data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper, PQRS, has PQ = 20 and QR = 15. The piece of paper is glued flat on the surface of a large cube so that Q and S are at vertices of the cube. The shortest distance from P to R, as measured through the cube, can be calculated using the 3D geometry of ... | math | {
"style": "rule",
"ground_truth": "18.4"
} | {
"index": 3000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular plot $ABCD$ is divided into two rectangles as shown in the diagram and is contracted to two households, Jia and Yi. Jia's vegetable greenhouse has the same area as Yi’s chicken farm, while the remaining part of Jia's area is 96 acres more than Yi's. Given that $BF... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 3001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular pool table has vertices at $(0,0)(12,0)(0,10)$, and $(12,10)$. There are pockets only in the four corners. A ball is hit from $(0,0)$ along the line $y=x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bou... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 3002,
"solution": "Consider the tiling of the plane with the $12 \\times 10$ rectangle to form a grid. Then the reflection of the ball off a wall is equivalent to traveling along the straight line $y=x$ into another $12 \\times 10$ rectangle. Hence we want to find the number of walls of the grid that the... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has 4 green faces, 2 yellow faces, and 6 blue faces. What's the probability that when it is rolled, a blue face will be facing up?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has 6 faces, 12 edges, and 8 vertices. If a new pyramid is added using one of its rectangular faces as the base, calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting shape after the fusion of the prism and pyrami... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 3004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has a volume of $12 \\mathrm{~cm}^{3}$. A new prism is formed by doubling the length, doubling the width, and tripling the height of the original prism. What is the volume of this new prism?"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 3005,
"solution": "Suppose that the original prism has length $\\ell \\mathrm{cm}$, width $w \\mathrm{~cm}$, and height $h \\mathrm{~cm}$. Since the volume of this prism is $12 \\mathrm{~cm}^{3}$, then $\\ell w h=12$. The new prism has length $2 \\ell \\mathrm{cm}$, width $2 w \\mathrm{~cm}$, and height ... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has a volume of \\(8 \\mathrm{~cm}^{3}\\), total surface area of \\(32 \\mathrm{~cm}^{2}\\), and its length, width, and height are in geometric progression. Find the sum of all its edge lengths (in cm)."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 3006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has dimensions 10 inches by 3 inches by 30 inches. If a cube has the same volume as this prism, what is the surface area of the cube, in square inches?"
}
] | math | {
"style": "rule",
"ground_truth": "6 \\times 900^{2/3}"
} | {
"index": 3007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has dimensions 8 inches by 2 inches by 32 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches?"
}
] | math | {
"style": "rule",
"ground_truth": "384"
} | {
"index": 3008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has dimensions of 1 by 1 by 2. Calculate the sum of the areas of all triangles whose vertices are also vertices of this rectangular prism, and express the sum in the form $m + \\sqrt{n} + \\sqrt{p}$, where $m, n,$ and $p$ are integers. Find $m + n + p$."
}... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 3009,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has edges $a=b=8$ units and $c=27$ units. Divide the prism into four parts from which a cube can be assembled."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and vertices $Q_i$ and $Q_i'$ are opposite each other for $1 \\le i \\le 4$. The dimensions of the prism are given by lengths 2 along the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{14}}{2}"
} | {
"index": 3011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism has vertices at the corners and edges joining them similarly to a cube. The prism dimensions differ along each axis; therefore, no two adjoining sides are of the same length. If one side has a length ratio of 2:3 with another, and there are three dimensions ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 3012,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism measures 10-inches by 20-inches by 10-inches. What is the length, in inches, of the diagonal connecting point A and point B? Express your answer in simplest radical form. [asy]\nunitsize(0.75cm);\ndefaultpen(linewidth(0.7pt)+fontsize(10pt));\ndotfactor=4;\n\... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{6}"
} | {
"index": 3013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism measuring 20 cm by 14 cm by 12 cm has a small cube of 4 cm on each side removed from each corner. What percent of the original volume is removed?"
