id stringlengths 1 4 | media images listlengths 1 1 | messages stringlengths 84 1.58k | answer stringclasses 371
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|---|---|---|---|---|---|---|
2349 | [{"role": "user", "question": "A rectangle intersects a circle as shown: $AB=4$, $BC=5$, and $DE=3$. Then $EF$ equals:\n\n<image1>", "hint": "", "options": {"A": "$6$", "B": "$7$", "C": "$\\frac{20}{3}$", "D": "$8$", "E": "$9$"}, "choices": []}] | B | mcq | 5 | metric geometry - length | |
235 | [{"role": "user", "question": "Jan cannot draw very accurately but nevertheless he tried to produce a roadmap of his village. The relative position of the houses and the street crossings are all correct but three of the roads are actually straight and only Qurwik street is not. Who lives in Qurwik street?\n<image1>", "... | C | mcq | 1 | logic | |
2350 | [{"role": "user", "question": "A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\\triangle ABC$ is:\n<image1>", "hint": "", "options": {"A": "$120$", "B": "$144$", "C": "$150$", "D": "$216$", "E": "$144\\sqrt{5}$"}, "cho... | C | mcq | 5 | metric geometry - area | |
2351 | [{"role": "user", "question": "In the obtuse triangle $ABC$, $AM = MB, MD \\perp BC, EC \\perp BC$. If the area of $\\triangle ABC$ is 24, then the area of $\\triangle BED$ is\n\n<image1>", "hint": "", "options": {"A": "$9$", "B": "$12$", "C": "$15$", "D": "$18$", "E": "$\\text{not uniquely determined}$"}, "choices": [... | B | mcq | 5 | metric geometry - area | |
2352 | [{"role": "user", "question": "In an arcade game, the \"monster\" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $ 60^{\\circ}$. What is the perimeter of the monster in cm?\n\n<image1>", "hint": "", "options": {"A": "$\\pi + 2$", "B": "$2\\p... | E | mcq | 5 | metric geometry - length | |
2353 | [{"role": "user", "question": "In right $ \\triangle ABC$ with legs $ 5$ and $ 12$, arcs of circles are drawn, one with center $ A$ and radius $ 12$, the other with center $ B$ and radius $ 5$. They intersect the hypotenuse at $ M$ and $ N$. Then, $ MN$ has length:\n\n<image1>", "hint": "", "options": {"A": "$2$", "B... | D | mcq | 5 | metric geometry - length | |
2354 | [{"role": "user", "question": "The odd positive integers $1,3,5,7,\\cdots,$ are arranged into in five columns continuing with the pattern shown on the right. Counting from the left, the column in which $ 1985$ appears in is the\n\n<image1>", "hint": "", "options": {"A": "$\\text{ first}$", "B": "$\\text{ second}$", "C... | B | mcq | 5 | algebra | |
2355 | [{"role": "user", "question": "Pegs are put in a board $ 1$ unit apart both horizontally and vertically. A reubber band is stretched over $ 4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 5 | combinatorial geometry | |
2356 | [{"role": "user", "question": "Diagonal $ DB$ of rectangle $ ABCD$ is divided into $ 3$ segments of length $ 1$ by parallel lines $ L$ and $ L'$ that pass through $ A$ and $ C$ and are perpendicular to $ DB$. The area of $ ABCD$, rounded to the nearest tenth, is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 4.2 | numeric | 5 | metric geometry - area | |
2357 | [{"role": "user", "question": "In a circle with center $ O$, $ AD$ is a diameter, $ ABC$ is a chord, $ BO = 5$, and $ \\angle ABO = \\stackrel{\\frown}{CD} = 60^{\\circ}$. Then the length of $ BC$ is:\n\n<image1>", "hint": "", "options": {"A": "$3$", "B": "$3 + \\sqrt{3}$", "C": "$5 - \\frac{\\sqrt{3}}{2}$", "D": "$5$... | D | mcq | 5 | metric geometry - length | |
2358 | [{"role": "user", "question": "In $ \\triangle ABC$, we have $ \\angle C = 3 \\angle A$, $ a = 27$, and $ c = 48$. What is $ b$?\n\n<image1>", "hint": "", "options": {"A": "$33$", "B": "$35$", "C": "$37$", "D": "$39$", "E": "$\\text{not uniquely determined}$"}, "choices": []}] | B | mcq | 5 | metric geometry - length | |
2359 | [{"role": "user", "question": "$\\triangle ABC$ is a right angle at $C$ and $\\angle A = 20^\\circ$. If $BD$ is the bisector of $\\angle ABC$, then $\\angle BDC =$\n<image1>", "hint": "", "options": {"A": "$40^\\circ$", "B": "$45^\\circ$", "C": "$50^\\circ$", "D": "$55^\\circ$", "E": "$60^\\circ$"}, "choices": []}] | D | mcq | 4 | metric geometry - angle | |
