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2439 | [{"role": "user", "question": "Consider all triangles $ ABC$ satisfying the following conditions: $ AB = AC$, $ D$ is a point on $ \\overline{AC}$ for which $ \\overline{BD} \\perp \\overline{AC}$, $ AD$ and $ CD$ are integers, and $ BD^2 = 57$. \n<image1>\nAmong all such triangles, the smallest possible value of $ AC$... | 11 | numeric | 5 | metric geometry - length | |
244 | [{"role": "user", "question": "A clock has three hands in different lengths (for seconds, minutes and hours). We don't know the length of each hand but we know that the clock shows the correct time. At 12:55:30 the hands are in the positions shown on the right. What does the clockface look like at 8:10:00?\n<image1>\n<... | A | mcq | 5 | algebra | |
2440 | [{"role": "user", "question": "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 =$\n\n<image1>", "hint": "", "options": {"A... | D | mcq | 5 | metric geometry - length | |
2441 | [{"role": "user", "question": "The graph below shows a portion of the curve defined by the quartic polynomial $ P(x) = x^4 + ax^3 + bx^2 + cx + d$. Which of the following is the smallest?\n<image1>", "hint": "", "options": {"A": "$P( - 1)$", "B": "$\\text{The product of the zeros of }P$", "C": "$\\text{The product of t... | C | mcq | 4 | analytic geometry | |
2442 | [{"role": "user", "question": "If circular arcs $ AC$ and $ BC$ have centers at $ B$ and $ A$, respectively, then there exists a circle tangent to both $ \\stackrel{\\frown}{AC}$ and $ \\stackrel{\\frown}{BC}$, and to $ \\overline{AB}$. If the length of $ \\stackrel{\\frown}{BC}$ is $ 12$, then the circumference of the... | 27 | numeric | 5 | metric geometry - length | |
2443 | [{"role": "user", "question": "Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)\n<image1>... | 1680 | numeric | 3 | solid geometry | |
2444 | [{"role": "user", "question": "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?\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{5}$", "B... | C | mcq | 4 | metric geometry - angle | |
2445 | [{"role": "user", "question": "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\n<image1>", "hint": "", "options": {"A"... | D | mcq | 5 | metric geometry - length | |
2446 | [{"role": "user", "question": "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 ... | C | mcq | 5 | metric geometry - area | |
2447 | [{"role": "user", "question": "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$.\n<image1>", "hint": "", "options": {"A": "$54^\\circ$", "B": "$60^\\circ$", "C": "$72^\\circ$", "D": "$75^\\circ$", "E": "$... | D | mcq | 4 | metric geometry - angle | |
2448 | [{"role": "user", "question": "The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) = 6$ have?\n<image1>", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 4 | analytic geometry | |
2449 | [{"role": "user", "question": "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}$?\n... | C | mcq | 5 | metric geometry - length | |
245 | [{"role": "user", "question": "The water level in a port rises and falls on a certain day as shown in the diagram. How many hours on that day was the water level over $30 \\mathrm{~cm}$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 13 | numeric | 3 | statistics | |
2450 | [{"role": "user", "question": "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 $-4\\leq x \\leq 4$?\n<image1>\n<image2>\n<image3>\n<image4>\n<image5>", "hin... | B | mcq | 4 | analytic geometry | |
2451 | [{"role": "user", "question": "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$.\n\n<image1>", "hint": "", "options": {"A": "$32$", "B": "$16 + 16\\sqrt{3}$", "C": "$48... | D | mcq | 5 | metric geometry - area | |
2452 | [{"role": "user", "question": "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}$?\n<image1>", "hint": "", "opt... | B | mcq | 5 | metric geometry - length | |
2453 | [{"role": "user", "question": "Several figures can be made by attaching two equilateral triangles to the regular pentagon $ ABCDE$ in two of the five positions shown. How many non-congruent figures can be constructed in this way?\n<image1>", "hint": "", "options": {}, "choices": []}] | 2 | numeric | 5 | combinatorial geometry | |
2454 | [{"role": "user", "question": "Part of the graph of $ f(x) = x^3 + bx^2 + cx + d$ is shown. What is $ b$?\n<image1>", "hint": "", "options": {"A": "$-\\!4$", "B": "$-\\!2$", "C": "$0$", "D": "$2$", "E": "$4$"}, "choices": []}] | B | mcq | 4 | analytic geometry | |
2455 | [{"role": "user", "question": "Let $ ABCD$ be a rhombus with $ AC=16$ and $ BD=30$. Let $ N$ be a point on $ \\overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \\overline{AC}$ and $ \\overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ P... | 7 | numeric | 5 | metric geometry - length | |