}
] | math | {
"style": "rule",
"ground_truth": "15.24\\%"
} | {
"index": 3014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism with dimensions 1 cm by 1 cm by 2 cm has a dot marked in the center of the top face (1 cm by 2 cm face). It is sitting on a table, which is 1 cm by 2 cm face. The prism is rolled over its shorter edge (1 cm edge) on the table, without slipping, and stops onc... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"index": 3015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular prism with integer edge lengths is painted red on its entire surface and then cut into smaller cubes with edge length 1. Among these smaller cubes, 40 cubes have two red faces and 66 cubes have one red face. What is the volume of this rectangular prism?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 3016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular room has a perimeter of 42 feet and an area of 108 square feet. What is the length, in feet, of the shorter side?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 3017,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular room measures 12-feet by 6-feet. How many square yards of carpet are needed to cover the floor of the room?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 3018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular room measures 15-feet by 8-feet and has a triangular extension with a base of 8-feet and a height of 5-feet. How many square yards of carpet are needed to cover the entire floor of the room, including the triangular extension?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 3019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular sheet of paper is 20 cm long and 12 cm wide. It is folded along its diagonal. What is the perimeter of the shaded region?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 3020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{2}{\\sqrt{5} - 1}}"
} | {
"index": 3021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular solid has three adjacent faces with areas of $1$, $2$, and $2$, respectively. All the vertices of the rectangular solid are located on the same sphere. Find the volume of this sphere."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}\\pi"
} | {
"index": 3022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular swimming pool is 20 meters long, 15 meters wide, and 3 meters deep. Calculate its surface area."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 3023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular table $ 9$ rows $ \\times$ $ 2008$ columns is fulfilled with numbers $ 1$ , $ 2$ , ..., $ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$ . What i... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular table measures $12'$ in length and $9'$ in width and is currently placed against one side of a rectangular room. The owners desire to move the table to lay diagonally in the room. Determine the minimum length of the shorter side of the room, denoted as $S$, in fee... | math | {
"style": "rule",
"ground_truth": "15'"
} | {
"index": 3025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular table of dimensions \\( x \\) cm \\(\\times 80\\) cm is covered with identical sheets of paper of size \\( 5 \\) cm \\(\\times 8 \\) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimet... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 3026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular table of size \\( x \\) cm \\( \\times 80 \\) cm is covered with identical sheets of paper of size 5 cm \\( \\times 8 \\) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 3027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular table of size \\( x \\) cm by 80 cm is covered with identical sheets of paper of size 5 cm by 8 cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. ... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 3028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular table with dimensions $x$ cm $\\times 80$ cm is covered with identical sheets of paper measuring 5 cm $\\times 8$ cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of ... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 3029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular tank with a horizontal cross-sectional area of \\(S = 6 \\ \\text{m}^2\\) is filled with water up to a height of \\(H = 5 \\ \\text{m}\\). Determine the time it takes for all the water to flow out of the tank through a small hole at the bottom with an area of \\(s... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 3030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 3031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular yard contains three flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, with the parallel sides measuring $10$ meters and $20$ meters. What fraction of the yard is occupied by the flower beds?\nA) $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 3032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "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 flow... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 3033,
"solution": "1. **Identify the dimensions of the triangles:**\n The problem states that the yard contains two congruent isosceles right triangles and the parallel sides of the trapezoidal remainder are $15$ meters and $25$ meters. The difference in length between the parallel sides of the trapezo... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape. The parallel sides of the trapezoid have lengths $18$ and $30$ meters. What fraction of the yard is occupied by the flower beds?\n... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 3034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The yard's remaining area forms a trapezoidal shape, as shown. The lengths of the parallel sides of the trapezoid are $20$ and $30$ meters, respectively. What fraction of the yard i... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 3035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a high... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 3036,
"solution": "\nWe are given that the probability that a ball is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \\ldots$. We need to find the probability that the red ball is tossed into a higher-numbered bin than the green ball.\n\n#### Step-by-step Analysis:\n\n1. **Probability of Landing in th... |
DeepScaleR | [
{
"role": "user",
