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2169 | [{"role": "user", "question": "The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\\ge3$, $ F_n$ is constructed from $ F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n - 1}$ had on each side of its outside... | 761 | numeric | 3 | algebra | |
217 | [{"role": "user", "question": "In the diagram on the right we want to colour the fields with the colours A, B, C and D so that adjacent fields are always in different colours. (Even fields that share only one corner, count as adjacent.) Some fields have already been coloured in. In which colour can the grey field be co... | D | mcq | 5 | graph theory | |
2170 | [{"role": "user", "question": "Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger c... | C | mcq | 4 | metric geometry - area | |
2171 | [{"role": "user", "question": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $ 15$ and $ 25$ meters. What fraction of the yard is occupied by the flower beds?\... | C | mcq | 4 | metric geometry - area | |
2172 | [{"role": "user", "question": "Segment $ BD$ and $ AE$ intersect at $ C$, as shown, $ AB=BC=CD=CE$, and $ \\angle A=\\frac{5}{2}\\angle B$. What is the degree measure of $ \\angle D$?\n<image1>", "hint": "", "options": {}, "choices": []}] | 52.5 | numeric | 2 | metric geometry - angle | |
2173 | [{"role": "user", "question": "As shown below, convex pentagon $ ABCDE$ has sides $ AB = 3$, $ BC = 4$, $ CD = 6$, $ DE = 3$, and $ EA = 7$. The pentagon is originally positioned in the plane with vertex $ A$ at the origin and vertex $ B$ on the positive $ x$-axis. The pentagon is then rolled clockwise to the right alo... | C | mcq | 3 | algebra | |
2174 | [{"role": "user", "question": "Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $ (a,0)$ to $ (3,3)$, divides the entire region into two regions of equal area. What is $ a$?\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{2}... | C | mcq | 5 | analytic geometry | |
2175 | [{"role": "user", "question": "Triangle $ ABC$ has a right angle at $ B$, $ AB = 1$, and $ BC = 2$. The bisector of $ \\angle BAC$ meets $ \\overline{BC}$ at $ D$. What is $ BD$?\n<image1>", "hint": "", "options": {"A": "$\\frac{\\sqrt{3} - 1}{2}$", "B": "$\\frac{\\sqrt{5} - 1}{2}$", "C": "$\\frac{\\sqrt{5} + 1}{2}$", ... | B | mcq | 4 | metric geometry - length | |
2176 | [{"role": "user", "question": "A cubical cake with edge length $ 2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $ M$ is the midpoint of a top edge. The piece whose top is triangle $ B$ contains $ c$ cubic inches of cake and $ s$ square inches of icing... | B | mcq | 4 | metric geometry - area | |
2177 | [{"role": "user", "question": "The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in... | 100 | numeric | 2 | metric geometry - angle | |
2178 | [{"role": "user", "question": "Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?\n<image1>", "hint": "", "options": {"A": "$\\frac{5}{4}$", "B": "$\\frac{4}{3}$", "C": "$\\frac{3}{2}$", "D": "$2$", "E": "$... | B | mcq | 4 | metric geometry - length | |
2179 | [{"role": "user", "question": "Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?\n\n<image1>", "hint": "", "options": {"A": "$3 ... | C | mcq | 4 | metric geometry - area | |
218 | [{"role": "user", "question": "A (very small) ball is kicked off from point A on a square billiard table with side length $2 \\mathrm{~m}$. After moving along the shown path and touching the sides three times as indicated, the path ends in point $B$. How long is the path that the bal travels from A to B? (As indicated ... | B | mcq | 5 | metric geometry - length | |
2180 | [{"role": "user", "question": "The area of $\\triangle EBD$ is one third of the area of $3-4-5$ $ \\triangle ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{4}{3}$", "B": "$\\sqrt{5}$", "C": "$\\frac{9}{4}$", "D": "$\\frac{4\\sqrt{3}}{3}$", "E": "$\... | D | mcq | 4 | metric geometry - length | |
2181 | [{"role": "user", "question": "A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{\\sqrt{2} - 1}{2}... | A | mcq | 3 | combinatorial geometry | |