236 | [{"role": "user", "question": "A rectangular piece of paper is wrapped around a cylinder. Then an angled straight cut is made through the points $\\mathrm{X}$ and $\\mathrm{Y}$ of the cylinder as shown on the left. The lower part of the piece of paper is then unrolled. Which of the following pictures could show the res... | C | mcq | 1 | descriptive geometry | |
2360 | [{"role": "user", "question": "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?\n\n<image1>", "hint": "", "options": {"A": ... | C | mcq | 5 | algebra | |
2361 | [{"role": "user", "question": "In $\\triangle ABC$, $AB = 13$, $BC = 14$ and $CA = 15$. Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC$. The length of $HM$ is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 6.5 | numeric | 5 | metric geometry - length | |
2362 | [{"role": "user", "question": "In $\\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\\triangle PAB$ is similar to $\\triangle PCA$. The length of $PC$ is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 9 | numeric | 5 | metric geometry - length | |
2363 | [{"role": "user", "question": "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 $A$.\n<image1>\n\nA necessary and sufficient condition for the equality of the two shaded areas, given $0 < \\theta < \\fra... | B | mcq | 5 | metric geometry - area | |
2364 | [{"role": "user", "question": "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\n\n<image1>", "hint": "", "options": {"A": "$\\cos \\alpha$... | C | mcq | 5 | metric geometry - area | |
2365 | [{"role": "user", "question": "$ABCDE$ 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\n\n<image1>", "hint": "", "options": {"A": "$3$", "B": "$1... | C | mcq | 5 | metric geometry - length | |
2366 | [{"role": "user", "question": "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\n\n<image1>", "hint": "", "options": {"A": "$6$", "B": "$6\\frac{1}{2}$", "C": "$7$", "D": "$7\\frac{1}{2}$", "E": "$8$"}, "choices": ... | E | mcq | 5 | metric geometry - length | |
2367 | [{"role": "user", "question": "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 $w$.\n<image1>", "hint": "", "options": {"A": "$w-y-z$", "B": "$w-2y-2z$", "C": "$180-w-y-z \\$", "D": "$2w-y-z$", "E": "$180-w+y+z$"}, "... | A | mcq | 4 | metric geometry - angle | |
2368 | [{"role": "user", "question": "In the figure the sum of the distances $AD$ and $BD$ is\n<image1>", "hint": "", "options": {"A": "$\\text{between 10 and 11}$", "B": "$12$", "C": "$\\text{between 15 and 16}$", "D": "$\\text{between 16 and 17}$", "E": "$17$"}, "choices": []}] | C | mcq | 5 | metric geometry - length | |
2369 | [{"role": "user", "question": "$ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Then $\\sin \\theta=$\n<image1>", "hint": "", "options": {"A": "$\\frac{\\sqrt{5}}{5}$", "B": "$\\frac{3}{5}$", "C": "$\\frac{\\sqrt{10}}{5}$", "D": "$\\frac{4}{5}$", "E": "$\\text{none of these}$"}, "choi... | B | mcq | 4 | metric geometry - angle | |
237 | [{"role": "user", "question": "Determine the area of the quadrilateral PQRS pictured on the right, where $\\mathrm{PS}=\\mathrm{RS}$, $\\angle \\mathrm{PSR}=\\angle \\mathrm{PQR}=90^{\\circ}, \\mathrm{ST} \\perp \\mathrm{PQ}$, and $\\mathrm{ST}=5$.\n<image1>", "hint": "", "options": {}, "choices": []}] | 25 | numeric | 5 | metric geometry - area | |
2370 | [{"role": "user", "question": "There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is $441 \\text{cm}^2$. What is the area (in $\\text{cm}^2$) of the square inscribed in the same $\\triangle ABC$ as shown in ... | 392 | numeric | 5 | metric geometry - area | |
2371 | [{"role": "user", "question": "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 $AC$.) ... | E | mcq | 5 | metric geometry - length | |
2372 | [{"role": "user", "question": "Four rectangular paper strips of length $10$ and width $1$ are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?\n<image1>", "hint": "", "options": {}, "choices": []}] | 36 | numeric | 5 | metric geometry - area | |
2373 | [{"role": "user", "question": "An $8'\\text{ X }10'$ table sits in the corner of a square room, as in Figure 1 below. The owners desire to move the table to the position shown in Figure 2. The side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without ... | 13 | numeric | 5 | metric geometry - length | |