2456 | [{"role": "user", "question": "The graph of the line $ y = mx + b$ is shown. Which of the following is true?\n<image1>", "hint": "", "options": {"A": "mb < - 1", "B": "- 1 < mb < 0", "C": "mb = 0", "D": "0 < mb < 1", "E": "mb > 1"}, "choices": []}] | B | mcq | 4 | analytic geometry | |
2457 | [{"role": "user", "question": "In the overlapping triangles $ \\triangle{ABC}$ and $ \\triangle{ABE}$ sharing common side $ AB$, $ \\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 ... | 4 | numeric | 5 | metric geometry - area | |
2458 | [{"role": "user", "question": "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}$?\n\n<image1>", "hint": "", "options": {"A": "$... | D | mcq | 5 | metric geometry - length | |
2459 | [{"role": "user", "question": "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$?\n<image1>", "hint": "", "options": {"A": "$\\f... | D | mcq | 5 | metric geometry - length | |
246 | [{"role": "user", "question": "In a list of five numbers the first number is 2 and the last one is 12. The product of the first three numbers is 30 , of the middle three 90 and of the last three 360. What is the middle number in that list?\n<image1>", "hint": "", "options": {}, "choices": []}] | 5 | numeric | 5 | algebra | |
2460 | [{"role": "user", "question": "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 $ \\over... | D | mcq | 5 | metric geometry - area | |
2461 | [{"role": "user", "question": "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... | B | mcq | 5 | metric geometry - area | |
2462 | [{"role": "user", "question": "Three circles of radius $ s$ are drawn in the first quadrant of the $ xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $ x$-axis, and the third is tangent to the first circle and the $ y$-axis. A circle of radius $ r > s$ is tangent to... | 9 | numeric | 5 | metric geometry - length | |
2463 | [{"role": "user", "question": "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$?\n<image1>", "h... | A | mcq | 3 | solid geometry | |
2464 | [{"role": "user", "question": "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?\n<image1>", "hint": "", "options": {"A": "$12\\pi$", "B": "$\\frac{25\\pi}{2}$", "C": "$13\\pi$", "D": "$\\frac{27\\pi}{... | E | mcq | 5 | metric geometry - area | |
2465 | [{"role": "user", "question": "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 t... | B | mcq | 5 | metric geometry - length | |
2466 | [{"role": "user", "question": "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$?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{908}{199}$", "B": "$\\frac{909}{... | E | mcq | 4 | analytic geometry | |
2467 | [{"role": "user", "question": "Isosceles $ \\triangle ABC$ has a right angle at $ C$. Point $ P$ is inside $ \\triangle ABC$, such that $ PA = 11, PB = 7,$ and $ PC = 6$. Legs $ \\overline{AC}$ and $ \\overline{BC}$ have length $ s = \\sqrt{a + b\\sqrt{2}}$, where $ a$ and $ b$ are positive integers. What is $ a + b... | 127 | numeric | 5 | metric geometry - length | |
2468 | [{"role": "user", "question": "The point $ O$ is the center of the circle circumscribed about $ \\triangle ABC$, with $ \\angle BOC = 120^\\circ$ and $ \\angle AOB = 140^\\circ$, as shown. What is the degree measure of $ \\angle ABC$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 50 | numeric | 4 | metric geometry - angle | |
2469 | [{"role": "user", "question": "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$?\n<image1>", "hint": "", "options"... | D | mcq | 5 | metric geometry - length | |
247 | [{"role": "user", "question": "A rectangular piece of paper $A B C D$ with the measurements $4 \\mathrm{~cm} \\times 16 \\mathrm{~cm}$ is folded along the line $\\mathrm{MN}$ so that point $C$ coincides with point $A$ as shown. How big is the area of the quadrilateral ANMD'?\n<image1>", "hint": "", "options": {"A": "$2... | C | mcq | 5 | transformation geometry | |
2470 | [{"role": "user", "question": "The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\\le|x|\\le7$, $ 3\\le|y|\\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 225 | numeric | 4 | analytic geometry | |
2471 | [{"role": "user", "question": "Rectangle $ ABCD$, pictured below, shares $50\\%$ of its area with square $ EFGH$. Square $ EFGH$ shares $20\\%$ of its area with rectangle $ ABCD$. What is $ \\frac{AB}{AD}$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 10 | numeric | 5 | metric geometry - length | |
2472 | [{"role": "user", "question": "A big $ L$ is formed as shown. What is its area?\n<image1>", "hint": "", "options": {}, "choices": []}] | 22 | numeric | 5 | metric geometry - area | |