"content": "A reflection takes $\\begin{pmatrix} -1 \\\\ 7 \\end{pmatrix}$ to $\\begin{pmatrix} 5 \\\\ -5 \\end{pmatrix}.$ Which vector does the reflection take $\\begin{pmatrix} -4 \\\\ 3 \\end{pmatrix}$ to?"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 \\\\ -5 \\end{pmatrix}"
} | {
"index": 3037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A refrigerator is offered at sale at $250.00 less successive discounts of 20% and 15%. The sale price of the refrigerator is:"
}
] | math | {
"style": "rule",
"ground_truth": "77\\% of 250.00"
} | {
"index": 3038,
"solution": "1. **Calculate the price after the first discount:** \n The original price of the refrigerator is $250.00$. The first discount is $20\\%$ of the original price. Therefore, the price after the first discount is calculated as follows:\n \\[\n 250.00 \\times (1 - 0.20) = 250.00 \\t... |
DeepScaleR | [
{
"role": "user",
"content": "A region $S$ in the complex plane is defined by \\begin{align*}\nS = \\{x + iy: - 1\\le x\\le1, - 1\\le y\\le1\\}.\n\\end{align*}A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\\left(\\frac34 + \\frac34i\\right)z$ is also in ... | math | {
"style": "rule",
"ground_truth": "\\frac 79"
} | {
"index": 3039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $2/\\pi$, as shown. What is the perimeter of this region? [asy]\npath a=(10,0)..(5,5)--(5,-5)..cycle;\npath b=(0,10)..(5,5)--(-5,5)..cycle;\npath c=(-10,0)..(-5,5)--(-5,-5)..cycle;\... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 3040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A region is bounded by semicircular arcs created on each side of an equilateral triangle, where each side measures \\(1/\\pi\\). Calculate the total perimeter of this shaped region."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 3041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular $17$ -gon with vertices $V_1, V_2, . . . , V_{17}$ and sides of length $3$ has a point $ P$ on $V_1V_2$ such that $V_1P = 1$ . A chord that stretches from $V_1$ to $V_2$ containing $ P$ is rotated within the interior of the heptadecagon around $V_2$ ... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 3042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular $2015$ -simplex $\\mathcal P$ has $2016$ vertices in $2015$ -dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\\mathcal P$ that are closer to its center than any of its vertice... | math | {
"style": "rule",
"ground_truth": "520"
} | {
"index": 3043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular $2018$ -gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$ -gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ( $2$ numbers are neighboring if the vertices they ar... | math | {
"style": "rule",
"ground_truth": "2 \\times 1008!"
} | {
"index": 3044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular $n$-gon $P_{1} P_{2} \\ldots P_{n}$ satisfies $\\angle P_{1} P_{7} P_{8}=178^{\\circ}$. Compute $n$."
}
] | math | {
"style": "rule",
"ground_truth": "630"
} | {
"index": 3045,
"solution": "Let $O$ be the center of the $n$-gon. Then $$\\angle P_{1} O P_{8}=2\\left(180^{\\circ}-\\angle P_{1} P_{7} P_{8}\\right)=4^{\\circ}=\\frac{360^{\\circ}}{90}$$ which means the arc $\\widehat{P_{1} P_{8}}$ that spans 7 sides of the $n$-gon also spans $1 / 90$ of its circumcircle. Thus $... |
DeepScaleR | [
{
"role": "user",
"content": "A regular $n$-gon has $n$ diagonals, its perimeter is $p$, and the sum of the lengths of all the diagonals is $q$. What is $\\frac{p}{q} + \\frac{q}{p}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 3046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular $n$-gon is inscribed in a circle with radius $R$, and its area is equal to $3 R^{2}$. Find $n$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular 100-sided polygon is placed on a table, with the numbers $1, 2, \\ldots, 100$ written at its vertices. These numbers are then rewritten in order of their distance from the front edge of the table. If two vertices are at an equal distance from the edge, the left number... | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"index": 3048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular 100-sided polygon is placed on a table, with vertices labeled from 1 to 100. The numbers are written down in the order they appear from the front edge of the table. If two vertices are equidistant from the edge, the number to the left is written down first, followed b... | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"index": 3049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form\n\\[a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6},\\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 3050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a ... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 3051,
"solution": "Begin as in solution 2, drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon. Apply law of cosines on $\\theta = {30^{\\circ}, 60^{\\circ}, 90^{\\circ}, 120^{\\circ}, 150^{\\circ}, 180^{\\circ}}$ to get $d^2 = 288 - 288 \\cos \\theta$ where $d$ is the di... |
DeepScaleR | [
{
"role": "user",
"content": "A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 3052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular 15-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 3053,
"solution": "1. **Determine the Lines of Symmetry, $L$:**\n A regular polygon with an odd number of sides, such as a regular 15-gon, has lines of symmetry that pass through each vertex and bisect the opposite side. This is because each line of symmetry aligns a vertex with the midpoint of the sid... |
DeepScaleR | [
{
"role": "user",
"content": "A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "338350"
} | {
"index": 3054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular 2015-gon \\( A_{1} A_{2} \\cdots A_{2015} \\) is inscribed in a unit circle \\( O \\). What is the probability that for any two distinct vertices \\( A_{i}, A_{j} \\), the magnitude \\( \\left|\\overrightarrow{O A_{i}}+\\overrightarrow{O A_{j}}\\right| \\geqslant 1 \\... | math | {
"style": "rule",
"ground_truth": "671/1007"
} | {
"index": 3055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular 2017-gon \\( A_1 A_2 \\cdots A_{2017} \\) is inscribed in a unit circle \\( O \\). If two different vertices \\( A_i \\) and \\( A_j \\) are chosen randomly, what is the probability that \\( \\overrightarrow{O A_i} \\cdot \\overrightarrow{O A_j} > \\frac{1}{2} \\)?"
... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 3056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular decagon $A_{0} A_{1} A_{2} \\cdots A_{9}$ is given in the plane. Compute $\\angle A_{0} A_{3} A_{7}$ in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "54^{\\circ}"
} | {
"index": 3057,
"solution": "Put the decagon in a circle. Each side subtends an arc of $360^{\\circ} / 10=36^{\\circ}$. The inscribed angle $\\angle A_{0} A_{3} A_{7}$ contains 3 segments, namely $A_{7} A_{8}, A_{8} A_{9}, A_{9} A_{0}$, so the angle is $108^{\\circ} / 2=54^{\\circ}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular decagon $B_1B_2B_3B_4B_5B_6B_7B_8B_9B_{10}$ is inscribed in a circle of area $1$ square units. Point $Q$ lies inside the circle such that the region bounded by $\\overline{QB_1},\\overline{QB_2},$ and the minor arc $\\widehat{B_1B_2}$ of the circle has area $\\tfrac{1... | math | {
"style": "rule",
"ground_truth": "110\\sqrt{3}"
} | {
"index": 3058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular decagon is formed by connecting three sequentially adjacent vertices of the decagon. Find the probability that all three sides of the triangle are also sides of the decagon."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 3059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular decagon is given. A triangle is formed by connecting three randomly chosen vertices of the decagon. Calculate the probability that none of the sides of the triangle is a side of the decagon."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 3060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular dodecagon $P_{1} P_{2} \\cdots P_{12}$ is inscribed in a unit circle with center $O$. Let $X$ be the intersection of $P_{1} P_{5}$ and $O P_{2}$, and let $Y$ be the intersection of $P_{1} P_{5}$ and $O P_{4}$. Let $A$ be the area of the region bounded by $X Y, X P_{2}... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 3061,
"solution": "The area of sector $O P_{2} P_{4}$ is one sixth the area of the circle because its angle is $60^{\\circ}$. The desired area is just that of the sector subtracted by the area of equilateral triangle $O X Y$. Note that the altitude of this triangle is the distance from $O$ to $P_{1} P_{5... |
DeepScaleR | [
{
"role": "user",
"content": "A regular dodecagon ($12$ sides) is inscribed in a circle with radius $r$ inches. The area of the dodecagon, in square inches, is:"
}
] | math | {
"style": "rule",
"ground_truth": "3r^2"
} | {
"index": 3062,
"solution": "To find the area of a regular dodecagon inscribed in a circle, we can break the dodecagon into 12 congruent isosceles triangles, each having a vertex at the center of the circle.\n\n1. **Calculate the central angle of each triangle:**\n A regular dodecagon has 12 sides, so the centra... |
DeepScaleR | [
{
"role": "user",
"content": "A regular dodecagon \\( Q_1 Q_2 \\dotsb Q_{12} \\) is drawn in the coordinate plane with \\( Q_1 \\) at \\( (4,0) \\) and \\( Q_7 \\) at \\( (2,0) \\). If \\( Q_n \\) is the point \\( (x_n,y_n) \\), compute the numerical value of the product\n\\[\n(x_1 + y_1 i)(x_2 + y_2 i)(x_3... | math | {
"style": "rule",
"ground_truth": "531440"
} | {
"index": 3063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular dodecagon \\(Q_1 Q_2 \\dotsb Q_{12}\\) is drawn in the coordinate plane with \\(Q_1\\) at \\((1,0)\\) and \\(Q_7\\) at \\((3,0)\\). If \\(Q_n\\) is the point \\((x_n,y_n),\\) compute the numerical value of the product\n\\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \\dots... | math | {
"style": "rule",
"ground_truth": "4095"
} | {
"index": 3064,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular dodecagon \\(Q_1 Q_2 \\ldots Q_{12}\\) is drawn in the coordinate plane with \\(Q_1\\) at \\((2,0)\\) and \\(Q_7\\) at \\((4,0)\\). If \\(Q_n\\) is the point \\((x_n, y_n)\\), compute the numerical value of the product:\n\\[\n(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \\... | math | {
"style": "rule",
"ground_truth": "531440"
} | {
"index": 3065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 3066,
"solution": "We can achieve 6 by projecting onto a plane perpendicular to an edge of the dodecaheron. Indeed, if we imagine viewing the dodecahedron in such a direction, then 4 of the faces are projected to line segments (namely, the two faces adjacent to the edge and the two opposite faces), and o... |
DeepScaleR | [
{
"role": "user",
"content": "A regular football match is being played. A draw is possible. The waiting time for the next goal is independent of previous events in the match. It is known that the expected total number of goals in football matches of these teams is 2.8. Find the probability that an even numb... | math | {