2182 | [{"role": "user", "question": "In the given circle, the diameter $\\overline{EB}$ is parallel to $\\overline{DC}$, and $\\overline{AB}$ is parallel to $\\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4:5$. What is the degree measure of angle $BCD$?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 130 | numeric | 2 | metric geometry - angle | |
2183 | [{"role": "user", "question": "Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\\triangle ABC$?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{6}$", "B": "$\\frac{1}{5}$", "C": "$\\frac{2}{9}$", "D": "$\\frac{1}{3}$", "E": "$\\frac{\\sqrt{2}}{4}$"}, "cho... | B | mcq | 4 | metric geometry - area | |
2184 | [{"role": "user", "question": "The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\\frac{2\\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve?\n\n<image1>", "hint": "", "opt... | E | mcq | 4 | metric geometry - area | |
2185 | [{"role": "user", "question": "A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\\ :\\ 1$. What is the area of the rectangle?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 200 | numeric | 4 | metric geometry - area | |
2186 | [{"role": "user", "question": "Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?\n\n<image1>", "hint": "", "options": {"A": "$10\\pi+4\\sqrt{3}$", "B": "$13\\pi-\\sqrt{3}$", "C": "$12\\pi+\\sqrt{3}$", "D": "$10\\pi+9$", "E":... | A | mcq | 4 | metric geometry - area | |
2187 | [{"role": "user", "question": "A bug travels from $A$ to $B$ along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?\n\n<image1>", "hin... | 2400 | numeric | 4 | graph theory | |
2188 | [{"role": "user", "question": "Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \\overline{BC} $, and the area of $ \\bigtriangleup ABE $ is $ 40 $. What is $ BE $? \n<image1>", "hint": "", "options": {}, "choices": []}] | 8 | numeric | 4 | metric geometry - length | |
2189 | [{"role": "user", "question": "In $\\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\\overline{AB}$, $\\overline{BC}$, and $\\overline{AC}$, respectively, such that $\\overline{DE}$ and $\\overline{EF}$ are parallel to $\\overline{AC}$ and $\\overline{AB}$, respectively. What is the perime... | 56 | numeric | 4 | metric geometry - length | |
219 | [{"role": "user", "question": "In the diagram to the right a $2 \\times 2 \\times 2$ cube is made up of four transparent $1 \\times 1 \\times 1$ cubes and four non-transparent black $1 \\times 1 \\times 1$ cubes. They are placed in a way so that the entire big cube is nontransparent; i.e. looking at it from the front t... | 9 | numeric | 3 | solid geometry | |
2190 | [{"role": "user", "question": "In $\\triangle ABC$, medians $\\overline{AD}$ and $\\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$\n<image1>", "hint": "", "options": {}, "choices": []}] | 13.5 | numeric | 4 | metric geometry - area | |
2191 | [{"role": "user", "question": "The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many... | 1152 | numeric | 4 | combinatorics | |
2192 | [{"role": "user", "question": "A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?\n\n<image1>", "hint": ""... | C | mcq | 4 | metric geometry - area | |
2193 | [{"role": "user", "question": "Equilateral $\\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{12+3\\sqrt{3}}4$", "B": "$\\frac{9}{2}$", "C": "$3+\\sqrt{3}$", "D": "$\\frac{6+3\\sqrt{... | C | mcq | 4 | metric geometry - area | |
2194 | [{"role": "user", "question": "In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\\overline{BC}$, $\\overline{CD}$, and $\\overline{AD}$, respectively. Point $H$ is the midpoint of $\\overline{GE}$. What is the area of the shaded region?\n\n<image1>", "hint": "", "options": {"A": "$\\f... | E | mcq | 4 | metric geometry - area | |
2195 | [{"role": "user", "question": "Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\\overline{XY}$ contained in the cube with edge length $3$?\n<image1>", "hint": "", "options": {"A": "$\\frac{3\\sqrt{33}}5$", "B": "$2\\sqrt{3}$", "C": "$\\frac{2\\sqrt{33}}3$"... | A | mcq | 2 | solid geometry | |
2196 | [{"role": "user", "question": "A rectangular piece of paper whose length is $\\sqrt{3}$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded ... | C | mcq | 4 | transformation geometry | |