2473 | [{"role": "user", "question": "There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?\n\n<image1>", "hint": "", "options": {"A": "$(C, A, E, D, B)$", "B": "$(C, A, D, E, B)$", "C": "$(C, D, E, A, B) \\ [1ex]$", "D": "$(C, E, A, D, B)$", "E": "$(C, E, D, A, B... | E | mcq | 1 | descriptive geometry | |
2474 | [{"role": "user", "question": "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, ... | A | mcq | 3 | solid geometry | |
2475 | [{"role": "user", "question": "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?\n\n<image1>", "hint": "", "options": {"A": "$29\\sqrt{3}$",... | A | mcq | 5 | metric geometry - length | |
2476 | [{"role": "user", "question": "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 ... | B | mcq | 5 | combinatorial geometry | |
2477 | [{"role": "user", "question": "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... | C | mcq | 5 | metric geometry - length | |
2478 | [{"role": "user", "question": "Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\\widetilde{AB}$, $\\widetilde{AD}$, $\\widetilde{AE}$, $\\widetilde{BC}$, $\\widetilde{BD}$, $\\widetilde{CD}$, $\\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a rou... | 16 | numeric | 5 | graph theory | |
2479 | [{"role": "user", "question": "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 $h$?\n\n<image1>", "hint": "", "optio... | A | mcq | 3 | solid geometry | |
248 | [{"role": "user", "question": "How big is the angle $\\alpha$ in the regular five-sided star shown?\n<image1>", "hint": "", "options": {"A": "$24^{\\circ}$", "B": "$30^{\\circ}$", "C": "$36^{\\circ}$", "D": "$45^{\\circ}$", "E": "$72^{\\circ}$"}, "choices": []}] | C | mcq | 4 | metric geometry - angle | |
2480 | [{"role": "user", "question": "Convex quadrilateral $ABCD$ has $AB = 3, BC = 4, CD = 13, AD = 12,$ and $\\angle ABC = 90^\\circ,$ as shown. What is the area of the quadrilateral?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 36 | numeric | 5 | metric geometry - area | |
2481 | [{"role": "user", "question": "In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{2}(\\sqrt{6}-2)$", "B": "$\\frac{1}{4}$", "C": "$2-\\sqrt{3}$", "D": "$\\frac{\\sqrt{3}}{6}$", "E": "$1-\\frac{\\sqrt{2}}{... | C | mcq | 5 | metric geometry - length | |
2482 | [{"role": "user", "question": "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... | D | mcq | 5 | algebra | |
2483 | [{"role": "user", "question": "The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is $\\frac{a-\\sqrt{2}}{b}$, where $a$ and $b... | 11 | numeric | 5 | metric geometry - length | |
2484 | [{"role": "user", "question": "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 $AF$ : $FD$?\n\n<image1>... | C | mcq | 5 | metric geometry - length | |
2485 | [{"role": "user", "question": "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... | D | mcq | 5 | metric geometry - length | |
2486 | [{"role": "user", "question": "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... | B | mcq | 5 | metric geometry - length | |
2487 | [{"role": "user", "question": "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?\n\n<ima... | C | mcq | 5 | metric geometry - area | |
2488 | [{"role": "user", "question": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and... | 552 | numeric | 5 | metric geometry - area | |
2489 | [{"role": "user", "question": "The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L,$ without traveling along any portion of a road more than once. (Paula is allowed to visit ... | 4 | numeric | 5 | combinatorics | |
249 | [{"role": "user", "question": "In the diagram we see a rose bed. White roses are growing in the squares that are equally big, red ones are in the big square and yellow ones in the right-angled triangle. The bed has width and height $16 \\mathrm{~m}$. How big is the area of the bed?\n<image1>", "hint": "", "options": {"... | C | mcq | 5 | metric geometry - area | |
2490 | [{"role": "user", "question": "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 $\\sin(2\\angle BAD)$?\n<image1>", "hint": "", "options": {"A":... | D | mcq | 4 | metric geometry - angle | |
2491 | [{"role": "user", "question": "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$?$\n\n<image1>", "hint": "", "options": {"A": "$17$", "B": "$15 + 2\\sqrt{2}$", "C": "$13 + 4\\sqrt{2}$", "... | C | mcq | 5 | metric geometry - length | |
2492 | [{"role": "user", "question": "In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry$?$\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 7 | numeric | 5 | combinatorial geometry | |
2493 | [{"role": "user", "question": "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?\n<image1>", "hint": "", "options": {"A": "4", "B": "$4\\sqrt{3}$", "C": "12", "D":... | E | mcq | 5 | metric geometry - length | |
2494 | [{"role": "user", "question": "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 $YC = 2$. Wh... | C | mcq | 5 | metric geometry - area | |
