data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A small fish is holding 17 cards, labeled 1 through 17, which he shuffles into a random order. Then, he notices that although the cards are not currently sorted in ascending order, he can sort them into ascending order by removing one card and putting it back in a different pos... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 3500,
"solution": "Instead of looking at moves which put the cards in order, we start with the cards in order and consider possible starting positions by backtracking one move: each of 17 cards can be moved to 16 new places. But moving card $k$ between card $k+1$ and card $k+2$ is equivalent to moving ca... |
DeepScaleR | [
{
"role": "user",
"content": "A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 3501,
"solution": "Line Segment $DE = \\frac{1}{n}$, so $EC = 1 - \\frac{1}{n} = \\frac{n-1}{n}$. Draw line segment $HE$ parallel to the corresponding sides of the small square, $HE$ has length $\\frac{1}{\\sqrt{1985}}$, as it is the same length as the sides of the square. Notice that $\\triangle CEL$ is... |
DeepScaleR | [
{
"role": "user",
"content": "A small square is entirely contained in a larger square, as shown. The side length of the small square is 3 units and the side length of the larger square is 7 units. What is the number of square units in the area of the black region?\n\n[asy]\nfill((0,0)--(21,0)--(21,21)--(0,2... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 3502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A small town has fewer than 6000 inhabitants. We know that there are $10\\%$ more girls than boys among the children, and $15\\%$ more men than women among the adults. There are $20\\%$ more children than adults in the town.\n\nHow many people live in the town?"
}
] | math | {
"style": "rule",
"ground_truth": "3311"
} | {
"index": 3503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A small village has $n$ people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, i... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 3504,
"solution": "The problem asks for the number of $n$ that divide $\\binom{n}{3}$, which happens exactly when $\\frac{(n-1)(n-2)}{2 \\cdot 3}$ is an integer. Regardless of the parity of $n,(n-1)(n-2)$ is always divisible by 2. Also, $(n-1)(n-2)$ is divisible by 3 if and only if $n$ is not a multiple ... |
DeepScaleR | [
{
"role": "user",
"content": "A smaller regular tetrahedron is formed by joining the midpoints of the edges of a larger regular tetrahedron. Determine the ratio of the volume of the smaller tetrahedron to the volume of the larger tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 3505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A smaller square was cut out from a larger square in such a way that one side of the smaller square lies on a side of the original square. The perimeter of the resulting octagon is $40\\%$ greater than the perimeter of the original square. By what percentage is the area of the ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 3506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A smooth sphere with a radius of 1 cm is dipped in red paint and released between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm respectively (this sphere is outside the smaller sphere but inside the larger one). Upon contact with both spheres, the sphere ... | math | {
"style": "rule",
"ground_truth": "83.25"
} | {
"index": 3507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A snail crawls from one tree to another. In half a day, it covered \\( l_{1}=5 \\) meters. Then, it got tired of this and turned back, crawling \\( l_{2}=4 \\) meters. It got tired and fell asleep. The next day, the same process repeats. The distance between the trees is \\( s=... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 3508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A snake of length $k$ is an animal which occupies an ordered $k$-tuple $\\left(s_{1}, \\ldots, s_{k}\\right)$ of cells in a $n \\times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_{i}$ and $s_{i+1}$ must share a side for $i=1, \\ldots, k-1$. If th... | math | {
"style": "rule",
"ground_truth": "4571.8706930"
} | {
"index": 3509,
"solution": "Let $n=30$. The snake can get stuck in only 8 positions, while the total number of positions is about $n^{2} \\times 4 \\times 3 \\times 3=36 n^{2}$. We can estimate the answer as $\\frac{36 n^{2}}{8}=4050$, which is good enough for 13 points. Let's try to compute the answer as precise... |
DeepScaleR | [
{
"role": "user",
"content": "A snowball with a temperature of $0^{\\circ} \\mathrm{C}$ is launched at a speed $v$ towards a wall. Upon impact, $k=0.02\\%$ of the entire snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $\\frac{v}{2}$? The s... | math | {
"style": "rule",
"ground_truth": "0.005"
} | {
"index": 3510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A soccer ball kicked vertically upward reaches, after $t$ seconds, a height of $s$ meters where $s = 180t - 20t^2$. Find the maximum height reached by the ball."
