id stringlengths 1 4 | media images listlengths 1 1 | messages stringlengths 84 1.58k | answer stringclasses 371
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2079 | [{"role": "user", "question": "In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The resul... | 750 | numeric | 3 | solid geometry | |
208 | [{"role": "user", "question": "The figure shows a circle with the diameter $A B$ and point $D$ on it. Find $d$.\n<image1>", "hint": "", "options": {}, "choices": []}] | 4 | numeric | 4 | analytic geometry | |
2080 | [{"role": "user", "question": "In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\\circ}$ around the central sq... | 429 | numeric | 5 | transformation geometry | |
2081 | [{"role": "user", "question": "A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\\frac{m\\sqrt{p}}{n}$, whe... | 113 | numeric | 5 | transformation geometry | |
2082 | [{"role": "user", "question": "The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along ... | 790 | numeric | 5 | metric geometry - length | |
2083 | [{"role": "user", "question": "On square $ABCD,$ points $E,F,G,$ and $H$ lie on sides $\\overline{AB},\\overline{BC},\\overline{CD},$ and $\\overline{DA},$ respectively, so that $\\overline{EG} \\perp \\overline{FH}$ and $EG=FH = 34$. Segments $\\overline{EG}$ and $\\overline{FH}$ intersect at a point $P,$ and the area... | 850 | numeric | 5 | metric geometry - area | |
2084 | [{"role": "user", "question": "A rectangle has sides of length $a$ and $36$. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown.... | 720 | numeric | 5 | metric geometry - area | |
2085 | [{"role": "user", "question": "Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\\angle ABD$ exceeds $\\angle AHG$ by $12^\\circ$. Find the degree measure of $\... | 58 | numeric | 4 | metric geometry - angle | |
2086 | [{"role": "user", "question": "In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\\overline{AD}$. Points $F$ and $G$ lie on $\\overline{CE}$, and $H$ and $J$ lie on $\\overline{AB}$ and $\\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\\overline{GH}$, and $M$ ... | 539 | numeric | 5 | metric geometry - area | |
2087 | [{"role": "user", "question": "A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\\overarc{AB}$ on that face measures $120^\\circ$. The b... | 53 | numeric | 3 | solid geometry | |
2088 | [{"role": "user", "question": "A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n\n<image1>", "hint": "", "options... | 384 | numeric | 3 | solid geometry | |
2089 | [{"role": "user", "question": "Circles $\\mathcal{P}$ and $\\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\\mathcal{P}$ and point $C$ is on $\\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\\ell$ through $A$ intersec... | 129 | numeric | 5 | metric geometry - area | |
209 | [{"role": "user", "question": "Each of the cubes in the figure has the length of an edge equal to 1. What is the length of the segment $A B$?\n<image1>", "hint": "", "options": {"A": "$\\sqrt{17}$", "B": "7", "C": "$\\sqrt{13}$", "D": "$\\sqrt{7}$", "E": "$\\sqrt{14}$"}, "choices": []}] | A | mcq | 3 | solid geometry | |
2090 | [{"role": "user", "question": "A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and al... | 810 | numeric | 3 | solid geometry | |
2091 | [{"role": "user", "question": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and will paint each of the six sections a solid color. Find the number of ways you can choose to paint each of the six sections if no two adjacent section can be... | 732 | numeric | 5 | graph theory | |
2092 | [{"role": "user", "question": "The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\\sqrt{3}$, $5$, and $\\sqrt{37}$, as shown, is $\\frac{m\\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is... | 145 | numeric | 5 | metric geometry - area | |
2093 | [{"role": "user", "question": "Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius... | 110 | numeric | 5 | metric geometry - length | |
2094 | [{"role": "user", "question": "The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point \\(A\\). At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner ... | 4 | numeric | 5 | combinatorics | |
2095 | [{"role": "user", "question": "Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC= DE = FG = HA = 11$ is formed by removing four $6-8-10$ triangles from the corners of a $23\\times 27$ rectangle with side $\\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\\over... | 184 | numeric | 5 | metric geometry - area | |
