data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \\times 2 \\times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 1500,
"solution": "To solve this problem, we use Burnside's Lemma, which is a powerful tool in combinatorics for counting distinct configurations under group actions, such as rotations in this case. The lemma states that the number of distinct configurations, up to symmetry, is the average number of poin... |
DeepScaleR | [
{
"role": "user",
"content": "A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that each face of the cube forms the base of a pyramid. The volumes of five of these pyramids are 200, 500, 1000, 1100, and 1400. What is the volume of t... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 1501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube is inscribed in a regular octahedron in such a way that its vertices lie on the edges of the octahedron. By what factor is the surface area of the octahedron greater than the surface area of the inscribed cube?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 1502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 1503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube is suspended in space with its top and bottom faces horizontal. The cube has one top face, one bottom face, and four side faces. Determine the number of ways to move from the top face to the bottom face, visiting each face at most once, without moving directly from the t... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1505,
"solution": "1. **Understanding the problem**: We are given a cube with an edge length of $3$ cm, and it is divided into smaller cubes with whole number edge lengths. We need to determine the number of smaller cubes, $N$.\n\n2. **Analyzing possible edge lengths of smaller cubes**: Since the edge le... |
DeepScaleR | [
{
"role": "user",
"content": "A cube of mass $m$ slides down the felt end of a ramp semicircular of radius $h$ , reaching a height $h/2$ at the opposite extreme.\nFind the numerical coefficient of friction $\\mu_k$ bretween the cube and the surface.\n\n*Proposed by Danilo Tejeda, Atlantida*"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{1 - \\left(\\frac{1}{2\\pi}\\right)^2}}"
} | {
"index": 1506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the sur... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{54}"
} | {
"index": 1507,
"solution": "To solve this problem, we need to minimize the white surface area on the larger $3$-inch cube constructed from $27$ smaller $1$-inch cubes. We have $21$ red cubes and $6$ white cubes.\n\n1. **Understanding the structure of the cube:**\n - The larger cube has dimensions $3 \\times 3 \... |
DeepScaleR | [
{
"role": "user",
"content": "A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 1508,
"solution": "1. **Calculate the total surface area of the larger cube**: \n The formula for the surface area of a cube is $6s^2$, where $s$ is the length of a side of the cube. For a cube with side length 3 inches, the surface area is:\n \\[\n 6 \\times 3^2 = 6 \\times 9 = 54 \\text{ square ... |
DeepScaleR | [
{
"role": "user",
"content": "A cube with a side length of 1 meter was cut into smaller cubes with a side length of 1 centimeter and arranged in a straight line. What is the length of the resulting line?"
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 1509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with a side length of 10 is divided into 1000 smaller cubes with a side length of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (in any of the three directions) is zero. In one of the small cubes (denoted as \\( A \... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 1510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with a side of 10 is divided into 1000 smaller cubes each with an edge of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (along any of the three directions) equals 0. In one of the cubes (denoted as A), the number 1 ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 1511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with an edge length of 1 and its circumscribed sphere intersect with a plane to form a cross section that is a circle and an inscribed equilateral triangle. What is the distance from the center of the sphere to the plane of the cross section?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{\\sqrt{3}}{6}$"
} | {
"index": 1512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with an edge length of 2 Chinese feet is cut 4 times horizontally and then 5 times vertically. What is the total surface area of all the small blocks after cutting, in square Chinese feet?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 1513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with an edge length of 2 decimeters is first cut 4 times horizontally, then 5 times vertically. The total surface area of all the small pieces after cutting is ____ square decimeters."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 1514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with an edge length of 4 units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of $h$ units. What is the value of $h$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 1515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with an edge length of 6 is cut into smaller cubes with integer edge lengths. If the total surface area of these smaller cubes is \\(\\frac{10}{3}\\) times the surface area of the original larger cube before cutting, how many of these smaller cubes have an edge length of... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 1516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with an edge length of 6 units has the same volume as a triangular-based pyramid with a base having equilateral triangle sides of 10 units and a height of $h$ units. What is the value of $h$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{216\\sqrt{3}}{25}"