}
] | math | {
"style": "rule",
"ground_truth": "405"
} | {
"index": 3511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A soccer ball rolls at $4 \\mathrm{~m} / \\mathrm{s}$ towards Marcos in a direct line from Michael. The ball is $15 \\mathrm{~m}$ ahead of Michael who is chasing it at $9 \\mathrm{~m} / \\mathrm{s}$. Marcos is $30 \\mathrm{~m}$ away from the ball and is running towards it at $8... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 3512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the su... | math | {
"style": "rule",
"ground_truth": "122"
} | {
"index": 3513,
"solution": "There are $0-3$ substitutions. The number of ways to sub any number of times must be multiplied by the previous number. This is defined recursively. The case for $0$ subs is $1$, and the ways to reorganize after $n$ subs is the product of the number of new subs ($12-n$) and the players... |
DeepScaleR | [
{
"role": "user",
"content": "A soccer team has 16 members. We need to select a starting lineup including a goalkeeper, a defender, a midfielder, and two forwards. However, only 3 players can play as a goalkeeper, 5 can play as a defender, 8 can play as a midfielder, and 4 players can play as forwards. In h... | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"index": 3514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-me... | math | {
"style": "rule",
"ground_truth": "\\binom{2k+1}{3} - k(2k+1)"
} | {
"index": 3515,
"solution": "\nLet \\( A \\) be the number of three-member subsets such that the three distinct pairs among them use different languages. We aim to find the maximum possible value of \\( A \\).\n\nGiven that the social club has \\( 2k+1 \\) members, each fluent in \\( k \\) languages, and that no t... |
DeepScaleR | [
{
"role": "user",
"content": "A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 3516,
"solution": "1. **Identify the games they lost and won:** The problem states that the team lost by one run in exactly five games. Since they lost by one run, their score in those games must be one less than their opponents' score. Additionally, in the games they won, they scored twice as many runs ... |
DeepScaleR | [
{
"role": "user",
"content": "A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 3517,
"solution": "1. **Identify the games where the team lost by one run**: The problem states that the team lost by one run in exactly five games. Since they cannot score twice as many runs as their opponents and still lose by one run, these games must be the ones where they scored odd numbers of runs.... |
DeepScaleR | [
{
"role": "user",
"content": "A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume is removed?"
}
] | math | {
"style": "rule",
"ground_truth": "18\\%"
} | {
"index": 3518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid box is 20 cm by 15 cm by 12 cm. A new solid is formed by removing a cube 4 cm on a side from each of the top four corners of this box. After that, four cubes, 2 cm on a side, are placed on each lower corner of the box. What percent of the original volume has been altere... | math | {
"style": "rule",
"ground_truth": "6.22\\%"
} | {
"index": 3519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?"
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 3520,
"solution": "To solve this problem, we need to understand the structure of the solid after the smaller cubes are removed from each corner of the larger cube.\n\n1. **Original Cube Characteristics**:\n - The original cube has a side length of $3$.\n - It has $8$ vertices (corners).\n - It has ... |
DeepScaleR | [
{
"role": "user",
"content": "A solid cube of side length $2$ is removed from each corner of a larger solid cube of side length $4$. Find the number of edges of the remaining solid."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 3521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid cube of side length $4$ has $12$ edges, and a solid cube of side length $2$ has $12$ edges. The remaining solid is created by removing $8$ solid cubes of side length $2$ from the corners of the solid cube of side length $4$. Calculate the total number of edges of the re... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid cube of side length 4 cm is cut into two pieces by a plane that passed through the midpoints of six edges. Find the surface area of each half cube created."