2374 | [{"role": "user", "question": "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?\n\n<image1>", "hint": "", "options": {"A": "A", "B": "B",... | C | mcq | 3 | statistics | |
2375 | [{"role": "user", "question": "$ABC$ 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\n<im... | C | mcq | 5 | metric geometry - area | |
2376 | [{"role": "user", "question": "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 ... | E | mcq | 5 | metric geometry - length | |
2377 | [{"role": "user", "question": "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?\n\n<image1>", "hint": "", "options": {"A": "AB=3, CD=1", "B": "AB=5, CD=2", "C": "AB=7, CD=3", "D": "A... | D | mcq | 5 | metric geometry - area | |
2378 | [{"role": "user", "question": "A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 81 | numeric | 5 | transformation geometry | |
2379 | [{"role": "user", "question": "In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD = BC = 5, AB = 4,$ and $DC = 10$. The point $C$ is on $\\overline{DF}$ and $B$ is the midpoint of hypotenuse $\\overline{DE}$ in the right triangle $DEF$. Then $CF =$\n<image1>", "hint": "", "options": {}, "choices": []... | 4.0 | numeric | 5 | metric geometry - length | |
238 | [{"role": "user", "question": "Michael wants to write whole numbers into the empty fields of the $3 \\times 3$ table on the right so that the sum of the numbers in each $2 \\times 2$ square equals 10. Four numbers have already been written down. Which of the following values could be the sum of the remaining five numbe... | E | mcq | 1 | logic | |
2380 | [{"role": "user", "question": "Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 430 | numeric | 5 | combinatorial geometry | |
2381 | [{"role": "user", "question": "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 $\\angle MHC =$\n\n<image1>", "hint": "", "options": {"A": "$15^\\circ$", "B": "$22.5^\\circ$", "C": "$30^\\circ$", "D": "$40^\\... | C | mcq | 4 | metric geometry - angle | |
2382 | [{"role": "user", "question": "Two strips of width 1 overlap at an angle of $\\alpha$ as shown. The area of the overlap (shown shaded) is\n\n<image1>", "hint": "", "options": {"A": "$\\sin \\alpha$", "B": "$\\frac{1}{\\sin \\alpha}$", "C": "$\\frac{1}{1 - \\cos \\alpha}$", "D": "$\\frac{1}{\\sin^2 \\alpha}$", "E": "$\\... | B | mcq | 5 | metric geometry - area | |
2383 | [{"role": "user", "question": "In $\\triangle ABC$, $AB = 5, BC = 7, AC = 9$ and $D$ is on $\\overline{AC}$ with $BD = 5$. Find the ratio of $AD: DC$.\n\n<image1>", "hint": "", "options": {"A": "4:3", "B": "7:5", "C": "11:6", "D": "13:5", "E": "19:8"}, "choices": []}] | E | mcq | 5 | metric geometry - length | |
2384 | [{"role": "user", "question": "A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up $36\\%$ of the area of the fla... | 2 | numeric | 5 | metric geometry - area | |
2385 | [{"role": "user", "question": "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 ... | D | mcq | 5 | algebra | |
2386 | [{"role": "user", "question": "Let $ABCD$ be a parallelogram with $\\angle ABC=120^\\circ$, $AB=16$ and $BC=10$. Extend $\\overline{CD}$ through $D$ to $E$ so that $DE=4$. \n<image1>\nIf $\\overline{BE}$ intersects $\\overline{AD}$ at $F$, then $FD$ is closest to", "hint": "", "options": {}, "choices": []}] | 2 | numeric | 5 | metric geometry - length | |
2387 | [{"role": "user", "question": "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 $x=$\n\n<image1>", "hint": "", "options": {"A": "$\\frac{3}{... | A | mcq | 4 | metric geometry - angle | |
2388 | [{"role": "user", "question": "$ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 4.2 | numeric | 5 | metric geometry - length | |
2389 | [{"role": "user", "question": "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... | A | mcq | 5 | algebra | |
239 | [{"role": "user", "question": "How many graphs of the functions $y=x^{2}, y=-x^{2}, y=+\\sqrt{x}, y=-\\sqrt{x}$, $y=+\\sqrt{-x}, y=-\\sqrt{-x}, y=+\\sqrt{|x|}, y=-\\sqrt{|x|}$ are included in the sketch on the right?\n<image1>", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 4 | analytic geometry | |
2390 | [{"role": "user", "question": "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\n\n<image1>", "hint": "", "options": {"A": "288", "B": "291", "C": "294", "D": "297... | E | mcq | 5 | metric geometry - area | |