2197 | [{"role": "user", "question": "Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5:2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length o the square window?\n\n<image1>", "hint": "", "opti... | 26 | numeric | 4 | metric geometry - length | |
2198 | [{"role": "user", "question": "Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\\triangle ABC$?\n\n<image1>", "hint": "", "options": {"A": "$2\\sqrt{3}$", "B": "$3\\sqrt{3}$", "C": "$1+3\\sqrt{2}$", "D": "$2+2\\sqrt{3}$", "E": "$3+2\\sqrt{3}$"}, "choices": []}] | B | mcq | 4 | metric geometry - area | |
2199 | [{"role": "user", "question": "In rectangle $ABCD$, $DC = 2CB$ and points $E$ and $F$ lie on $\\overline{AB}$ so that $\\overline{ED}$ and $\\overline{FD}$ trisect $\\angle ADC$ as shown. What is the ratio of the area of $\\triangle DEF$ to the area of rectangle $ABCD$?\n\n<image1>", "hint": "", "options": {"A": "$\\fr... | A | mcq | 4 | metric geometry - area | |
22 | [{"role": "user", "question": "What time is it now, if after 6 hours and 30 minutes the clock will show 4:00?\n<image1>", "hint": "", "options": {"A": "10:00", "B": "10:30", "C": "2:30", "D": "22:10", "E": "21:30"}, "choices": []}] | E | mcq | 2 | arithmetic | |
220 | [{"role": "user", "question": "An equilateral triangle with side length 3 and a circle with radius 1 have the same centre. What is the perimeter of the figure that is created when the two are being put together?\n<image1>", "hint": "", "options": {"A": "$6+\\pi$", "B": "$3+2 \\pi$", "C": "$9+\\frac{\\pi}{3}$", "D": "$3... | A | mcq | 5 | metric geometry - length | |
2200 | [{"role": "user", "question": "Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{1+\\sqrt{2}}4$", "B": "$\\frac{\\sqrt{5}-1}2$", "C": "$\\frac{\\sqrt{3}+1}4$", "D": "$\\frac... | B | mcq | 4 | metric geometry - length | |
2201 | [{"role": "user", "question": "A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?\n\n\n<image1>", "hint": "", "optio... | E | mcq | 2 | solid geometry | |
2202 | [{"role": "user", "question": "Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase? \n<image1>", "hint": "", "options": {}, "choices": []}] | 22 | numeric | 3 | algebra | |
2203 | [{"role": "user", "question": "The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise arou... | C | mcq | 4 | transformation geometry | |
2204 | [{"role": "user", "question": "The letter F shown below is rotated $90^\\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?\n<image1><image2>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": [... | E | mcq | 4 | transformation geometry | |
2205 | [{"role": "user", "question": "The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\\frac{3}{2}$ and center $(0,\\frac{3}{2})$ t... | B | mcq | 5 | analytic geometry | |
2206 | [{"role": "user", "question": "The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?\n\n<image1>", "hint": "", "options": {"A": "$\\frac{75}{12}$", "B": "$10$", "C": "$12$", "D": "$10\\sqrt{2}$", "E": "$15$"}, "choices": []}] | B | mcq | 2 | solid geometry | |
2207 | [{"role": "user", "question": "In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$?\n<image1>", "hint": "", "options": {"A": "$3$", "B": "$12-4\\sqrt{5}$", "C": "$\\frac{5+2\\sqrt{5}}{3}$", "D": "$1+\\sqrt{5}$", "E": "$\\frac{11+11\\sqrt{5}}{10}$"}, "choices": []}] | D | mcq | 4 | metric geometry - length | |
2208 | [{"role": "user", "question": "A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner recta... | 2 | numeric | 4 | metric geometry - length | |
2209 | [{"role": "user", "question": "What is the area of the shaded region of the given $8 \\times 5$ rectangle?\n\n<image1>", "hint": "", "options": {"A": "$4\\frac{3}{5}$", "B": "$5$", "C": "$5\\frac{1}{4}$", "D": "$6\\frac{1}{2}$", "E": "$8$"}, "choices": []}] | D | mcq | 4 | metric geometry - area | |
221 | [{"role": "user", "question": "The adjacent diagram illustrates the graphs of the two functions f and g. How can we describe the relationship between f and g?\n<image1>", "hint": "", "options": {"A": "$g(x-2)=-f(x)$", "B": "$g(x)=f(x+2)$", "C": "$g(x)=-f(-x+2)$", "D": "$g(-x)=-f(-x-2)$", "E": "$g(2-x)=-f(x)$"}, "choice... | A | mcq | 4 | analytic geometry | |