"style": "rule",
"ground_truth": "0.502"
} | {
"index": 3067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon $A B C D E F$ has side length 1 and center $O$. Parabolas $P_{1}, P_{2}, \\ldots, P_{6}$ are constructed with common focus $O$ and directrices $A B, B C, C D, D E, E F, F A$ respectively. Let $\\chi$ be the set of all distinct points on the plane that lie on a... | math | {
"style": "rule",
"ground_truth": "35 \\sqrt{3}"
} | {
"index": 3068,
"solution": "Recall the focus and the directrix are such that the parabola is the locus of points equidistant from the focus and the directrix. We will consider pairs of parabolas and find their points of intersections (we label counterclockwise): (1): $P_{1} \\cap P_{2}$, two parabolas with direct... |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon $ABCDEF$ has sides of length 2. Find the area of $\\bigtriangleup ADF$. Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 3069,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\\bigtriangleup ACE$. Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"index": 3070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon PROFIT has area 1. Every minute, greedy George places the largest possible equilateral triangle that does not overlap with other already-placed triangles in the hexagon, with ties broken arbitrarily. How many triangles would George need to cover at least $90 \... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 3071,
"solution": "It's not difficult to see that the first triangle must connect three non-adjacent vertices (e.g. POI), which covers area $\\frac{1}{2}$, and leaves three 30-30-120 triangles of area $\\frac{1}{6}$ each. Then, the next three triangles cover $\\frac{1}{3}$ of the respective small triangl... |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon \\( A B C D E K \\) is inscribed in a circle of radius \\( 3 + 2\\sqrt{3} \\). Find the radius of the circle inscribed in the triangle \\( B C D \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 3072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon \\( K L M N O P \\) is inscribed in an equilateral triangle \\( A B C \\) such that the points \\( K, M, O \\) lie at the midpoints of the sides \\( A B, B C, \\) and \\( A C \\), respectively. Calculate the area of the hexagon \\( K L M N O P \\) given that t... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 3073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon \\(ABCDEF\\) is inscribed in a circle with a radius of \\(3+\\sqrt{3}\\). Find the radius of the inscribed circle of triangle \\(ACD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 3074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon `LMNOPQ` has sides of length 4. Find the area of triangle `LNP`. Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"index": 3075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 3076,
"solution": "1. **Identify the formulas for the areas of the shapes**:\n - The area of an equilateral triangle ($A_{\\triangle}$) with side length $s_t$ is given by:\n \\[\n A_{\\triangle} = \\frac{s_t^2 \\sqrt{3}}{4}\n \\]\n - A regular hexagon can be divided into six equilateral t... |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon has a side length of 8 cm. Calculate the area of the shaded region formed by connecting two non-adjacent vertices to the center of the hexagon, creating a kite-shaped region.\n\n[asy]\nsize(100);\n\npair A,B,C,D,E,F,O;\nA = dir(0); B = dir(60); C = dir(120); D... | math | {
"style": "rule",
"ground_truth": "16\\sqrt{3}"
} | {
"index": 3077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon has an area of $150\\sqrt{3}$ cm². If each side of the hexagon is decreased by 3 cm, by how many square centimeters is the area decreased?"
}
] | math | {
"style": "rule",
"ground_truth": "76.5\\sqrt{3}"
} | {
"index": 3078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon has one side along the diameter of a semicircle, and the two opposite vertices on the semicircle. Find the area of the hexagon if the diameter of the semicircle is 1."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3} / 26"
} | {
"index": 3079,
"solution": "The midpoint of the side of the hexagon on the diameter is the center of the circle. Draw the segment from this center to a vertex of the hexagon on the circle. This segment, whose length is $1 / 2$, is the hypotenuse of a right triangle whose legs have lengths $a / 2$ and $a \\sqrt{3}... |
DeepScaleR | [
{
"role": "user",
"content": "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?"