} | {
"index": 1517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with an edge length of \\( n \\) (where \\( n \\) is a positive integer) is painted red on its surface and then cut into \\( n^3 \\) smaller cubes with an edge length of 1. It is found that the number of small cubes with only one face painted red is exactly 12 times the ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 1518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Find the minimum value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 1519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with edge length 2 cm has a dot marked at the center of the top face. The cube is on a flat table and rolls without slipping, making a full rotation back to its initial orientation, with the dot back on top. Calculate the length of the path followed by the dot in terms o... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}\\pi"
} | {
"index": 1520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 1521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ or $C$, as shown. What is the area of quadrilateral $ABCD$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 1522,
"solution": "1. **Identify the Shape of Quadrilateral $ABCD$**:\n - The cube has side length $1$.\n - Vertices $A$ and $C$ are diagonally opposite in the cube, making $AC$ a space diagonal.\n - Points $B$ and $D$ are midpoints of two opposite edges not containing $A$ or $C$.\n - Since $B$ a... |
DeepScaleR | [
{
"role": "user",
"content": "A cube with side length $2$ is sliced by a plane that passes through a vertex $A$, the midpoint $M$ of an adjacent edge, and the midpoint $P$ of the face diagonal of the top face, not containing vertex $A$. Find the area of the triangle $AMP$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 1523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $ \\frac{r-\\sqrt{s}}{t}... | math | {
"style": "rule",
"ground_truth": "330"
} | {
"index": 1524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\\frac{r-\\sqrt{s}}{t}$... | math | {
"style": "rule",
"ground_truth": "330"
} | {
"index": 1525,
"solution": "Set the cube at the origin with the three vertices along the axes and the plane equal to $ax+by+cz+d=0$, where $a^2+b^2+c^2=1$. The distance from a point $(X,Y,Z)$ to a plane with equation $Ax+By+Cz+D=0$ is \\[\\frac{AX+BY+CZ+D}{\\sqrt{A^2+B^2+C^2}},\\] so the (directed) distance from ... |
DeepScaleR | [
{
"role": "user",
"content": "A cube with side length 2 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 for $1\\le i\\le 4,$ vertices $Q_i$ and $Q_i'$ are opposite to each other. A regular octahedron has one vertex in each of the s... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{2}}{3}"
} | {
"index": 1526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube with side length 2 is cut three times to form various polyhedral pieces. It is first cut through the diagonal planes intersecting at vertex W, then an additional cut is made that passes through the midpoint of one side of the cube and perpendicular to its adjacent face, ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 1527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube, all of whose surfaces are painted, is cut into $1000$ smaller cubes of the same size. Find the expected value $E(X)$, where $X$ denotes the number of painted faces of a small cube randomly selected."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 1528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\\overline{AB}$ and $\\overline{CD}$ are parallel edges of the cube, and $\\overline{AC}$ and $\\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $... | math | {
"style": "rule",
"ground_truth": "751"
} | {
"index": 1529,
"solution": "Denote $h(X)$ the distance from point $X$ to $\\mathcal{P}, h(A) = 0, h(B) = 2,$ $h(C) = 8, h(D) = 10, h(G) = h(I) = h(H) = 7, AB = a, AC = a \\sqrt{2}.$\n Let slope $AB$ to $\\mathcal{P}$ be $\\alpha.$ Notation is shown in the diagram. \\[\\tan \\alpha = \\frac {\\sin \\alpha}{\\cos \... |
DeepScaleR | [
{
"role": "user",
"content": "A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will stil... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cubic polynomial $p(x)$ satisfies\n\\[p(n) = \\frac{1}{n^2}\\]for $n = 1, 2, 3,$ and $4.$ Find $p(5).$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{12}"
} | {
"index": 1531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "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... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{5}"
} | {
"index": 1532,
"solution": "\n1. **Understanding the Problem:**\n - We have a cube with edge length $2$ inches, which is iced on the sides and the top.\n - The cube is cut into three pieces, and we focus on the piece with a triangular top view labeled as triangle $B$.\n - We need to find the sum of the volu... |
DeepScaleR | [
{
"role": "user",
"content": "A cubical cake with edge length 3 inches is iced on the sides and the top. It is cut vertically into four pieces, such that one of the cut starts at the midpoint of the top edge and ends at a corner on the opposite edge. The piece whose top is triangular contains an area $A$ an... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 1533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles such that two squares and two triangles alternate around each vertex. What is the volume of a cuboctahedron of side length 1?"