}
] | math | {
"style": "rule",
"ground_truth": "69"
} | {
"index": 3523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid cube of side length \\(4 \\mathrm{~cm}\\) is cut into two pieces by a plane that passed through the midpoints of six edges. To the nearest square centimetre, the surface area of each half cube created is:"
}
] | math | {
"style": "rule",
"ground_truth": "69"
} | {
"index": 3524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid has a triangular base with sides of lengths $s$, $s$, $s \\sqrt{2}$. Two opposite vertices of the triangle extend vertically upward by a height $h$ where $h = 3s$. Given $s = 2\\sqrt{2}$, what is the volume of this solid?"
}
] | math | {
"style": "rule",
"ground_truth": "24\\sqrt{2}"
} | {
"index": 3525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frus... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 3526,
"solution": "Our original solid $V$ has surface area $A_v = \\pi r^2 + \\pi r \\ell$, where $\\ell$ is the slant height of the cone. Using the Pythagorean Theorem or Pythagorean Triple knowledge, we obtain $\\ell = 5$ and lateral area $A_\\ell = 15\\pi$. The area of the base is $A_B = 3^2\\pi = 9\\... |
DeepScaleR | [
{
"role": "user",
"content": "A solid is formed by rotating a triangle with sides of lengths 3, 4, and 5 around the line containing its shortest side. Find the surface area of this solid."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 3527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid rectangular block is created using $N$ congruent 1-cm cubes adhered face-to-face. When observing the block to maximize visibility of its surfaces, exactly $252$ of the 1-cm cubes remain hidden from view. Determine the smallest possible value of $N.$"
}
] | math | {
"style": "rule",
"ground_truth": "392"
} | {
"index": 3528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$"
}
] | math | {
"style": "rule",
"ground_truth": "384"
} | {
"index": 3529,
"solution": "The $231$ cubes which are not visible must lie below exactly one layer of cubes. Thus, they form a rectangular solid which is one unit shorter in each dimension. If the original block has dimensions $l \\times m \\times n$, we must have $(l - 1)\\times(m-1) \\times(n - 1) = 231$. The p... |
DeepScaleR | [
{
"role": "user",
"content": "A solid rectangular prism has dimensions 4 by 2 by 2. A 1 by 1 by 1 cube is cut out of the corner creating the new solid shown. What is the surface area of the new solid?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 3530,
"solution": "The original prism has four faces that are 4 by 2 rectangles, and two faces that are 2 by 2 rectangles. Thus, the surface area of the original prism is \\( 4(4 \\cdot 2)+2(2 \\cdot 2)=32+8=40 \\). When a 1 by 1 by cube is cut out, a 1 by 1 square is removed from each of three faces of ... |
DeepScaleR | [
{
"role": "user",
"content": "A solid right prism $ABCDEF$ has a height of $16,$ as shown. Also, its bases are equilateral triangles with side length $12.$ Points $X,$ $Y,$ and $Z$ are the midpoints of edges $AC,$ $BC,$ and $DC,$ respectively. A part of the prism above is sliced off with a straight cut thro... | math | {
"style": "rule",
"ground_truth": "48+9\\sqrt{3}+3\\sqrt{91}"
} | {
"index": 3531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid right prism $ABCDEF$ has a height of 16, as shown. Also, its bases are equilateral triangles with side length 12. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. Determine the perimeter of triangle $XYZ$. [asy]\npair A, B, C, D, E... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 3532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid right prism $ABCDEF$ has a height of 20, as illustrated. The bases are equilateral triangles with a side length of 14. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. Determine the perimeter of triangle $XYZ$."
}
] | math | {
"style": "rule",
"ground_truth": "7 + 2\\sqrt{149}"
} | {
"index": 3533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 10. Points $V$, $W$, and $X$ are the midpoints of edges $PR$, $RQ$, and $QT$, respectively. Determine the perimeter of triangle $VWX$."
}
] | math | {
"style": "rule",
"ground_truth": "5 + 10\\sqrt{5}"
} | {
"index": 3534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 15. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RS$, respectively. Determine the perimeter of triangle $MNO$."