2210 | [{"role": "user", "question": "Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inch... | A | mcq | 4 | metric geometry - length | |
2211 | [{"role": "user", "question": "Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\\overline{AB}$ so that $EB=1$, point $G$ lies on $\\overline{BC}$ so that $CG=1$. and point $F$ lies on $\\overline{CD}$ so that $DF=2$. Segments $\\overline{AG}$ and $\\overline{AC}$ intersect $\\overline{EF}$ at $Q$ and $P$, re... | D | mcq | 4 | metric geometry - length | |
2212 | [{"role": "user", "question": "Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?\n<image1>", "hint": "", "options": {}, "choices": []}] | 78 | numeric | 4 | metric geometry - area | |
2213 | [{"role": "user", "question": "In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are pos... | 12 | numeric | 4 | combinatorics | |
2214 | [{"role": "user", "question": "All of the triangles in the diagram below are similar to iscoceles triangle $ABC$, in which $AB=AC$. Each of the 7 smallest triangles has area 1, and $\\triangle ABC$ has area 40. What is the area of trapezoid $DBCE$?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 24 | numeric | 4 | metric geometry - area | |
2215 | [{"role": "user", "question": "A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?\n<image1>", "hint": "", "options": {"A": "$1+\\frac{1}{2} \\sqrt{2}$", "B": "$\\sqrt{3}$", "C": "$\\frac{7}{4}$", "D": "$\\frac{... | D | mcq | 4 | transformation geometry | |
2216 | [{"role": "user", "question": "Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is ... | 69 | numeric | 4 | metric geometry - length | |
2217 | [{"role": "user", "question": "Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the fi... | D | mcq | 4 | metric geometry - area | |
2218 | [{"role": "user", "question": "In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle... | 19 | numeric | 4 | metric geometry - area | |
2219 | [{"role": "user", "question": "Sara makes a staircase out of toothpicks as shown:<image1>\nThis is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?", "hint": "", "options": {}, "choices": []}] | 12 | numeric | 3 | algebra | |
222 | [{"role": "user", "question": "In the diagram on the right we see the birdô-eye view and front elevation of a solid that is defined by flat surfaces (i.e. view from obove and the front respectively). Bird' s-Eye View (view from above): <image1>. Front Elevation (view from the front): <image2>.\nWhich of the outlines I ... | D | mcq | 1 | descriptive geometry | |
2220 | [{"role": "user", "question": "In the rectangular parallelpiped shown, $AB = 3, BC= 1,$ and $CG = 2$. Point $M$ is the midpoint of $\\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?\n\n<image1>", "hint": "", "options": {"A": "$1$", "B": "$\\frac{4}{3}$", "C": "$\\frac{3}{2}$"... | E | mcq | 2 | solid geometry | |
2221 | [{"role": "user", "question": "A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper ar... | A | mcq | 2 | solid geometry | |
2222 | [{"role": "user", "question": "The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n<image1>\nHow many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform ... | 2 | numeric | 4 | transformation geometry | |
2223 | [{"role": "user", "question": "The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$\n\n<image1>", "hint": "", "options": {"A"... | A | mcq | 4 | metric geometry - area | |
2224 | [{"role": "user", "question": "The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the ... | B | mcq | 4 | metric geometry - area | |
2225 | [{"role": "user", "question": "As shown in the figure, line segment $\\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2$. Three semicircles of radius $1,$ $\\overarc{AEB},\\overarc{BFC},$ and $\\overarc{CGD},$ have their diameters on $\\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ ... | 17 | numeric | 4 | metric geometry - area | |
2226 | [{"role": "user", "question": "Triangle $AMC$ is isoceles with $AM = AC$. Medians $\\overline{MV}$ and $\\overline{CU}$ are perpendicular to each other, and $MV=CU=12$. What is the area of $\\triangle AMC?$\n<image1>", "hint": "", "options": {}, "choices": []}] | 96 | numeric | 4 | metric geometry - area | |