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{2} / 3"
} | {
"index": 1534,
"solution": "We can construct a cube such that the vertices of the cuboctahedron are the midpoints of the edges of the cube. Let $s$ be the side length of this cube. Now, the cuboctahedron is obtained from the cube by cutting a tetrahedron from each corner. Each such tetrahedron has a base in the f... |
DeepScaleR | [
{
"role": "user",
"content": "A cuboid has a diagonal $A A^{\\prime}$. In what ratio does the plane passing through the endpoints $B, C, D$ of the edges originating from vertex $A$ divide the $A A^{\\prime}$ diagonal?"
}
] | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"index": 1535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$ , and $L$ . What is the smallest possible integer value for $L$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 1536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuboid has dimensions of 2 units by 2 units by 2 units. It has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3', P_4'.$ Vertices $P_2, P_3,$ and $P_4$ are adjacent to $P_1$, and vertices $P_i' (i = 1,2,3,4)$ are opposite to $P_i$. A regular octahedron has one vertex in each of ... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{2}}{3}"
} | {
"index": 1537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuboid with dimensions corresponding to length twice the cube's edge is painted with stripes running from the center of one edge to the center of the opposite edge, on each of its six faces. Each face's stripe orientation (either horizontal-center or vertical-center) is chose... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 1538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuckoo clock chimes \"cuckoo\" as many times as the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock at 9:05 and started turning the minute hand until the clock advanced by 7 hours. How many times did the clock chime... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 1539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuckoo clock chimes \"cuckoo\" on the hour, with the number of \"cuckoo\" calls equal to the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand with his finger un... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 1540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuckoo clock chimes the number of times corresponding to the current hour (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it was 9:05 and started moving the minute hand forward until the clock read 7 hours later. How many times did the cuckoo... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 1541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuckoo clock is on the wall. At the beginning of every hour, the cuckoo makes a number of \"cuckoo\" sounds equal to the hour displayed by the hour hand (for example, at 19:00 the cuckoo makes 7 sounds). One morning, Maxim approached the clock when it showed 9:05. He started ... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 1542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuckoo clock produces a number of \"cuckoo\" sounds equal to the hour it indicates (for example, at 19:00, it sounds \"cuckoo\" 7 times). One morning, Maxim approaches the clock at 9:05 and starts turning the minute hand forward until the clock shows 7 hours later. How many \... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 1543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuckoo clock rings \"cuckoo\" every hour, with the number of rings corresponding to the hour shown by the hour hand (e.g., at 7:00, it rings 7 times). One morning, Maxim approached the clock at 9:05 and started moving the minute hand until 7 hours had passed. How many times d... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 1544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cuckoo clock strikes the number of times corresponding to the current hour (for example, at 19:00, it strikes 7 times). One morning, Max approached the clock when it showed 9:05. He started turning the minute hand until it moved forward by 7 hours. How many times did the cuck... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 1545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A curious archaeologist is holding a competition where participants must guess the age of a unique fossil. The age of the fossil is formed from the six digits 2, 2, 5, 5, 7, and 9, and the fossil's age must begin with a prime number."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 1546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A curve C is established in the polar coordinate system with the coordinate origin O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of the curve C is given by $$ρ^{2}= \\frac {12}{4-cos^{2}\\theta }$$\n\n1. Find the rectangular coordi... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 1547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A curve is described parametrically by\n\\[(x,y) = (2 \\cos t - \\sin t, 4 \\sin t).\\]The graph of the curve can be expressed in the form\n\\[ax^2 + bxy + cy^2 = 1.\\]Enter the ordered triple $(a,b,c).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{4}, \\frac{1}{8}, \\frac{5}{64} \\right)"
} | {
"index": 1548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cyclist initially traveled at a speed of 20 km/h. After covering one-third of the distance, the cyclist looked at the clock and decided to increase the speed by 20%. With the new speed, the cyclist traveled the remaining part of the distance. What is the average speed of the ... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 1549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cyclist is riding on a track at a constant speed. It is known that at 11:22, he covered a distance that is 1.4 times greater than the distance he covered at 11:08. When did he start?"