}
] | math | {
"style": "rule",
"ground_truth": "32.5"
} | {
"index": 3535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid right prism $PQRSTU$ has a height of 20. Its bases are equilateral triangles with side length 10. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RT$, respectively. Calculate the perimeter of triangle $MNO$."
}
] | math | {
"style": "rule",
"ground_truth": "5 + 10\\sqrt{5}"
} | {
"index": 3536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid triangular prism is made up of 27 identical smaller solid triangular prisms. The length of every edge of each of the smaller prisms is 1. If the entire outer surface of the larger prism is painted, what fraction of the total surface area of all the smaller prisms is pai... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 3537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A solid wooden rectangular prism measures $3 \\times 5 \\times 12$. The prism is cut in half by a vertical cut through four vertices, creating two congruent triangular-based prisms. What is the surface area of one of these triangular-based prisms?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 3538,
"solution": "Consider the triangular-based prism on the front of the rectangular prism. This prism has five faces: a rectangle on the front, a rectangle on the left, a triangle on the bottom, a triangle on the top, and a rectangle on the back. The rectangle on the front measures $3 \\times 12$ and ... |
DeepScaleR | [
{
"role": "user",
"content": "A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if th... | math | {
"style": "rule",
"ground_truth": "394"
} | {
"index": 3539,
"solution": "Let $P_k$ be the probability of emptying the bag when it has $k$ pairs in it. Let's consider the possible draws for the first three cards:\nCase 1. We draw a pair on the first two cards. The second card is the same as the first with probability $\\frac {1}{2k - 1}$, then we have $k - 1... |
DeepScaleR | [
{
"role": "user",
"content": "A sorcerer is concocting a healing potion. In order to do so, he must have one of three roots, and one of five minerals. However, one of the minerals is incompatible with two of the roots. Otherwise, he does not foresee any complications with other combinations of roots and min... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 3540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A space probe travels $5,555,555,555,555$ kilometers in its journey towards a distant star. After reaching its destination, it has traveled $3,333,333,333,333$ kilometers from its last refueling point. How many kilometers did the probe travel before the last refueling point?\nA... | math | {
"style": "rule",
"ground_truth": "2,222,222,222,222"
} | {
"index": 3541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average n... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 3542,
"solution": "1. **Assumptions and Initial Setup**:\n Let's assume there are $100$ people in the audience for simplicity of calculation. The total duration of the talk is $60$ minutes.\n\n2. **Distribution of Audience**:\n - **20% heard the entire talk**: This is $20\\%$ of $100$ people, which e... |
DeepScaleR | [
{
"role": "user",
"content": "A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 3543,
"solution": "Without loss of generality, assume that the $8$ numbers on Sharon's cards are $1$, $1$, $2$, $3$, $4$, $5$, $6$, and $7$, in that order, and assume the $8$ colors are red, red, and six different arbitrary colors. There are ${8\\choose2}-1$ ways of assigning the two red cards to the $8$... |
DeepScaleR | [
{
"role": "user",
"content": "A special six-sided die has its faces numbered 1 through 6 and has the property that rolling each number \\( x \\) is \\( x \\) times as likely as rolling a 1. For example, the probability of rolling a 5 is 5 times the probability of rolling a 1, while the probability of rollin... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 3544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere has a volume of \\( 288\\pi \\) cubic inches. Determine the surface area of the sphere. Also, if the sphere were to be perfectly cut in half, what would be the circumference of the flat circular surface of one of the halves? Express your answers in terms of \\( \\pi \\... | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"index": 3545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(2, 3, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 3, -8),$ with radius $r.$ Find $r.$"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{15}"
} | {
"index": 3546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(2,4,0)$ with radius 1. The sphere also intersects the $yz$-plane in a circle centered at $(0,4,-7),$ with radius $r.$ Find $r.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{46}"
} | {
"index": 3547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(3, 5, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 5, -8),$ with radius $r.$ Find $r.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{59}"
} | {
"index": 3548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with a radius of 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{59}"
} | {
"index": 3549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{59}"
} | {
"index": 3550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0)$, $(0,4,0),(0,0,6)$, but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 3551,