2096 | [{"role": "user", "question": "In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\\fracmn$, where $m$ and $n$ are relatively prime positiv... | 109 | numeric | 5 | metric geometry - area | |
2097 | [{"role": "user", "question": "Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\\overline{BD} \\perp \\overline{BC}$. The line $\\ell$ through $D$ parallel to line $BC$ intersects sides $\\overline{AB}$ and $\\overline{AC}$ at points $E$ and $F$, respecti... | 336 | numeric | 5 | metric geometry - length | |
2098 | [{"role": "user", "question": "Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\\mathcal{P}$ and $\\mathcal{Q}$. The intersection of planes $\\mathcal{P}$ and $\\mathcal{Q}$ is the line $\\ell$. The distance from line $\\ell$ ... | 335 | numeric | 3 | solid geometry | |
2099 | [{"role": "user", "question": "Let $\\triangle ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\\triangle ABC$ at $A$ and the line perpendicular to $GO$ at $G$. Let $Y$ be the intersection of lines $XG$ and $BC$. Given that the me... | 592 | numeric | 4 | metric geometry - angle | |
21 | [{"role": "user", "question": "Which of the figures shown bellow cannot be cut out of the figure illustrated nearby?\n<image1>\n<image2>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": []}] | C | mcq | 1 | combinatorial geometry | |
210 | [{"role": "user", "question": "We take three points from the grid so that they were collinear. How many possibilities do we have?\n<image1>", "hint": "", "options": {}, "choices": []}] | 20 | numeric | 5 | combinatorics | |
2100 | [{"role": "user", "question": "Let $ABCD$ be a parallelogram with $\\angle BAD < 90^{\\circ}$. A circle tangent to sides $\\overline{DA}$, $\\overline{AB}$, and $\\overline{BC}$ intersects diagonal $\\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the ... | 150 | numeric | 5 | metric geometry - area | |
2101 | [{"role": "user", "question": "Let $\\ell_A$ and $\\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \\allowbreak A_3, \\allowbreak \\ldots, \\allowbreak A_m$ lie on $\\ell_A$, and distinct points $B_1, B_2, B_3, \\ldots, B_n$ lie on $\\ell_B$. Additionally, when segm... | 244 | numeric | 5 | combinatorics | |
2102 | [{"role": "user", "question": "Two externally tangent circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\\Omega$ passing through $O_1$ and $O_2$ intersects $\\omega_1$ at $B$ and $C$ and $\\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, ... | 140 | numeric | 5 | metric geometry - area | |
2103 | [{"role": "user", "question": "Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\\sqrt{21}$ and $\\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\\fracmn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n... | 125 | numeric | 3 | solid geometry | |
2104 | [{"role": "user", "question": "The following analog clock has two hands that can move independently of each other.\n<image1>\nInitially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while... | 608 | numeric | 5 | combinatorics | |
2105 | [{"role": "user", "question": "Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\\overline{AB}$ also lies inside this L-shaped region can... | 35 | numeric | 5 | metric geometry - area | |
2106 | [{"role": "user", "question": "Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$. One way to do this is shown below. Find the number of positi... | 144 | numeric | 5 | combinatorics | |
2107 | [{"role": "user", "question": "A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\\overline{AB}$ and $\\overline{CD}$ are parallel edges of the cube, and $\\overline{AC}$ and $\\overline{BD}$ are diagonals of the faces of the cube. Vertex $A$ of the cube is set on a horizontal plane $\\mathcal P$ so t... | 751 | numeric | 3 | solid geometry | |
2108 | [{"role": "user", "question": "Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?\n\n$\\mathrm{a.}\\ \\text{the length of the segment} MN$\n\n$\\mathrm{b.}\\ \\text{the perimeter o... | 1 | numeric | 4 | metric geometry - area | |
2109 | [{"role": "user", "question": "In rectangle $ ABCD$, $ AD = 1$, $ P$ is on $ \\overline{AB}$, and $ \\overline{DB}$ and $ \\overline{DP}$ trisect $ \\angle ADC$. What is the perimeter of $ \\triangle BDP$?\n<image1>", "hint": "", "options": {"A": "$3 + \\frac{\\sqrt{3}}{3}$", "B": "$2 + \\frac{4\\sqrt{3}}{3}$", "C": "$... | B | mcq | 4 | metric geometry - length | |