}
] | math | {
"style": "rule",
"ground_truth": "10:33"
} | {
"index": 1550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cyclist rode 96 km 2 hours faster than expected. At the same time, he covered 1 km more per hour than he expected to cover in 1 hour 15 minutes. What was his speed?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 1551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How m... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder has a height of $10$ and a radius of $3.$ Determine the total surface area, including the two ends, of the cylinder. [asy]\ndraw(ellipse((5,0),5,2),black+linewidth(1));\ndraw(ellipse((5,10),5,2),black+linewidth(1));\ndraw((1,8.95)--(5,10),black+linewidth(1));\ndraw((... | math | {
"style": "rule",
"ground_truth": "78\\pi"
} | {
"index": 1553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder has a radius of 2 inches and a height of 3 inches. What is the radius of a sphere that has the same volume as this cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{9}"
} | {
"index": 1554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder has a radius of 3 cm and a height of 8 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2}"
} | {
"index": 1556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that fits inside the cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{61}"
} | {
"index": 1557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{61}"
} | {
"index": 1558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside this cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{61}"
} | {
"index": 1559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder has both its front and left views as rectangles with length 4 and height 3. Calculate the surface area of this cylinder."
}
] | math | {
"style": "rule",
"ground_truth": "20\\pi"
} | {
"index": 1560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston that has moved inside the cylinder by $h$ meters.\n\nHint: The equation of state for the gas is ... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 1561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder of base radius 1 is cut into two equal parts along a plane passing through the center of the cylinder and tangent to the two base circles. Suppose that each piece's surface area is $m$ times its volume. Find the greatest lower bound for all possible values of $m$ as ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 1562,
"solution": "Let $h$ be the height of the cylinder. Then the volume of each piece is half the volume of the cylinder, so it is $\\frac{1}{2} \\pi h$. The base of the piece has area $\\pi$, and the ellipse formed by the cut has area $\\pi \\cdot 1 \\cdot \\sqrt{1+\\frac{h^{2}}{4}}$ because its area ... |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a truncated cone with a volume of 91 from the original cone. Find the volume of the original cone."
}
] | math | {
"style": "rule",
"ground_truth": "94.5"
} | {
"index": 1563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 91. Find the volume of the original cone."
}
] | math | {
"style": "rule",
"ground_truth": "94.5"
} | {
"index": 1564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 1565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheres of radius $x$ are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{37}-75}{4}"
} | {
"index": 1566,
"solution": "Let $O$ be the center of the large sphere, and let $O_{1}, O_{2}, O_{3}$ be the centers of the small spheres. Consider $G$, the center of equilateral $\\triangle O_{1} O_{2} O_{3}$. Then if the radii of the small spheres are $r$, we have that $O G=8+r$ and $O_{1} O_{2}=O_{2} O_{3}=O_{3... |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical barrel with a radius of 5 feet and a height of 15 feet is full of water. A solid cube with side length 7 feet is set into the barrel so that one edge of the cube is vertical. Calculate the square of the volume of water displaced, $v^2$, when the cube is fully subm... | math | {
"style": "rule",
"ground_truth": "117649"
} | {
"index": 1567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n[asy] import three; im... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"index": 1568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n[asy] import three; im... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"index": 1569,
"solution": "We can use the same method as in Solution 2 to find the side length of the equilateral triangle, which is $4\\sqrt3$. From here, its area is \\[\\dfrac{\\bigl(4\\sqrt3\\bigr)^2\\sqrt3}4=12\\sqrt3.\\] The leg of the isosceles right triangle is $\\dfrac{4\\sqrt3}{\\sqrt2}=2\\sqrt6$, and ... |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical can has a circumference of 24 inches and a height of 7 inches. A spiral strip is painted on the can such that it winds around the can precisely once, reaching from the bottom to the top. However, instead of reaching directly above where it started, it ends 3 inche... | math | {
"style": "rule",
"ground_truth": "\\sqrt{778}"
} | {
"index": 1570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical glass is half full of lemonade. The ratio of lemon juice to water in the lemonade is 1:11. If the glass is 6 inches tall and has a diameter of 2 inches, what is the volume of lemon juice in the glass? Express your answer as a decimal to the nearest hundredth."
}... | math | {
"style": "rule",
"ground_truth": ".79"
} | {
"index": 1571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical glass vessel is heated to a temperature of $t = 1200^{\\circ}$C, causing part of the air to be expelled. After covering the vessel with a well-sealing glass plate, it is left to cool to room temperature at $t^{\\prime}=16^{\\circ}$C. The remaining air inside becom... | math | {
"style": "rule",
"ground_truth": "19.6\\%"
} | {
"index": 1572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical hole of 6 cm in length is drilled through a sphere, with the axis of the cylinder passing through the center of the sphere. What is the remaining volume? (Note: The volume of a spherical cap is $\\pi h^{2}(R-h / 3)$, where $R$ is the radius of the sphere and $h$ i... | math | {
"style": "rule",
"ground_truth": "36 \\pi"
} | {
"index": 1573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\\circ$ angle with the plane of the first cut.... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 1574,
"solution": "The volume of the wedge is half the volume of a cylinder with height $12$ and radius $6$. (Imagine taking another identical wedge and sticking it to the existing one). Thus, $V=\\dfrac{6^2\\cdot 12\\pi}{2}=216\\pi$, so $n=\\boxed{216}$.\n Part 1: \n Part 2: \n - AMBRIGGS"
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet?"