2391 | [{"role": "user", "question": "Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\\overline{AB}$. The line through $D$ parallel to $\\overline{AC}$ meets $\\overline{CB}$ extended at $E$. If $\\frac{DE}... | 128 | numeric | 5 | metric geometry - length | |
2392 | [{"role": "user", "question": "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\n<image1>", "hint": "", "options": {"A": "$1.44... | B | mcq | 5 | metric geometry - area | |
2393 | [{"role": "user", "question": "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\n\n<image1>", "... | C | mcq | 5 | metric geometry - area | |
2394 | [{"role": "user", "question": "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\n<image1>", "hint": "", "options": {"A": "$\\frac{8}{5}$", "B": "$\\frac{7}{20}\\sqrt{21}$", ... | B | mcq | 5 | metric geometry - length | |
2395 | [{"role": "user", "question": "A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 2601 | numeric | 5 | algebra | |
2396 | [{"role": "user", "question": "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 uni... | E | mcq | 5 | metric geometry - area | |
2397 | [{"role": "user", "question": "The ratio of the radii of two concentric circles is $1:3$. If $\\overline{AC}$ is a diameter of the larger circle, $\\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is\n\n<image1>", "hint": "", "opt... | 18 | numeric | 5 | metric geometry - length | |
2398 | [{"role": "user", "question": "Part of an \"$n$-pointed regular star\" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_{1}$, $A_{2}$, $\\ldots$, $A_{n}$ are congruent and angles $B_{1}$, $B_{2}$, $\\ldots$, $B_{n}$ are congruent. If the acute angle at $A_{1}$ is $10^{\\circ}$ ... | 36 | numeric | 4 | metric geometry - angle | |
2399 | [{"role": "user", "question": "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 $\\sta... | B | mcq | 5 | metric geometry - area | |
24 | [{"role": "user", "question": "A trip of the pupils to the zoo took 135 minutes.\n<image1>\nHow many hours and minutes does it make?", "hint": "", "options": {"A": "3 h 5 min", "B": "2 h 15 min", "C": "1 h 35 min", "D": "2 h 35 min", "E": "3 h 35 min"}, "choices": []}] | B | mcq | 2 | arithmetic | |
240 | [{"role": "user", "question": "The rear window wiper of a car is made in a way so that the rod $r$ and the wiper blade $\\mathrm{w}$ are equally long and are connected at an angle $\\alpha$. The wiper rotates around the centre of rotation $\\mathrm{O}$ and wipes over the area shown on the right. Calculate the angle $\\... | B | mcq | 4 | metric geometry - angle | |
2400 | [{"role": "user", "question": "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 $\\angle BED=$\n<image1>", "hint": "", "options": {"A": "$50^{\\circ}$", "B": "$55^{\\circ}$", "C": "$60^{\\circ}$", "D": "$65^{\\ci... | D | mcq | 4 | metric geometry - angle | |
2401 | [{"role": "user", "question": "The convex pentagon $ABCDE$ has $\\angle A=\\angle B=120^{\\circ}$, $EA=AB=BC=2$ and $CD=DE=4$. What is the area of $ABCDE$?\n<image1>", "hint": "", "options": {"A": "$10$", "B": "$7\\sqrt{3}$", "C": "$15$", "D": "$9\\sqrt{3}$", "E": "$12\\sqrt{5}$"}, "choices": []}] | B | mcq | 5 | metric geometry - area | |
2402 | [{"role": "user", "question": "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 tha... | A | mcq | 5 | metric geometry - area | |
2403 | [{"role": "user", "question": "Twenty cubical blocks are arranged as shown. First, $10$ are arranged in a triangular pattern; then a layer of $6$, arranged in a triangular pattern, is centered on the $10$; then a layer of $3$, arranged in a triangular pattern, is centered on the $6$; and finally one block is centered o... | 114 | numeric | 3 | solid geometry | |
2404 | [{"role": "user", "question": "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 $AX=$\n<image1>", "hint": "", "options": {"A": "$\\cos 6^{\\circ}\\cos 12^{\\circ} \\sec 18^{\\circ}$", "B": "$\\cos 6^{\\circ}\\... | B | mcq | 5 | metric geometry - length | |
2405 | [{"role": "user", "question": "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 t... | E | mcq | 5 | combinatorics | |
2406 | [{"role": "user", "question": "The sides of $\\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ tr... | 12 | numeric | 5 | transformation geometry | |