"solution": "Let $(x, y, z)$ be the center of the sphere. By the given condition, we have $$(x-2)^{2}+y^{2}+z^{2}=x^{2}+(y-4)^{2}+z^{2}=x^{2}+y^{2}+(z-6)^{2}$$ Subtracting $x^{2}+y^{2}+z^{2}$ yields $$x^{2}-(x-2)^{2}=y^{2}-(y-4)^{2}=z^{2}-(z-6)^{2}$$ or $$4(x-1)=8(y-2)=12(z-3)$$ Therefore $(x-1, y-... |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is cut into four congruent wedges. The circumference of the sphere is $12\\pi$ inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of $\\pi$.\n\nNote: To measure the circumference, take the largest circle on the surface o... | math | {
"style": "rule",
"ground_truth": "72\\pi"
} | {
"index": 3552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is cut into three equal wedges. The circumference of the sphere is $18\\pi$ inches. What is the volume of the intersection between one wedge and the top half of the sphere? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "162\\pi"
} | {
"index": 3553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a cone whose axial cross-section is an equilateral triangle. Find the volume of the cone if the volume of the sphere is \\( \\frac{32\\pi}{3} \\ \\text{cm}^3 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "24 \\pi"
} | {
"index": 3554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a cone, and the surface area of the sphere is equal to the area of the base of the cone. Find the cosine of the angle at the vertex in the axial section of the cone."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"index": 3555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "81\\sqrt{3}"
} | {
"index": 3556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a cube, and the cube has a surface area of 24 square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 3557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a cube, and the cube has a surface area of 54 square meters. A second cube is then inscribed within the sphere. A third, smaller sphere is then inscribed within this second cube. What is the surface area of the second cube and the volume of the third sp... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}\\pi}{2}"
} | {
"index": 3558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a cube. Given that one edge of the cube is 6 inches, how many cubic inches are in the volume of the inscribed sphere? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 3559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a cube. The edge of the cube is 10 inches. Calculate both the volume and the surface area of the sphere. Express your answer for the volume in terms of \\(\\pi\\)."
}
] | math | {
"style": "rule",
"ground_truth": "100\\pi"
} | {
"index": 3560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a cube. What is the ratio of the volume of the inscribed sphere to the volume of the cube? Express your answer as a common fraction in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 3561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a right circular cylinder. The height of the cylinder is 12 inches, and the diameter of its base is 10 inches. Find the volume of the inscribed sphere. Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{500}{3} \\pi"
} | {
"index": 3562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a right cone with base radius $12$ cm and height $24$ cm, as shown. The radius of the sphere can be expressed as $a\\sqrt{c} - a$ cm. What is the value of $a + c$? [asy]\nimport three; size(120); defaultpen(linewidth(1)); pen dashes = linetype(\"2 2\") ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 3563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\\sqrt{d} - g$ cm, where $g = b + 6$. What is the value of $b + d$?"
}
] | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"index": 3564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a right cone with base radius \\(15\\) cm and height \\(30\\) cm. Determine the radius of the sphere if it can be expressed as \\(b\\sqrt{d} - b\\) cm. What is the value of \\(b + d\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"index": 3565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in a right cone with base radius \\(15\\) cm and height \\(30\\) cm. Find the radius \\(r\\) of the sphere, which can be expressed as \\(b\\sqrt{d} - b\\) cm. What is the value of \\(b + d\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"index": 3566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "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?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{5}}{2}"
} | {
"index": 3567,
"solution": "1. **Define Variables:**\n Let $R$ be the radius of the larger base, $r$ be the radius of the smaller base, and $s$ be the radius of the sphere inscribed in the truncated cone.\n\n2. **Relationship between Radii and Sphere:**\n By the geometric mean theorem (a consequence of the Py... |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 3568,
"solution": "Note that the center is obviously at some point $(r,r,r)$ since it must be equidistant from the x-y, x-z, and y-z planes. Now note that the point-plane distance formula is $\\widehat{n}\\cdot \\vec{d}$ (the direction vector of the normal to the plane dotted with the distance between a ... |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is inside a cube with an edge length of $3$, and it touches all $12$ edges of the cube. Find the volume of the sphere."