211 | [{"role": "user", "question": "In the figure each asterisk stands for one digit. The sum of the digits of the product is equal to\n<image1>", "hint": "", "options": {}, "choices": []}] | 16 | numeric | 5 | algebra | |
2110 | [{"role": "user", "question": "Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 20201 | numeric | 3 | algebra | |
2111 | [{"role": "user", "question": "There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?\n\n<image1>", "hint": "", "options": {"A": "$... | B | mcq | 4 | combinatorics | |
2112 | [{"role": "user", "question": "The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.\n\n<image1>", "hint": "", "options": {"A": "$\\frac{4\\sqrt{5}}{3}$", "B": "$\\frac{5\\sqrt{5}}{3}$", "C": "$\\frac{12\\sqrt{5}}{7}$", "... | B | mcq | 4 | metric geometry - length | |
2113 | [{"role": "user", "question": "How many of the twelve pentominoes pictured below have at least one line of symmetry?\n<image1>", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 4 | counting | |
2114 | [{"role": "user", "question": "Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains $ 8$ unit squares. The second ring contains $ 16$ unit squares.\n<image1>\nIf we continue this process, the number of unit squares in the $ 100^\\text... | C | mcq | 3 | algebra | |
2115 | [{"role": "user", "question": "Which of the cones listed below can be formed from a $ 252^\\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides?\n<image1>\n<image2>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": []}] | C | mcq | 2 | solid geometry | |
2116 | [{"role": "user", "question": "The plane is tiled by congruent squares and congruent pentagons as indicated.\n<image1>\nThe percent of the plane that is enclosed by the pentagons is closest to", "hint": "", "options": {"A": "50", "B": "52", "C": "54", "D": "56", "E": "58"}, "choices": []}] | D | mcq | 4 | metric geometry - area | |
2117 | [{"role": "user", "question": "In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $ v$, $ w$, $ x$, $ y$, and $ z$. Find $ y + z$.\n<image1>", "hint": "", "options": {}, "choices": []}] | 46 | numeric | 3 | algebra | |
2118 | [{"role": "user", "question": "Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.\n<image1>", "hint": "", "options": {... | C | mcq | 4 | metric geometry - area | |
2119 | [{"role": "user", "question": "Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct?\n\n<image1>", "hint": "",... | A | mcq | 4 | metric geometry - area | |
212 | [{"role": "user", "question": "A circle is inscribed in the triangle $A B C$ (see the figure), $A C=5, A B=6, B C=3$. The segment $E D$ is tangent to the circle. The perimeter of the triangle $A D E$ is\n<image1>", "hint": "", "options": {}, "choices": []}] | 8 | numeric | 5 | metric geometry - length | |
2120 | [{"role": "user", "question": "In trapezoid $ ABCD$ with bases $ AB$ and $ CD$, we have $ AB=52$, $ BC=12$, $ CD=39$, and $ DA=5$. The area of $ ABCD$ is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 210 | numeric | 4 | metric geometry - area | |
2121 | [{"role": "user", "question": "Circles of radius $ 2$ and $ 3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.\n\n<image1>", "hint": "", "options": {"A": "$3\\pi$", "B": "$4\\pi$", "C": "$6\\pi$", "D": "$9\\pi$", "E": "$12\\pi$"}, "choices": []... | E | mcq | 4 | metric geometry - area | |
2122 | [{"role": "user", "question": "The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing... | 6 | numeric | 2 | solid geometry | |
2123 | [{"role": "user", "question": "A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{6}\\pi - \\frac... | C | mcq | 4 | metric geometry - area | |
2124 | [{"role": "user", "question": "In rectangle $ ABCD$, we have $ AB=8$, $ BC=9$, $ H$ is on $ \\overline{BC}$ with $ BH=6$, $ E$ is on $ \\overline{AD}$ with $ DE=4$, line $ EC$ intersects line $ AH$ at $ G$, and $ F$ is on line $ AD$ with $ \\overline{GF}\\perp\\overline{AF}$. Find the length $ GF$.\n<image1>", "hint": ... | 20 | numeric | 4 | metric geometry - length | |
2125 | [{"role": "user", "question": "A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $ 3$ rows of small congruent equilateral triangles, with $ 5$ small triangles in the base row. How many toothpicks would be needed to construct... | C | mcq | 3 | algebra | |
