data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out ... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 36100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The s... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 36101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve congruent disks are placed on a circle $C^{}_{}$ of radius 1 in such a way that the twelve disks cover $C^{}_{}$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure ... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 36102,
"solution": "We wish to find the radius of one circle, so that we can find the total area.\n Notice that for them to contain the entire circle, each pair of circles must be tangent on the larger circle. Now consider two adjacent smaller circles. This means that the line connecting the radii is a s... |
DeepScaleR | [
{
"role": "user",
"content": "Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to color the dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.)"... | math | {
"style": "rule",
"ground_truth": "7983360"
} | {
"index": 36103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to construct this dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the oth... | math | {
"style": "rule",
"ground_truth": "7983360"
} | {
"index": 36104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve congruent rectangles are placed together to make a rectangle $PQRS$. What is the ratio $PQ: QR$?\n\nA) $2: 3$ \nB) $3: 4$ \nC) $5: 6$ \nD) $7: 8$ \nE) $8: 9$"
}
] | math | {
"style": "rule",
"ground_truth": "8 : 9"
} | {
"index": 36105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve delegates, four each from three different countries, randomly select chairs at a round table that seats twelve people. Calculate the probability that each delegate sits next to at least one delegate from another country, and express this probability as a fraction $\\frac... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 36106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve mayoral candidates each made a statement about how many times lies had been told before their turn. The first candidate said, \"Before me, one lie was told.\" The second candidate said, \"Now, two lies have been told.\" The third candidate said, \"Now, three lies have be... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve million added to twelve thousand equals"
}
] | math | {
"style": "rule",
"ground_truth": "12012000"
} | {
"index": 36108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve million added to twelve thousand equals what number?"
}
] | math | {
"style": "rule",
"ground_truth": "12012000"
} | {
"index": 36109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve people arrive at dinner, but the circular table only seats eight. If two seatings, such that one is a rotation of the other, are considered the same, then in how many different ways can we choose eight people, divide them into two groups of four each, and seat each group... | math | {
"style": "rule",
"ground_truth": "1247400"
} | {
"index": 36110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve points are spaced around a $3 \\times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{11}"
} | {
"index": 36111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth."
}
] | math | {
"style": "rule",
"ground_truth": "0.230"
} | {
"index": 36112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve students in Mrs. Stephenson's class have brown eyes. Twenty students in the class have a lunch box. Of Mrs. Stephenson's 30 students, what is the least possible number of students who have brown eyes and a lunch box?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 36113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and a die with 8 faces numbered 1 to 8 is rolled. Determine the probability that the product of the numbers on the tile and the die will be a square."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{48}"
} | {
"index": 36114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and an 8-sided die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?\nA) $\\frac{11}{96}$\nB) $\\frac{17}{96}$\nC) $\\frac{21... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{96}"
} | {
"index": 36115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and an eight-sided die (numbered 1 to 8) is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?\nA) $\\frac{1}{12}$\nB) $\\frac{1}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 36116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve tiles numbered $1$ through $12$ are turned up at random, and an 8-sided die (sides numbered from 1 to 8) is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a square."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{48}"
} | {
"index": 36117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve tiles numbered $1$ through $12$ are turned up at random, and an eight-sided die is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{96}"
} | {
"index": 36118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered ... | math | {
"style": "rule",
"ground_truth": "114"
} | {
"index": 36119,
"solution": "To solve this problem, we need to understand how the numbers assigned to each block in the pyramid structure contribute to the number on the top block. The structure is built in layers with each block in the upper layers resting on three blocks from the layer below. The number on each... |
DeepScaleR | [
{
"role": "user",
"content": "Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points."
}... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 36120,
"solution": "Let $a$, $b$, and $c$ be the vertex of a triangle that satisfies this problem, where $a > b > c$. \\[a - b = p_1\\] \\[b - c = p_2\\] \\[a - c = p_3\\]\n$p_3 = a - c = a - b + b - c = p_1 + p_2$. Because $p_3$ is the sum of two primes, $p_1$ and $p_2$, $p_1$ or $p_2$ must be $2$. Let ... |
DeepScaleR | [
{
"role": "user",
"content": "Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each o... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 36121,
"solution": "We simplify this problem by using complementary counting and fixing one knight in place. Then, either a knight can sit two spaces apart from the fixed knight, or a knight can sit more than two spaces apart from the fixed knight. The probability is then $\\frac{24\\left(23\\right)-\\le... |
DeepScaleR | [
{
"role": "user",
"content": "Twenty gremlins and fifteen imps are at the Annual Mischief Convention. The imps have had a lot of in-fighting lately and refuse to shake hands with each other, but they readily shake hands with all of the gremlins. Meanwhile, all the gremlins are quite friendly and shake hands... | math | {
"style": "rule",
"ground_truth": "490"
} | {
"index": 36122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and d... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty kindergarten children are arranged in a line at random, consisting of 11 girls and 9 boys. Find the probability that there are no more than five girls between the first and the last boys in the line."
}
] | math | {
"style": "rule",
"ground_truth": "0.058"
} | {
"index": 36124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty pairs of integers are formed using each of the integers \\( 1, 2, 3, \\ldots, 40 \\) once. The positive difference between the integers in each pair is 1 or 3. If the resulting differences are added together, what is the greatest possible sum?"
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 36125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty people, including \\( A, B, \\) and \\( C \\), sit randomly at a round table. What is the probability that at least two of \\( A, B, \\) and \\( C \\) sit next to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "17/57"
} | {
"index": 36126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty percent less than 60 is one-third more than what number?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 36127,
"solution": "1. **Calculate 20% less than 60**: \n To find 20% less than a number, we subtract 20% of the number from the number itself. \n \\[\n 60 - 0.20 \\times 60 = 60 - 12 = 48\n \\]\n Alternatively, we can calculate this as 80% of 60:\n \\[\n 0.80 \\times 60 = 48\n \\]\n or... |
DeepScaleR | [
{
"role": "user",
"content": "Twenty points are equally spaced around the circumference of a circle. Kevin draws all the possible chords that connect pairs of these points. How many of these chords are longer than the radius of the circle but shorter than its diameter?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 36128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a $3\\times 3 \\times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\\frac{p^a}... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 36129,
"solution": "We can consider the orientation of each of the individual cubes independent=== The unit cube at the center of our large cube has no exterior faces, so all of its orientations work.\n For the six unit cubes and the centers of the faces of the large cube, we need that they show an orang... |
DeepScaleR | [
{
"role": "user",
"content": "Twenty switches in an office computer network are to be connected so that each switch has a direct connection to exactly three other switches. How many connections will be necessary?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 36130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty tiles are numbered 1 through 20 and are placed into box $A$. Twenty other tiles numbered 11 through 30 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than 15 and the tile from box $B$ is eithe... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{50}"
} | {
"index": 36131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty tiles are numbered 1 through 20 and are placed into box $C$. Twenty other tiles numbered 15 through 34 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 18 and the tile from box $D$ is eithe... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{40}"
} | {
"index": 36132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty-eight 4-inch wide square posts are evenly spaced with 4 feet between adjacent posts to enclose a rectangular field. The rectangle has 6 posts on each of the longer sides (including the corners). What is the outer perimeter, in feet, of the fence?"