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2}\\pi"
} | {
"index": 3569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere is tangent to all the edges of the pyramid \\( SABC \\), specifically the lateral edges \\( SA, SB, \\) and \\( SC \\) at the points \\( A', B' \\), and \\( C' \\) respectively. Find the volume of the pyramid \\( SA'B'C' \\), given that \\( AB = BC = SB = 5 \\) and \\(... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{59}}{15}"
} | {
"index": 3570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere passes through two adjacent vertices of a unit cube and touches the planes of the faces that do not contain these vertices. What is the radius of this sphere?"
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\frac{\\sqrt{7}}{2}"
} | {
"index": 3571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere with a radius of $1$ is placed inside a cone and touches the base of the cone. The minimum volume of the cone is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{8\\pi}{3}"
} | {
"index": 3572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere with a radius of \\(\\sqrt{3}\\) has a cylindrical hole drilled through it; the axis of the cylinder passes through the center of the sphere, and the diameter of the base of the cylinder is equal to the radius of the sphere. Find the volume of the remaining part of the... | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\pi}{2}"
} | {
"index": 3573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere with center $O$ has radius $10$. A right triangle with sides $8, 15,$ and $17$ is situated in 3D space such that each side is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?\n- **A)** $\\sqrt{84}$\n- **B)** $\\sqrt{85}$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{91}"
} | {
"index": 3574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "$2\\sqrt{5}$"
} | {
"index": 3575,
"solution": "1. **Understanding the Problem**: A sphere with radius $6$ is tangent to all sides of a triangle with sides $15, 15, 24$. We need to find the distance from the center of the sphere, $O$, to the plane of the triangle.\n\n2. **Triangle's Inradius**: The triangle's sides are tangent to th... |
DeepScaleR | [
{
"role": "user",
"content": "A sphere with radius $r$ is inside a cone, the cross section of which is an equilateral triangle inscribed in a circle. Find the ratio of the total surface area of the cone to the surface area of the sphere."
}
] | math | {
"style": "rule",
"ground_truth": "9:4"
} | {
"index": 3576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere with radius $r$ is inside a cone, whose axial section is an equilateral triangle with the sphere inscribed in it. The ratio of the total surface area of the cone to the surface area of the sphere is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "9:4"
} | {
"index": 3577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sphere with volume $V$ is inside a closed right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $AB \\perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$. Find the maximum value of $V$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{2}"
} | {
"index": 3578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spherical balloon collapses into a wet horizontal surface and settles into a shape of a hemisphere while keeping the same volume. The minor radius of the original balloon, when viewed as an ellipsoid due to unequal pressure distribution, was $4\\sqrt[3]{3}$ cm. Find the major... | math | {
"style": "rule",
"ground_truth": "8\\sqrt[3]{3}"
} | {
"index": 3579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spherical decoration was suspended in a cylindrical container when the water inside it froze. The decoration was removed (without breaking the ice), leaving a hole 30 cm across at the top and 10 cm deep. If the water froze up to a height of 5 cm from the top of the sphere, wh... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{13}"
} | {
"index": 3580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spherical scoop of vanilla ice cream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the ice cream spreads out uniformly forming a cylindrical region 8 inches in radius. Assuming the density of the ice cream remains constant,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 3581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\\sqrt[3]{2}$ cm, find the radius of the original bubble."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 3582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {16!}{2^8}"
} | {
"index": 3583,
"solution": "\nTo solve this problem, we need to determine the number of ways the spider can put on its socks and shoes such that each sock is put on before its corresponding shoe on each leg.\n\n#### Step-by-step Analysis:\n\n1. **Total Items to Wear:** The spider has 8 legs, and for each leg, it ... |
DeepScaleR | [
{
"role": "user",
"content": "A spider is making a web between $n>1$ distinct leaves which are equally spaced around a circle. He chooses a leaf to start at, and to make the base layer he travels to each leaf one at a time, making a straight line of silk between each consecutive pair of leaves, such that no... | math | {
"style": "rule",
"ground_truth": "n 2^{n-2}"
} | {
"index": 3584,