2495 | [{"role": "user", "question": "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 $\\overli... | B | mcq | 5 | metric geometry - length | |
2496 | [{"role": "user", "question": "Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC$. See the figure, which also shows the relative locations of th... | 81 | numeric | 5 | metric geometry - length | |
2497 | [{"role": "user", "question": "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 ... | B | mcq | 5 | combinatorial geometry | |
2498 | [{"role": "user", "question": "In $\\triangle{ABC}$ medians $\\overline{AD}$ and $\\overline{BE}$ intersect at $G$ and $\\triangle{AGE}$ is equilateral. Then $\\cos(C)$ can be written as $\\frac{m\\sqrtp}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the squa... | 44 | numeric | 4 | metric geometry - angle | |
2499 | [{"role": "user", "question": "The figure below depicts a regular 7-gon inscribed in a unit circle.\n<image1>\nWhat is the sum of the 4th powers of the lengths of all 21 of its edges and diagonals?", "hint": "", "options": {}, "choices": []}] | 147 | numeric | 5 | metric geometry - length | |
25 | [{"role": "user", "question": "A wooden block has 8 vertices. One vertex is cut off now (see the picture).\n<image1>\nHow many vertices has the block now?", "hint": "", "options": {}, "choices": []}] | 10 | numeric | 1 | solid geometry | |
250 | [{"role": "user", "question": "A right-angled triangle with side lengths $a=8, b=15$ and $c=17$ is given. How big is the radius $r$ of the inscribed semicircle shown?\n<image1>", "hint": "", "options": {}, "choices": []}] | 4.8 | numeric | 5 | metric geometry - length | |
2500 | [{"role": "user", "question": "Four regular hexagons surround a square with a side length $1$, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as $m\\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisi... | -4 | numeric | 5 | metric geometry - area | |
2501 | [{"role": "user", "question": "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 P... | A | mcq | 4 | metric geometry - angle | |
2502 | [{"role": "user", "question": "Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below:\n<image1>\n\nEach row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the un... | 5 | numeric | 5 | algebra | |
2503 | [{"role": "user", "question": "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 th... | E | mcq | 3 | solid geometry | |
2504 | [{"role": "user", "question": "What is the area of polygon $ ABCDEF$?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 46 | numeric | 3 | metric geometry - area | |
2505 | [{"role": "user", "question": "<image1>\n\nThe bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?", "hint": "", "options": {"A": "$\\frac{1}{2}$", "B": "$\\frac{2}{3}$", "C": "$\\frac... | C | mcq | 2 | statistics | |
2506 | [{"role": "user", "question": "<image1>\n\nA \"stair-step\" figure is made up of alternating black and white squares in each row. Rows $ 1$ through $ 4$ are shown. All rows begin and end with a white square. The number of black squares in the $ 37$th row is", "hint": "", "options": {}, "choices": []}] | 36 | numeric | 4 | algebra | |
2507 | [{"role": "user", "question": "<image1>\n\nA 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:", "hint": "", "options": {"A": "$\\text{Z}$", "B": "$\\text{U}$", "C": "$\\... | E | mcq | 4 | solid geometry | |
2508 | [{"role": "user", "question": "<image1>\n\nIn a magic triangle, each of the six whole numbers $ 10-15$ is placed in one of the circles so that the sum, $ S$, of the three numbers on each side of the triangle is the same. The largest possible value for $ S$ is", "hint": "", "options": {}, "choices": []}] | 39 | numeric | 4 | algebra | |
2509 | [{"role": "user", "question": "<image1>\n\nFive 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 car... | A | mcq | 4 | logic | |
251 | [{"role": "user", "question": "The clock shown has a rectangular clock face, the hands however move as usual in a constant circular pattern. How big is the distance $x$ of the digits 1 and 2 (in $\\mathrm{cm}$ ), if the distance between the numbers 8 and 10 is given as $12 \\mathrm{~cm}$?\n<image1>", "hint": "", "optio... | C | mcq | 5 | metric geometry - length | |
2510 | [{"role": "user", "question": "<image1>\n\nUsing only the paths and the directions shown, how many different routes are there from $ M$ to $ N$?", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 3 | graph theory | |
2511 | [{"role": "user", "question": "<image1>\n\nThe 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?", "hint": "", ... | D | mcq | 2 | statistics | |