2407 | [{"role": "user", "question": "A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle?\n<image1>", "hint": "", "options": {}, "choices": []}] | 15 | numeric | 5 | metric geometry - area | |
2408 | [{"role": "user", "question": "Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is\n<image1>", "hint": "", "options": {}, "choices": []}] | 175 | numeric | 5 | metric geometry - area | |
2409 | [{"role": "user", "question": "In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is $56$. The area of the region bounded by the polygon is\n<image1>", "hint": "", "options": {}, "choices": []}] | 100 | numeric | 5 | metric geometry - area | |
241 | [{"role": "user", "question": "We have three horizontal lines and three parallel, sloped lines. Both of the circles shown touch four of the lines. X, Y and Z are the areas of the grey regions. $\\mathrm{D}$ is the area of the parallelogram PQRS. At least how many of the areas $\\mathrm{X}, \\mathrm{Y}, \\mathrm{Z}$ and... | 1 | numeric | 5 | metric geometry - area | |
2410 | [{"role": "user", "question": "In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\\angle A =$\n<image1>", "hint": "", "options": {"A": "$30^{\\circ}$", "B": "$36^{\\circ}$", "C": "$48^{\\circ}$", "D": "$60^{\\circ}$", "E": "$72^{\\circ}$"}, "choices": []}] | B | mcq | 4 | metric geometry - angle | |
2411 | [{"role": "user", "question": "Triangle $ABC$ is inscribed in a circle, and $\\angle B = \\angle C = 4\\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$\n<image1>", "hint": "", "options": {}, "choices": []}] | 9 | numeric | 4 | metric geometry - angle | |
2412 | [{"role": "user", "question": "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\n<image1>", "hint": "", "options":... | E | mcq | 4 | analytic geometry | |
2413 | [{"role": "user", "question": "A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 5 | numeric | 4 | metric geometry - angle | |
2414 | [{"role": "user", "question": "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 $AB/BC=$\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{2}\\csc{\\frac{1}{4}}$", "B": "$2\\cos{\\frac{1}{2}}$", "C": "... | A | mcq | 5 | metric geometry - length | |
2415 | [{"role": "user", "question": "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 $x$?\n\n<image1>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": []}] | C | mcq | 3 | solid geometry | |
2416 | [{"role": "user", "question": "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 $BD =$\n\n<image1>", "hint": "", "options": {"A": "$5$", "B": "$\\frac{16}{3}$", "C": "$\\frac... | C | mcq | 5 | metric geometry - length | |
2417 | [{"role": "user", "question": "Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 16 | numeric | 5 | combinatorial geometry | |
2418 | [{"role": "user", "question": "Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins o... | 4 | numeric | 5 | algebra | |
2419 | [{"role": "user", "question": "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\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{6}$", "B": "$\\frac{1}{4}$", "C": "$\\f... | C | mcq | 5 | metric geometry - area | |
242 | [{"role": "user", "question": "In the (x,y)-plane the co-ordinate axes are positioned as usual. Point $A(1,-10)$ which is on the parabola $y=a x^{2}+b x+c$ was marked. Afterwards the co-ordinate axis and the majority of the parabola were deleted. Which of the following statements could be false?\n<image1>", "hint": "",... | E | mcq | 4 | analytic geometry | |
2420 | [{"role": "user", "question": "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\n<image1>", "hint": "", "option... | C | mcq | 5 | metric geometry - area | |
2421 | [{"role": "user", "question": "Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\\sqrt{a}$ where $a$ is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 184 | numeric | 5 | metric geometry - length | |
2422 | [{"role": "user", "question": "A large cube is formed by stacking $27$ unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is\n<image1>", "hint": "", "options": {}, "choices": []}] | 19 | numeric | 3 | solid geometry | |
2423 | [{"role": "user", "question": "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... | B | mcq | 5 | metric geometry - area | |