"solution": "There are $n$ ways to choose a starting vertex, and at each vertex he has only two choices for where to go next: the nearest untouched leaf in the clockwise direction, and the nearest untouched leaf in the counterclockwise direction. For, if the spider visited a leaf which is not neare... |
DeepScaleR | [
{
"role": "user",
"content": "A spider is on the edge of a ceiling of a circular room with a radius of 65 feet. The spider moves straight across the ceiling to the opposite edge, passing through the circle's center. It then moves directly to another point on the edge of the circle, not passing through the c... | math | {
"style": "rule",
"ground_truth": "220 + 20\\sqrt{22}"
} | {
"index": 3585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spinner has eight congruent sections, each labeled with numbers 1 to 8. Jane and her brother each spin this spinner once. Jane wins if the non-negative difference of their numbers is less than three; otherwise, her brother wins. Determine the probability of Jane winning. Expr... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{32}"
} | {
"index": 3586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spinner has four sections labeled 1, 2, 3, and 4, each section being equally likely to be selected. If you spin the spinner three times to form a three-digit number, with the first outcome as the hundreds digit, the second as the tens digit, and the third as the unit digit, w... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 3587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spinner is divided into 8 equal sectors numbered from 1 to 8. Jane and her sister each spin the spinner once. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her sister wins. What is the probability that Jane wins?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{16}"
} | {
"index": 3588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spinner is divided into six congruent sectors, numbered from 1 to 6. Jane and her brother each spin the spinner once. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 3589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spinner is divided into three equal sections labeled 0, 1, and 5. If you spin the spinner three times, use the outcomes to form a three-digit number (hundreds, tens, units), what is the probability that the resulting number is divisible by 5?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 3590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spinner was created by drawing five radii from the center of a circle. The first four radii divide the circle into four equal wedges. The fifth radius divides one of the wedges into two parts, one having twice the area of the other. The five wedges are labeled with the wedge ... | math | {
"style": "rule",
"ground_truth": "7/12"
} | {
"index": 3591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spinner with seven congruent sectors numbered from 1 to 7 is used. If Jane and her brother each spin the spinner once, and Jane wins if the absolute difference of their numbers is less than 4, what is the probability that Jane wins? Express your answer as a common fraction."
... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{49}"
} | {
"index": 3592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spiral staircase turns $180^\\circ$ as it rises 8 feet. The radius of the staircase is 4 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "14.9"
} | {
"index": 3593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A spiral staircase turns $270^\\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "17.3"
} | {
"index": 3594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sports conference has 12 teams in two divisions of 6 each. How many games are in a complete season for the conference if each team must play every other team in its own division three times and every team in the other division twice?"
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 3595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sports conference has 14 teams in two divisions of 7. How many games are in a complete season for the conference if each team must play every other team in its own division twice and every team in the other division once?"
}
] | math | {
"style": "rule",
"ground_truth": "133"
} | {
"index": 3596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sports equipment store purchaser needs to purchase a total of 100 basketballs and volleyballs from the manufacturer, with a total payment not exceeding $11,815. The wholesale prices from the two types of balls and the retail prices at the store are shown in the table below. S... | math | {
"style": "rule",
"ground_truth": "2,600"
} | {
"index": 3597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sports league consists of 16 teams divided into 2 divisions of 8 teams each. Throughout the season, each team plays every other team in its own division three times and every team in the other division twice. How many games are in a complete season for the league?"
}
] | math | {
"style": "rule",
"ground_truth": "296"
} | {
"index": 3598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sports league has 20 teams split into four divisions of 5 teams each. How many games are there in a complete season if each team plays every other team in its own division three times and every team in the other divisions once?"
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"index": 3599,
"solution": ""
} |
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