2512 | [{"role": "user", "question": "<image1>\n\nGiven that all angles shown are marked, the perimeter of the polygon shown is", "hint": "", "options": {"A": "$14$", "B": "$20$", "C": "$28$", "D": "$48$", "E": "$\\text{cannot be determined from the information given}$"}, "choices": []}] | C | mcq | 2 | metric geometry - length | |
2513 | [{"role": "user", "question": "<image1>\n\nA bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $ 25 \\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold i... | 2.5 | numeric | 2 | statistics | |
2514 | [{"role": "user", "question": "<image1>\n\nSuppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 4 | solid geometry | |
2515 | [{"role": "user", "question": "<image1>\n\nThe 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$, what is the value of the ratio of the area of the shaded region to the... | B | mcq | 3 | metric geometry - area | |
2516 | [{"role": "user", "question": "The area of the rectangular region is\n\n<image1>", "hint": "", "options": {"A": "$\\text{.088 m}^2$", "B": "$\\text{.62 m}^2$", "C": "$\\text{.88 m}^2$", "D": "$\\text{1.24 m}^2$", "E": "$\\text{4.22 m}^2$"}, "choices": []}] | A | mcq | 3 | metric geometry - area | |
2517 | [{"role": "user", "question": "The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is\n<image1>", "hint": "", "options": {}, "choices": []}] | 20 | numeric | 4 | solid geometry | |
2518 | [{"role": "user", "question": "What fraction of the large $12$ by $18$ rectangular region is shaded?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{108}$", "B": "$\\frac{1}{18}$", "C": "$\\frac{1}{12}$", "D": "$\\frac{2}{9}$", "E": "$\\frac{1}{3}$"}, "choices": []}] | C | mcq | 3 | metric geometry - area | |
2519 | [{"role": "user", "question": "$\\text{ABCD}$ 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\n\n<image1>", "hint": "", "options": {"A": "$4\\text{ and }5$", "B": "$5\\text{ and }6$", "C": "$6... | D | mcq | 3 | metric geometry - area | |
252 | [{"role": "user", "question": "Renate wants to glue together a number of ordinary dice (whose number of points on opposite sides always adds up to 7) to form a \"dicebar\" as shown. Doing this she only wants to glue sides together with an equal number of points. She wants to make sure that the sum of all points on the ... | E | mcq | 1 | arithmetic | |
2520 | [{"role": "user", "question": "The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of\n<image1>", "hint": "", "options": {"A": "10.05", "B": "10.15", "C": "10.25", "D": "10.3", "E": "10.6"}, "choices": []}] | B | mcq | 2 | metric geometry - length | |
2521 | [{"role": "user", "question": "The figure consists of alternating light and dark squares.\n<image1>\nThe number of dark squares exceeds the number of light squares by", "hint": "", "options": {}, "choices": []}] | 11 | numeric | 4 | algebra | |
2522 | [{"role": "user", "question": "If $ \\angle\\text{CBD} $ is a right angle, then this protractor indicates that the measure of $ \\angle\\text{ABC} $ is approximately\n\n<image1>", "hint": "", "options": {"A": "$20^\\circ$", "B": "$40^\\circ$", "C": "$50^\\circ$", "D": "$70^\\circ$", "E": "$120^\\circ$"}, "choices": []}... | C | mcq | 5 | metric geometry - angle | |
2523 | [{"role": "user", "question": "An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 3 | numeric | 4 | combinatorial geometry | |
2524 | [{"role": "user", "question": "<image1>\n\nPlacing no more than one $x$ in each small square, what is the greatest number of $x$'s that can be put on the grid shown without getting three $x$'s in a row vertically, horizontally, or diagonally?", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 2 | combinatorics | |
2525 | [{"role": "user", "question": "The shaded region formed by the two intersecting perpendicular rectangles, in square units, is\n\n<image1>", "hint": "", "options": {"A": "$23$", "B": "$38$", "C": "$44$", "D": "$46$", "E": "$\\text{unable to be determined from the information given}$"}, "choices": []}] | C | mcq | 3 | metric geometry - area | |
2526 | [{"role": "user", "question": "The glass gauge on a cylindrical coffee maker shows that there are 45 cups left when the coffee maker is $36\\%$ full. How many cups of coffee does it hold when it is full?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 130 | numeric | 4 | solid geometry | |
2527 | [{"role": "user", "question": "<image1>\n\nThe square in the first diagram \"rolls\" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram $4$?\n\n<image2>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": []... | E | mcq | 3 | transformation geometry | |
2528 | [{"role": "user", "question": "If the markings on the number line are equally spaced, what is the number $\\text{y}$?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 12 | numeric | 4 | algebra |
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