}
] | math | {
"style": "rule",
"ground_truth": "54\\sqrt{3}-18\\pi"
} | {
"index": 3080,
"solution": "1. **Calculate the area of the hexagon**: \n A regular hexagon can be divided into 6 equilateral triangles. Each side of the hexagon is given as 6. The area \\( A \\) of an equilateral triangle with side length \\( s \\) is given by the formula:\n \\[\n A = \\frac{\\sqrt{3}}{4} s... |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 3081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon is inscribed in a circle of radius 2 units. In square units, what is the area of the hexagon? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"index": 3082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12, what is the area of the equilateral triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 3083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon is inscribed in another regular hexagon such that each vertex of the inscribed hexagon divides a side of the original hexagon into two parts in the ratio 2:1. Find the ratio of the area of the inscribed hexagon to the area of the larger hexagon."
}
] | math | {
"style": "rule",
"ground_truth": "7/9"
} | {
"index": 3084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs?"
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}-\\pi"
} | {
"index": 3085,
"solution": "1. **Identify the Geometry of the Problem:**\n A regular hexagon with side length $s = 1$ is inscribed in a circle. This means that each side of the hexagon subtends a $60^\\circ$ angle at the center of the circle.\n\n2. **Calculate the Radius of the Circle:**\n The radius $r$ of t... |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z} \\ | \\ z \\in R\\right\\rbrace$.... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3} + 2 \\pi"
} | {
"index": 3086,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z}|z \\in R\\right\\rbrace$. Then th... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 3087,
"solution": "We can describe the line parallel to the imaginary axis $x=\\frac{1}{2}$ using polar coordinates as $r(\\theta)=\\dfrac{1}{2\\cos{\\theta}},$\n which rearranges to $z=\\left(\\dfrac{1}{2\\cos{\\theta}}\\right)(cis{\\theta})\\implies \\frac{1}{z}=2\\cos{\\theta}cis(-\\theta).$\n Denote ... |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\\sqrt7$ .\n\nNotes: ''Using a ruler construct points $\\ldots$ '' means: N... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"index": 3088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon with side length 1 has an arbitrary interior point that is reflected over the midpoints of its six sides. Calculate the area of the hexagon formed in this way."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{2}"
} | {
"index": 3089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexagon's center and vertices together make 7 points. Calculate the number of triangles that can be formed using any 3 of these points as vertices."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 3090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular hexahedron with an edge length of $1$ is cut by planes passing through the common vertex of three edges and their respective midpoints. After removing the $8$ triangular pyramids, the volume of the remaining convex polyhedron is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 3091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top ve... | math | {
"style": "rule",
"ground_truth": "810"
} | {
"index": 3092,
"solution": "Go to 2020 AMC 10A #19, and connect all of the centers of the faces on the dodecahedron to get the icosahedron. The answer is $\\boxed{810}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular octagon $ABCDEFGH$ has its sides' midpoints connected to form a smaller octagon inside it. Determine the fraction of the area of the larger octagon $ABCDEFGH$ that is enclosed by this smaller octagon."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3093,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\\bigtriangleup ADG$. Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "4+3\\sqrt{2}"
} | {
"index": 3094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular octagon has a side length of 8 cm. What is the number of square centimeters in the area of the shaded region formed by diagonals connecting alternate vertices (forming a square in the center)?"
}
] | math | {
"style": "rule",
"ground_truth": "192 + 128\\sqrt{2}"
} | {
"index": 3095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"index": 3096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular octagon is inscribed in a circle of radius 2 units. What is the area of the octagon? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "16 \\sqrt{2} - 8(2)"
} | {
"index": 3097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected with... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2}, 4+2 \\sqrt{2}"
} | {
"index": 3098,
"solution": "A player ends up with a right angle iff they own two diametrically opposed vertices. Under optimal play, the game ends in a draw: on each of Bob's turns he is forced to choose the diametrically opposed vertex of Alice's most recent choice, making it impossible for either player to win.... |
DeepScaleR | [
{
"role": "user",
"content": "A regular octagon's perimeter is given as $P=16\\sqrt{2}$. If $R_i$ denotes the midpoint of side $V_iV_{i+1}$ (with $V_8V_1$ as the last side), calculate the area of the quadrilateral $R_1R_3R_5R_7$.\nA) $8$\nB) $4\\sqrt{2}$\nC) $10$\nD) $8 + 4\\sqrt{2}$\nE) $12$"
}
] | math | {
"style": "rule",
"ground_truth": "8 + 4\\sqrt{2}"
} | {
"index": 3099,
"solution": ""
} |
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