}
] | math | {
"style": "rule",
"ground_truth": "$48\\pi - 36 \\sqrt {3}$"
} | {
"index": 1575,
"solution": "1. **Understanding the Geometry**: The cylindrical tank is lying on its side, and we are given that the radius $r = 4$ feet and the height (length of the cylinder) $h = 9$ feet. The water fills up to a depth of $2$ feet from the bottom of the cylinder.\n\n2. **Cross-Section Analysis**:... |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical tank with radius 6 feet and height 7 feet is lying on its side. The tank is filled with water to a depth of 3 feet. Find the volume of water in the tank, in cubic feet."
}
] | math | {
"style": "rule",
"ground_truth": "84\\pi - 63\\sqrt{3}"
} | {
"index": 1576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical water tank is $\\frac{1}{5}$ full. If three liters were added, the tank would be $\\frac{1}{4}$ full. How many liters does the tank hold when it is full?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 1577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cylindrical water tank, placed horizontally, has an interior length of 15 feet and an interior diameter of 8 feet. If the surface area of the water exposed is 60 square feet, find the depth of the water in the tank."
}
] | math | {
"style": "rule",
"ground_truth": "4 - 2\\sqrt{3}"
} | {
"index": 1578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i\\geq1$, let $s_i=i\\ (\\textrm{mod}\\ 2018)$ and $r_i=2i\\ (\\textrm{mod}\\ 201... | math | {
"style": "rule",
"ground_truth": "505"
} | {
"index": 1579,
"solution": "\nTo solve this problem, we need to analyze the movement of couples on the circumference and calculate how many remain at the end of the process.\n\nInitially, we have 2018 couples placed at points labeled from 0 to 2017 on a circumference. For each minute \\( i \\), two operations are... |
DeepScaleR | [
{
"role": "user",
"content": "A dark drawer contains $90$ red socks, $70$ green socks, $50$ blue socks, and $30$ purple socks. Someone randomly selects socks without seeing their color. What is the smallest number of socks that they must pick to guarantee at least $12$ pairs? A pair of socks consists of tw... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 1580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dark room contains 120 red socks, 100 green socks, 70 blue socks, 50 yellow socks, and 30 black socks. A person randomly selects socks from the room without the ability to see their colors. What is the smallest number of socks that must be selected to guarantee that the selec... | math | {
"style": "rule",
"ground_truth": "146"
} | {
"index": 1581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?\n[asy] unitsize(10mm); defaultpen(linewidth(.8pt... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2} - 1}{2}"
} | {
"index": 1582,
"solution": "1. **Assume the side length of the octagon**: Let the side length of the octagon be $x$. \n\n2. **Calculate the area of the center square**: The center square is formed by connecting the midpoints of the sides of the octagon. Since the square is regular, its side length is $x$, and thu... |
DeepScaleR | [
{
"role": "user",
"content": "A dart board is shaped like a regular dodecagon divided into regions, with a square at its center. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?\n**A)** $\\frac{2+\\... | math | {
"style": "rule",
"ground_truth": "\\frac{2 - \\sqrt{3}}{6}"
} | {
"index": 1583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 1584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $8 on each radio sold. If the overall profit was $72, then the least possible value of $n$ for the given in... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1585,
"solution": "1. **Calculate the cost and selling price of the radios:**\n - The dealer bought $n$ radios for $d$ dollars, so the cost per radio is $\\frac{d}{n}$.\n - He sold two radios at half their cost, which is $\\frac{1}{2} \\times \\frac{d}{n} = \\frac{d}{2n}$ each. Therefore, the total i... |
DeepScaleR | [
{
"role": "user",
"content": "A dealer plans to sell a new type of air purifier. After market research, the following pattern was discovered: When the profit per purifier is $x$ (unit: Yuan, $x > 0$), the sales volume $q(x)$ (unit: hundred units) and $x$ satisfy the following relationship: If $x$ does not e... | math | {
"style": "rule",
"ground_truth": "240000"
} | {
"index": 1586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dealer sold 200 cars, and the data for some of those sales are recorded in this table. If the rest of the cars she sold were Hondas, how many Hondas did she sell?\n\n\\begin{tabular}{ |c | c|}\n\\hline \\textbf{Type of Car} & \\textbf{$\\%$ of Total Cars Sold} \\\\ \\hline\nA... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 1587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A decagon is inscribed in a rectangle such that the vertices of the decagon divide each side of the rectangle into five equal segments. The perimeter of the rectangle is 160 centimeters, and the ratio of the length to the width of the rectangle is 3:2. What is the number of squ... | math | {
"style": "rule",
"ground_truth": "1413.12"
} | {
"index": 1588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck consists of 32 cards divided into 4 suits, each containing 8 cards. In how many ways can we choose 6 cards such that all four suits are represented among them?"