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 36133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty-five cards are randomly arranged in a grid. Five of these cards have a 0 on one side and a 1 on the other side. The remaining twenty cards either have a 0 on both sides or a 1 on both sides. Loron chooses one row or one column and flips over each of the five cards in tha... | math | {
"style": "rule",
"ground_truth": "9:16"
} | {
"index": 36134,
"solution": "The given arrangement has 14 zeroes and 11 ones showing. Loron can pick any row or column in which to flip the 5 cards over. Furthermore, the row or column that Loron chooses can contain between 0 and 5 of the cards with different numbers on their two sides. Of the 5 rows and 5 column... |
DeepScaleR | [
{
"role": "user",
"content": "Twenty-five people who always tell the truth or always lie are standing in a queue. The man at the front of the queue says that everyone behind him always lies. Everyone else says that the person immediately in front of them always lies. How many people in the queue always lie?... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 36135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty-four 4-inch wide square posts are evenly spaced with 5 feet between adjacent posts to enclose a square field, as shown. What is the outer perimeter, in feet, of the fence? Express your answer as a mixed number. [asy]\nunitsize(2mm);\ndefaultpen(linewidth(.7pt));\ndotfact... | math | {
"style": "rule",
"ground_truth": "129\\frac{1}{3}"
} | {
"index": 36136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty-seven increased by twice a number is 39. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{17}"
} | {
"index": 36138,
"solution": "Once we have assigned Zack and Mihir teams, there are 8 spots for more players on Zack's team and 9 for more players on the third team. Andrew is equally likely to occupy any of these spots, so our answer is $\\frac{8}{17}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, th... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 36139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two $4 \\times 4$ squares are randomly placed on an $8 \\times 8$ chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap?"
}
] | math | {
"style": "rule",
"ground_truth": "529/625"
} | {
"index": 36140,
"solution": "$529 / 625$. Each square has 5 horizontal $\\cdot 5$ vertical $=25$ possible positions, so there are 625 possible placements of the squares. If they do not overlap, then either one square lies in the top four rows and the other square lies in the bottom four rows, or one square lies i... |
DeepScaleR | [
{
"role": "user",
"content": "Two 18-24-30 triangles in the plane share the same circumcircle as well as the same incircle. What's the area of the region common to both the triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 36141,
"solution": "Notice, first of all, that $18-24-30$ is 6 times $3-4-5$, so the triangles are right. Thus, the midpoint of the hypotenuse of each is the center of their common circumcircle, and the inradius is $\\frac{1}{2}(18+24-30)=6$. Let one of the triangles be $A B C$, where $\\angle A<\\angle ... |
DeepScaleR | [
{
"role": "user",
"content": "Two 24-sided dice have the following configurations: 5 purple sides, 8 blue sides, 10 red sides, and 1 gold side. What is the probability that when both dice are rolled, they will show the same color?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{95}{288}"
} | {
"index": 36142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two 5-digit positive integers are formed using each of the digits from 0 through 9 once. What is the smallest possible positive difference between the two integers?"
}
] | math | {
"style": "rule",
"ground_truth": "247"
} | {
"index": 36143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two 6-sided dice, one red and one green, are rolled. What is the probability that the red die shows an odd number and the green die shows a number that is a perfect square?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac16"
} | {
"index": 36144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two 600 mL pitchers contain orange juice. One pitcher is 1/3 full and the other pitcher is 2/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?"
}... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{30}"
} | {
"index": 36145,
"solution": "1. **Calculate the amount of orange juice in each pitcher:**\n - The first pitcher is $\\frac{1}{3}$ full of orange juice. Since the total capacity of the pitcher is $600$ mL, the amount of orange juice in the first pitcher is:\n \\[\n 600 \\times \\frac{1}{3} = 200 \\text{ ... |
DeepScaleR | [
{
"role": "user",
"content": "Two 8-sided dice are tossed (each die has faces numbered 1 to 8). What is the probability that the sum of the numbers shown on the dice is either a prime or a multiple of 4?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{39}{64}"
} | {
"index": 36146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 36147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two Wei Qi teams, $A$ and $B$, each comprising 7 members, compete against each other. Players from each team face off in sequence. The first game is between the first player of each team. The loser is eliminated, and the winner moves on to face the next player of the opposing t... | math | {
"style": "rule",
"ground_truth": "3432"
} | {
"index": 36148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two \\(1 \\times 1\\) squares are removed from a \\(5 \\times 5\\) grid as shown. Determine the total number of squares of various sizes on the grid."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 36149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two \\(10 \\times 24\\) rectangles are inscribed in a circle as shown. Find the shaded area."