2126 | [{"role": "user", "question": "Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $ \\$$1 each, begonias... | 108 | numeric | 3 | algebra | |
2127 | [{"role": "user", "question": "Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the followi... | D | mcq | 4 | metric geometry - length | |
2128 | [{"role": "user", "question": "Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \\overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle b... | E | mcq | 4 | metric geometry - area | |
2129 | [{"role": "user", "question": "In rectangle $ ABCD$, $ AB=5$ and $ BC=3$. Points $ F$ and $ G$ are on $ \\overline{CD}$ so that $ DF=1$ and $ GC=2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \\triangle{AEB}$.\n<image1>", "hint": "", "options": {"A": "$10$", "B": "$\\frac{21}{2}$", "C": "$12$", "D": "$... | D | mcq | 4 | metric geometry - area | |
213 | [{"role": "user", "question": "The square $A B C D$ has a side of length 1 and $M$ is the midpoint of $A B$. The area of the shaded region is\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{24}$", "B": "$\\frac{1}{16}$", "C": "$\\frac{1}{8}$", "D": "$\\frac{1}{12}$", "E": "$\\frac{2}{13}$"}, "choices": []}] | D | mcq | 5 | metric geometry - area | |
2130 | [{"role": "user", "question": "A regular octagon $ ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ ABEF$?\n<image1>", "hint": "", "options": {"A": "$1-\\frac{\\sqrt{2}}{2}$", "B": "$\\frac{\\sqrt{2}}{4}$", "C": "$\\sqrt{2}-1$", "D": "$\\frac{1}{2}$", "E": "$\\frac{1+\\sqrt{2}}{4}$"}, "choi... | D | mcq | 4 | metric geometry - area | |
2131 | [{"role": "user", "question": "A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{21}$", "B": "$\\frac{1}{14}$", "C": "$\\... | C | mcq | 4 | combinatorics | |
2132 | [{"role": "user", "question": "In the figure, $ \\angle EAB$ and $ \\angle ABC$ are right angles. $ AB = 4, BC = 6, AE = 8$, and $ \\overline{AC}$ and $ \\overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \\triangle ADE$ and $ \\triangle BDC$?\n<image1>", "hint": "", "options": {}, "choices... | 4 | numeric | 4 | metric geometry - area | |
2133 | [{"role": "user", "question": "The $ 5\\times 5$ grid shown contains a collection of squares with sizes from $ 1\\times 1$ to $ 5\\times 5$. How many of these squares contain the black center square?\n<image1>", "hint": "", "options": {}, "choices": []}] | 19 | numeric | 4 | combinatorics | |
2134 | [{"role": "user", "question": "A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?\n\n<image1>", "hint": "", "options": {"A":... | C | mcq | 2 | solid geometry | |
2135 | [{"role": "user", "question": "Points $E$ and $F$ are located on square $ABCD$ so that $\\Delta BEF$ is equilateral. What is the ratio of the area of $\\Delta DEF$ to that of $\\Delta ABE$?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{4}{3}$", "B": "$\\frac{3}{2}$", "C": "$\\sqrt{3}$", "D": "$2$", "E": "$1+\\sqr... | D | mcq | 4 | metric geometry - area | |
2136 | [{"role": "user", "question": "Two distinct lines pass through the center of three concentric circles of radii $3$, $2$, and $1$. The area of the shaded region in the diagram is $8/13$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\\pi$ radians is $18... | B | mcq | 2 | metric geometry - angle | |
2137 | [{"role": "user", "question": "Square $ABCD$ has side length 2. A semicircle with diameter $AB$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $AD$ at $E$. What is the length of $CE$?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{2+\\sqrt{5}}2$", "B": "$\\sqrt{5}$", "... | D | mcq | 4 | metric geometry - length | |
2138 | [{"role": "user", "question": "An annulus is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \\overline{OX}$ be a radius of the larger circle, let $ \\overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \\overline{OY}$ be the r... | A | mcq | 4 | metric geometry - area | |
2139 | [{"role": "user", "question": "Three circles of radius $ 1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{2 + \\sqrt{6}}{3}$", "B": "$2$", "C": "$\\frac{2 + 3\\sqrt{2}}{3}$", "D": "$\\frac{3 + 2... | D | mcq | 4 | metric geometry - length | |
214 | [{"role": "user", "question": "We used metal rods to build this nice ensemble. We know there are 61 octagons. How many rods are there?\n<image1>", "hint": "", "options": {}, "choices": []}] | 446 | numeric | 5 | algebra | |