}
] | math | {
"style": "rule",
"ground_truth": "415744"
} | {
"index": 1589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck consists of six red cards and six green cards, each with labels $A$, $B$, $C$, $D$, $E$ corresponding to each color. Two cards are dealt from this deck. A winning pair consists of cards that either share the same color or the same label. Calculate the probability of draw... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{66}"
} | {
"index": 1590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck of 100 cards is labeled $1,2, \\ldots, 100$ from top to bottom. The top two cards are drawn; one of them is discarded at random, and the other is inserted back at the bottom of the deck. This process is repeated until only one card remains in the deck. Compute the expect... | math | {
"style": "rule",
"ground_truth": "\\frac{467}{8}"
} | {
"index": 1591,
"solution": "Note that we can just take averages: every time you draw one of two cards, the EV of the resulting card is the average of the EVs of the two cards. This average must be of the form $$2^{\\bullet} \\cdot 1+2^{\\bullet} \\cdot 2+2^{\\bullet} \\cdot 3+\\cdots+2^{\\bullet} \\cdot 100$$ whe... |
DeepScaleR | [
{
"role": "user",
"content": "A deck of 100 cards is numbered from 1 to 100, each card having the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 1592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck of 100 cards is numbered from 1 to 100. Each card has the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by 2. ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 1593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck of 60 cards, divided into 5 suits of 12 cards each, is shuffled. In how many ways can we pick three different cards in sequence? (Order matters, so picking card A, then card B, then card C is different from picking card B, then card A, then card C.)"
}
] | math | {
"style": "rule",
"ground_truth": "205320"
} | {
"index": 1594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\\frac{1}{3}$. When $4$ black cards are added to the deck, the probability of choosing red becomes $\\frac{1}{4}$. How many cards were in the deck originally?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1595,
"solution": "1. **Identify the initial conditions**: Let $r$ be the number of red cards and $b$ be the number of black cards in the original deck. The probability of choosing a red card is given as $\\frac{1}{3}$. Therefore, we have the equation:\n \\[\n \\frac{r}{r+b} = \\frac{1}{3}\n \\]\n ... |
DeepScaleR | [
{
"role": "user",
"content": "A deck of cards now contains 54 cards, including two jokers, one being a red joker and the other a black joker, along with the standard 52 cards. In how many ways can we pick two different cards such that at least one of them is a joker? (Order matters.)"
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 1596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck of fifty-two cards consists of four $1$'s, four $2$'s, ..., four $13$'s. Two matching pairs (two sets of two cards with the same number) are removed from the deck. After removing these cards, find the probability, represented as a fraction $m/n$ in simplest form, where $... | math | {
"style": "rule",
"ground_truth": "299"
} | {
"index": 1597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck of fifty-two cards consists of four $1$'s, four $2$'s,..., and four $13$'s. A matching pair (two cards with the same number) is removed from the deck. Determine the probability that two randomly selected cards from the remaining deck also form a pair, and express the res... | math | {
"style": "rule",
"ground_truth": "1298"
} | {
"index": 1598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form... | math | {
"style": "rule",
"ground_truth": "758"
} | {
"index": 1599,
"solution": "There are ${38 \\choose 2} = 703$ ways we can draw two cards from the reduced deck. The two cards will form a pair if both are one of the nine numbers that were not removed, which can happen in $9{4 \\choose 2} = 54$ ways, or if the two cards are the remaining two cards of the number t... |
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