2227 | [{"role": "user", "question": "As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bo... | 810 | numeric | 4 | combinatorics | |
2228 | [{"role": "user", "question": "A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?\n<image1>", "hint": "", "options": {"... | C | mcq | 2 | solid geometry | |
2229 | [{"role": "user", "question": "As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length $2$ so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?\n\n<ima... | D | mcq | 4 | metric geometry - area | |
223 | [{"role": "user", "question": "The sum of the number in each line, column and diagonal in the Ămagic squareñon the right is always constant. Only two numbers are visible. Which number is missing in field $a$?\n<image1>", "hint": "", "options": {"A": "16", "B": "51", "C": "54", "D": "55", "E": "110"}, "choices": []}] | D | mcq | 1 | logic | |
2230 | [{"role": "user", "question": "In square $ABCD$, points $E$ and $H$ lie on $\\overline{AB}$ and $\\overline{DA}$, respectively, so that $AE=AH$. Points $F$ and $G$ lie on $\\overline{BC}$ and $\\overline{CD}$, respectively, and points $I$ and $J$ lie on $\\overline{EH}$ so that $\\overline{FI} \\perp \\overline{EH}$ an... | B | mcq | 4 | metric geometry - length | |
2231 | [{"role": "user", "question": "As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\\angle CDE = 110^\\circ$. Point $F$ lies on $\\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\\angle AFE?$\n<image1>", "hint": "... | 170 | numeric | 2 | metric geometry - angle | |
2232 | [{"role": "user", "question": "A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border... | 84 | numeric | 4 | combinatorics | |
2233 | [{"role": "user", "question": "What is the area of the shaded figure shown below?\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 4 | metric geometry - area | |
2234 | [{"role": "user", "question": "A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?\n\n<im... | B | mcq | 4 | metric geometry - area | |
2235 | [{"role": "user", "question": "In square $ABCD$, points $P$ and $Q$ lie on $\\overline{AD}$ and $\\overline{AB}$, respectively. Segments $\\overline{BP}$ and $\\overline{CQ}$ intersect at right angles at $R$, with $BR=6$ and $PR=7$. What is the area of the square?\n\n<image1>", "hint": "", "options": {}, "choices": []}... | 117 | numeric | 4 | metric geometry - area | |
2236 | [{"role": "user", "question": "Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in the $24$-sided polygon shown in the fi... | 147 | numeric | 4 | transformation geometry | |
2237 | [{"role": "user", "question": "A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive int... | 67 | numeric | 4 | metric geometry - area | |
2238 | [{"role": "user", "question": "Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \\text{ cm}$ and $6 \\text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \\text{ cm}$, which sinks to t... | E | mcq | 2 | solid geometry | |
2239 | [{"role": "user", "question": "Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eight row and eight column) and places the other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest and the least number that ... | 367 | numeric | 3 | algebra | |
224 | [{"role": "user", "question": "In a rectangle JKLM the angle bisector in $\\mathrm{J}$ intersects the diagonal KM in $\\mathrm{N}$. The distance of $\\mathrm{N}$ to $\\mathrm{LM}$ is 1 and the distance of $\\mathrm{N}$ to $\\mathrm{KL}$ is 8. How long is LM?\n<image1>", "hint": "", "options": {"A": "$8+2 \\sqrt{2}$", "... | A | mcq | 5 | metric geometry - length | |
2240 | [{"role": "user", "question": "The figure below is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\\sqrt{m}+\\sqrt{n},$ where $m$ and $n$ are positive integers. What is $m+n?$\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 23 | numeric | 4 | metric geometry - area | |
2241 | [{"role": "user", "question": "A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\\overline{AD}$ at point $C'$, and edge $\\overline{BC}$ intersects edge $\\overline{AB}$ at point $E$. Suppose that $C'D=\\f... | A | mcq | 4 | transformation geometry | |