2140 | [{"role": "user", "question": "In right triangle $ \\triangle ACE$, we have $ AC = 12$, $ CE = 16$, and $ EA = 20$. Points $ B$, $ D$, and $ F$ are located on $ \\overline{AC}$, $ \\overline{CE}$, and $ \\overline{EA}$, respectively, so that $ AB = 3$, $ CD = 4$, and $ EF = 5$. What is the ratio of the area of $ \\tria... | E | mcq | 4 | metric geometry - area | |
2141 | [{"role": "user", "question": "In $ \\triangle ABC$ points $ D$ and $ E$ lie on $ \\overline{BC}$ and $ \\overline{AC}$, respectively. If $ \\overline{AD}$ and $ \\overline{BE}$ intersect at $ T$ so that $ AT/DT = 3$ and $ BT/ET = 4$, what is $ CD/BD$?\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{8}$", "B": "$\... | D | mcq | 4 | metric geometry - length | |
2142 | [{"role": "user", "question": "A circle of radius $ 1$ is internally tangent to two circles of radius $ 2$ at points $ A$ and $ B$, where $ AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles?\n<image1>... | B | mcq | 4 | metric geometry - area | |
2143 | [{"role": "user", "question": "Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \\sqrt{50}$ and $ BE = 1$. What is the area of the inner square $ EFGH$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 36 | numeric | 4 | metric geometry - length | |
2144 | [{"role": "user", "question": "The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $ 2$?\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{3}\\pi+\\frac{\\sqrt{3}}{2}$", ... | B | mcq | 4 | metric geometry - length | |
2145 | [{"role": "user", "question": "In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\\overline{AB}$,$\\overline{BC}$,$\\overline{CD}$,$\\overline{DE}$, and $\\overli... | 12 | numeric | 3 | algebra | |
2146 | [{"role": "user", "question": "Three one-inch squares are palced with their bases on a line. The center square is lifted out and rotated $ 45^\\circ$, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $ B$ from the line on w... | D | mcq | 4 | transformation geometry | |
2147 | [{"role": "user", "question": "Let $ \\overline{AB}$ be a diameter of a circle and $ C$ be a point on $ \\overline{AB}$ with $ 2 \\cdot AC = BC$. Let $ D$ and $ E$ be points on the circle such that $ \\overline{DC} \\perp \\overline{AB}$ and $ \\overline{DE}$ is a second diameter. What is the ratio of the area of $ \\t... | C | mcq | 4 | metric geometry - length | |
2148 | [{"role": "user", "question": "An $ 8$-foot by $ 10$-foot floor is tiled with square tiles of size $ 1$ foot by $ 1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $ 1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the ... | A | mcq | 4 | metric geometry - length | |
2149 | [{"role": "user", "question": "Equilateral $ \\triangle ABC$ has side length $ 2$, $ M$ is the midpoint of $ \\overline{AC}$, and $ C$ is the midpoint of $ \\overline{BD}$. What is the area of $ \\triangle CDM$?\n<image1>", "hint": "", "options": {"A": "$\\frac{\\sqrt{2}}{2}$", "B": "$\\frac{3}{4}$", "C": "$\\frac{\\sq... | C | mcq | 4 | metric geometry - length | |
215 | [{"role": "user", "question": "The suare in the diagram has side length 1. The radius of the small circle would\nthen be of the length\n<image1>", "hint": "", "options": {"A": "$\\sqrt{2}-1$", "B": "$\\frac{1}{4}$", "C": "$\\frac{\\sqrt{2}}{4}$", "D": "$1-\\frac{\\sqrt{2}}{2}$", "E": "$(\\sqrt{2}-1)^{2}$"}, "choices": ... | E | mcq | 5 | metric geometry - length | |
2150 | [{"role": "user", "question": "The $ 8\\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 3 | combinatorial geometry | |
2151 | [{"role": "user", "question": "Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet?\n\n<image1>", "hint": "", "options": {"A": "$\\text{ I, by... | C | mcq | 4 | metric geometry - area | |
2152 | [{"role": "user", "question": "A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from t... | 173 | numeric | 4 | metric geometry - length | |
2153 | [{"role": "user", "question": "A circle of radius 1 is tangent to a circle of radius 2. The sides of $ \\triangle ABC$ are tangent to the circles as shown, and the sides $ \\overline{AB}$ and $ \\overline{AC}$ are congruent. What is the area of $ \\triangle ABC$?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{35... | D | mcq | 4 | metric geometry - area | |
2154 | [{"role": "user", "question": "In rectangle $ ADEH$, points $ B$ and $ C$ trisect $ \\overline{AD}$, and points $ G$ and $ F$ trisect $ \\overline{HE}$. In addition, $ AH = AC = 2$. What is the area of quadrilateral $ WXYZ$ shown in the figure?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{2}$", "B": "$\\fra... | A | mcq | 4 | metric geometry - area | |