2424 | [{"role": "user", "question": "In rectangle $ABCD$, angle $C$ is trisected by $\\overline{CF}$ and $\\overline{CE}$, where $E$ is on $\\overline{AB}$, $F$ is on $\\overline{AD}$, $BE = 6,$ and $AF = 2$. Which of the following is closest to the area of the rectangle $ABCD$?\n<image1>", "hint": "", "options": {}, "choice... | 150 | numeric | 5 | metric geometry - area | |
2425 | [{"role": "user", "question": "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?\n<image1>", "hint": "", "options": {"A": "$\\displaystyle \\frac{1}{2}$", "B": "$\\displaystyle \\frac{\\sqrt{3}}{3}$", ... | D | mcq | 5 | metric geometry - area | |
2426 | [{"role": "user", "question": "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\n<image1>", "hint": "", "options": {"A": "$115^\\circ$", "B": "$120^\\circ$", "C": "$130^\\circ$", "D": "$135^\\circ$", "E": "$\\t... | D | mcq | 4 | metric geometry - angle | |
2427 | [{"role": "user", "question": "The adjacent sides of the decagon shown meet at right angles. What is its perimeter?\n<image1>", "hint": "", "options": {}, "choices": []}] | 44 | numeric | 5 | metric geometry - length | |
2428 | [{"role": "user", "question": "A rectangle with perimeter $ 176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?\n<image1>", "hint": "", "options": {}, "choices": []}] | 80 | numeric | 5 | metric geometry - length | |
2429 | [{"role": "user", "question": "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\n<image1>", "hint": "", "options": {"A": "$2$", "B": "$3 - \... | C | mcq | 5 | metric geometry - area | |
243 | [{"role": "user", "question": "An archer tries his art on the target shown below on the right. With each of his three arrows he always hits the target. How many different scores could he total with three arrows?\n<image1>", "hint": "", "options": {}, "choices": []}] | 19 | numeric | 5 | combinatorics | |
2430 | [{"role": "user", "question": "Medians $ BD$ and $ CE$ of triangle $ ABC$ are perpendicular, $ BD = 8$, and $ CE = 12$. The area of triangle $ ABC$ is\n<image1>", "hint": "", "options": {}, "choices": []}] | 64 | numeric | 5 | metric geometry - area | |
2431 | [{"role": "user", "question": "A circle with center $ O$ is tangent to the coordinate axes and to the hypotenuse of the $ 30^\\circ$-$ 60^\\circ$-$ 90^\\circ$ triangle $ ABC$ as shown, where $ AB = 1$. To the nearest hundredth, what is the radius of the circle?\n<image1>", "hint": "", "options": {}, "choices": []}] | 2.37 | numeric | 5 | metric geometry - length | |
2432 | [{"role": "user", "question": "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.\n<image1>\nThe volume of... | D | mcq | 3 | solid geometry | |
2433 | [{"role": "user", "question": "Triangle $ ABC$ and point $ P$ in the same plane are given. Point $ P$ is equidistant from $ A$ and $ B$, angle $ APB$ is twice angle $ ACB$, and $ \\overline{AC}$ intersects $ \\overline{BP}$ at point $ D$.\n<image1>\nIf $ PB = 3$ and $ PD = 2$, then $ AD\\cdot CD =$", "hint": "", "optio... | 5 | numeric | 5 | metric geometry - length | |
2434 | [{"role": "user", "question": "<image1>\nEach 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.\n<image2>\nWhich of the rectangles is i... | E | mcq | 5 | combinatorial geometry | |
2435 | [{"role": "user", "question": "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 $... | D | mcq | 5 | metric geometry - length | |
2436 | [{"role": "user", "question": "A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is 14. What is the area of the large square?\n<image1>", "hint": "", "options": {}, "choices": []}] | 49 | numeric | 5 | metric geometry - area | |
2437 | [{"role": "user", "question": "The figure shown is the union of a circle and two semicircles of diameters of $ a$ and $ b$, all of whose centers are collinear.\n<image1>\nThe ratio of the area of the shaded region to that of the unshaded region is", "hint": "", "options": {"A": "$\\sqrt{\\frac{a}{b}}$", "B": "$\\frac{a... | B | mcq | 5 | metric geometry - area | |
2438 | [{"role": "user", "question": "A $ 9\\times9\\times9$ cube is composed of twenty-seven $ 3\\times3\\times3$ cubes. The big cube is 'tunneled' as follows: First, the six $ 3\\times3\\times3$ cubes which make up the center of each face as well as the center of $ 3\\times3\\times3$ cube are removed. Second, each of the tw... | 1056 | numeric | 3 | solid geometry |
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