2242 | [{"role": "user", "question": "A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter... | 68 | numeric | 4 | metric geometry - area | |
2243 | [{"role": "user", "question": "Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into... | B | mcq | 4 | combinatorics | |
2244 | [{"role": "user", "question": "A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?\n<image1>", "hi... | 540 | numeric | 4 | combinatorics | |
2245 | [{"role": "user", "question": "Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\\sqrt{2}$ centimeters, as sho... | E | mcq | 4 | metric geometry - area | |
2246 | [{"role": "user", "question": "A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?\n\n<image... | B | mcq | 2 | solid geometry | |
2247 | [{"role": "user", "question": "Suppose that 13 cards numbered $1, 2, 3, \\dots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third p... | 8178 | numeric | 4 | combinatorics | |
2248 | [{"role": "user", "question": "Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-a... | 337 | numeric | 5 | analytic geometry | |
2249 | [{"role": "user", "question": "In rhombus $ABCD$, point $P$ lies on segment $\\overline{AD}$ such that $BP\\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?\n\n<image1>", "hint": "", "options": {"A": "$3\\sqrt{5}$", "B": "$10$", "C": "$6\\sqrt{5}$", "D": "$20$", "E": "$25$"}, "choices": []}] | D | mcq | 4 | metric geometry - area | |
225 | [{"role": "user", "question": "In the box are seven blockss. You want to rearrange the blocks so that another block can placed. What is the minimum number of blocks that have to be moved?\n<image1>", "hint": "", "options": {}, "choices": []}] | 3 | numeric | 5 | combinatorial geometry | |
2250 | [{"role": "user", "question": "The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?\n\n<image1>", "hint": "", "op... | D | mcq | 4 | metric geometry - area | |
2251 | [{"role": "user", "question": "Each square in a $5 \\times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:\n\nAny filled square with two or three filled neighbors remains fille... | 22 | numeric | 4 | transformation geometry | |
2252 | [{"role": "user", "question": "A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{5}$", "B": "$\\frac{1}{4}$", "C": "$2-\\s... | C | mcq | 4 | metric geometry - length | |
2253 | [{"role": "user", "question": "Each square in a $3\\times 3$ grid of squares is colored red, white, blue, or green so that every $2\\times 2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?\n<image1>", "hint": "", "options": {}, "choice... | 72 | numeric | 4 | combinatorics | |
2254 | [{"role": "user", "question": "Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the rad... | D | mcq | 4 | metric geometry - length | |
2255 | [{"role": "user", "question": "Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a bloc... | C | mcq | 4 | metric geometry - area | |
2256 | [{"role": "user", "question": "Square $ABCD$ is rotated $20^\\circ$ clockwise about its center to obtain square $EFGH$, as shown below. What is the degree measure of $\\angle EAB$?\n<image1>", "hint": "", "options": {"A": "$20^\\circ$", "B": "$30^\\circ$", "C": "$32^\\circ$", "D": "$35^\\circ$", "E": "$45^\\circ$"}, "... | B | mcq | 4 | transformation geometry | |
2257 | [{"role": "user", "question": "In the figure, it is given that angle $ C = 90^{\\circ}, \\overline{AD} = \\overline{DB}, DE \\perp AB, \\overline{AB} = 20$, and $ \\overline{AC} = 12$. The area of quadrilateral $ ADEC$ is:\n<image1>", "hint": "", "options": {"A": "$75$", "B": "$58\\frac{1}{2}$", "C": "$48$", "D": "$37\... | B | mcq | 5 | metric geometry - area | |
2258 | [{"role": "user", "question": "In the figure, $ \\overline{CD}, \\overline{AE}$ and $ \\overline{BF}$ are one-third of their respective sides. It follows that $ \\overline{AN_2}: \\overline{N_2N_1}: \\overline{N_1D} = 3: 3: 1$, and similarly for lines $ BE$ and $ CF$. Then the area of triangle $ N_1N_2N_3$ is:\n<image1... | C | mcq | 5 | metric geometry - area |
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