2155 | [{"role": "user", "question": "Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE = 5$. What is $ CD$?\n<image1>", "hint": "", "options": {"A": "$13$", "B": "$\\frac{44}{3... | B | mcq | 4 | metric geometry - length | |
2156 | [{"role": "user", "question": "A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $ 2/\\pi $, as shown. What is the perimeter of this region?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{4}\\pi$", "B": "$2$", "C": "$\\frac{8}\\pi$", "D": "$4$", "E": "$\\frac{16}{\\pi... | D | mcq | 4 | metric geometry - length | |
2157 | [{"role": "user", "question": "A square of area $40$ is inscribed in a semicircle as shown. What is the area of the semicircle?\n\n<image1>", "hint": "", "options": {"A": "$20\\pi$", "B": "$25\\pi$", "C": "$30\\pi$", "D": "$40\\pi$", "E": "$50\\pi$"}, "choices": []}] | B | mcq | 4 | metric geometry - area | |
2158 | [{"role": "user", "question": "Rhombus $ ABCD$ is similar to rhombus $ BFDE$. The area of rhombus $ ABCD$ is 24, and $ \\angle BAD = 60^\\circ$. What is the area of rhombus $ BFDE$?\n\n<image1>", "hint": "", "options": {"A": "$6$", "B": "$4\\sqrt{3}$", "C": "$8$", "D": "$9$", "E": "$6\\sqrt{3}$"}, "choices": []}] | C | mcq | 4 | metric geometry - area | |
2159 | [{"role": "user", "question": "A circle of radius 2 is centered at $ O$. Square $ OABC$ has side length 1. Sides $ \\overline{AB}$ and $ \\overline{CB}$ are extended past $ b$ to meet the circle at $ D$ and $ E$, respectively. What is the area of the shaded region in the figure, which is bounded by $ \\overline{BD}$, $... | A | mcq | 4 | metric geometry - area | |
216 | [{"role": "user", "question": "Each side of a triangle $A B C$ is being extended to the points $\\mathrm{P}, \\mathrm{Q}, \\mathrm{R}, \\mathrm{S}, \\mathrm{T}$ and $\\mathrm{U}$, so that $\\mathrm{PA}=\\mathrm{AB}=\\mathrm{BS}, \\mathrm{TC}=\\mathrm{CA}$ $=\\mathrm{AQ}$ and $\\mathrm{UC}=\\mathrm{CB}=\\mathrm{BR}$. Th... | 13 | numeric | 5 | metric geometry - area | |
2160 | [{"role": "user", "question": "A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?\n\n\n<image1>", "hint": "", "options": {"A": "$15$", "B"... | D | mcq | 4 | metric geometry - area | |
2161 | [{"role": "user", "question": "Circles with centers $ O$ and $ P$ have radii 2 and 4, respectively, and are externally tangent. Points $ A$ and $ B$ are on the circle centered at $ O$, and points $ C$ and $ D$ are on the circle centered at $ P$, such that $ \\overline{AD}$ and $ \\overline{BC}$ are common external tan... | B | mcq | 4 | metric geometry - area | |
2162 | [{"role": "user", "question": "Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom... | 2.7 | numeric | 4 | metric geometry - length | |
2163 | [{"role": "user", "question": "A cube with side length $ 1$ is sliced by a plane that passes through two diagonally opposite vertices $ A$ and $ C$ and the midpoints $ B$ and $ D$ of two opposite edges not containing $ A$ and $ C$, ac shown. What is the area of quadrilateral $ ABCD$?\n<image1>", "hint": "", "options": ... | A | mcq | 2 | solid geometry | |
2164 | [{"role": "user", "question": "A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Fur... | C | mcq | 4 | metric geometry - length | |
2165 | [{"role": "user", "question": "A circle of radius $ 2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{2}$", "B": "$\\frac{\\pi}{6}$", "C": "$\\frac{2}{\\pi}$",... | A | mcq | 4 | metric geometry - area | |
2166 | [{"role": "user", "question": "Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD=3$, and $ DC=4$. What is the area of $ \\triangle{ABC}$?\n<image1>", "hint": "", "options": {"A": "$4\\sqrt{3}$", "B": "$7\\sqrt{3}$", "C": "$21$", "D": "$14\\sqrt{3}$", "E": "$42$"}, "choice... | B | mcq | 4 | metric geometry - area | |
2167 | [{"role": "user", "question": "In quadrilateral $ ABCD$, $ AB = 5$, $ BC = 17$, $ CD = 5$, $ DA = 9$, and $ BD$ is an integer. What is $ BD$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 13 | numeric | 4 | metric geometry - length | |
2168 | [{"role": "user", "question": "Four congruent rectangles are placed as shown. The area of the outer square is $ 4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?\n<image1>", "hint": "", "options": {"A": "$3$", "B": "$\\sqrt{10}$", ... | A | mcq | 4 | metric geometry - length |
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