\frac{1}{3}$", "$\\frac{19}{36}$", "$\\frac{5}{9}$", "I don't know" ]	images/2787.jpg	B	null	3	metric geometry - area	B
2789	The letters $P$, $Q$, and $R$ are entered in a $20\times 20$ grid according to the pattern shown below. How many $P$s, $Q$s, and $R$s will appear in the completed table? <image1>	[ "$132~\\text{Ps}, 134~\\text{Qs}, 134~\\text{Rs}$", "$133~\\text{Ps}, 133~\\text{Qs}, 134~\\text{Rs}$", "$133~\\text{Ps}, 134~\\text{Qs}, 133~\\text{Rs}$", "$134~\\text{Ps}, 132~\\text{Qs}, 134~\\text{Rs}$", "$134~\\text{Ps}, 133~\\text{Qs}, 133~\\text{Rs}$", "I don't know" ]	images/2789.jpg	C	null	4	algebra	C
2791	An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac{2}{3}$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle? <image1>	[ "1:3", "3:8", "5:12", "7:16", "4:9", "I don't know" ]	images/2791.jpg	C	null	3	metric geometry - area	C
2792	Each square in a $3 \times 3$ grid is randomly filled with one of the $4$ gray-and-white tiles shown below on the right.<image1> What is the probability that the tiling will contain a large gray diamond in one of the smaller $2\times 2$ grids? Below is an example of one such tiling. <image2>	[ "$\\frac{1}{1024}$", "$\\frac{1}{256}$", "$\\frac{1}{64}$", "$\\frac{1}{16}$", "$\\frac{1}{4}$", "I don't know" ]	images/2792.jpg	C	null	4	combinatorial geometry	C

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