}
] | math | {
"style": "rule",
"ground_truth": "169\\pi - 380"
} | {
"index": 36150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron are equilateral triangles with a side length of 1 and form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane that co... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 36151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, each being equilateral triangles with side length 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane co... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 36152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, each of which is a regular triangle with a side length of 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 36153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, representing equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane conta... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"index": 36154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plan... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 36155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 36156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plan... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"index": 36157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane c... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"index": 36158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane conta... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 36159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane conta... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 36160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane containi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 36161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane conta... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"index": 36162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane contai... | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\sqrt{3}}{4}"
} | {
"index": 36163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two adjacent sides of a rhombus form a $60$-degree angle. If each side of the rhombus measures $2$ cm, what is the area of the rhombus, in square centimeters? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 36164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two advanced Level 3 students participated in a university chess tournament. Each participant plays against all others exactly once. A win is worth 1 point, a draw is worth 0.5 points, and a loss is worth 0 points. The total scores of the two Level 3 students sum up to 6.5 poin... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures $6\\sqrt2$ units, what is the sum of the lengths of the two remaining sides? Express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "28.4"
} | {
"index": 36166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two angles of an isosceles triangle measure $70^\\circ$ and $x^\\circ$. What is the sum of the three possible values of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 36167,
"solution": "We are given an isosceles triangle with two angles measuring $70^\\circ$ and $x^\\circ$. We need to find the sum of all possible values of $x$.\n\nAn isosceles triangle has at least two sides of equal length, and the angles opposite these sides are equal. The sum of the angles in any ... |
DeepScaleR | [
{
"role": "user",
"content": "Two ants crawled along their own closed routes on a $7 \\times 7$ board. Each ant crawled only along the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first and the ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 36168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. When the left ant first hits the line $y=\\frac{1}{2}$, what is its speed?"
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3} - 3"
} | {
"index": 36169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st term of sequence $... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 36170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two arithmetic sequences $C$ and $D$ both begin with 50. Sequence $C$ has a common difference of 12 and is increasing, while sequence $D$ has a common difference of 8 and is decreasing. What is the absolute value of the difference between the 31st term of sequence $C$ and the 3... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 36171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$ have the sums of the first $n$ terms as $S_{n}$ and $T_{n}$, respectively. It is known that $\\frac{{S}_{n}}{{T}_{n}}=\\frac{7n+2}{n+3}$. Find $\\frac{{a}_{7}}{{b}_{7}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{93}{16}"
} | {
"index": 36172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two arithmetic sequences, $C$ and $D$, begin with the terms 25 and 40 respectively, both with common differences of absolute value 15, with sequence $C$ increasing and sequence $D$ decreasing. What is the absolute value of the difference between the 60th terms of sequences $C$ ... | math | {
"style": "rule",
"ground_truth": "1755"
} | {
"index": 36173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two balls are randomly chosen from a box containing 20 balls numbered from 1 to 20. Calculate the probability that the sum of the numbers on the two balls is divisible by 3."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{95}"
} | {
"index": 36174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two balls, a red one and a green one, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is given by $p_k = \\frac{1}{k(k+1)}$ for $k = 1, 2, 3, ...$. Calculate the probability that the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 36175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two balls, one blue and one orange, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is $3^{-k}$ for $k = 1, 2, 3,...$. What is the probability that the blue ball is tossed into a high... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 36176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two boards, one 5 inches wide and the other 7 inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are later separated, what is the area of the unpainted region on the five-inch board? Assume the... | math | {
"style": "rule",
"ground_truth": "35\\sqrt{2}"
} | {
"index": 36177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two boards, one five inches wide and the other eight inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are separated, what is the area of the unpainted region on the five-inch board? (Neglect ... | math | {
"style": "rule",
"ground_truth": "40 \\sqrt{2}"
} | {
"index": 36178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two boards, one four inches wide and the other six inches wide, are nailed together to form an X. The angle at which they cross is 60 degrees. If this structure is painted and the boards are separated what is the area of the unpainted region on the four-inch board? (The holes c... | math | {
"style": "rule",
"ground_truth": "16\\sqrt{3}"
} | {
"index": 36179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two books of different subjects are taken from three shelves, each having 10 Chinese books, 9 math books, and 8 English books. Calculate the total number of different ways to do this."
}
] | math | {
"style": "rule",
"ground_truth": "242"
} | {
"index": 36180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two boxes of candies have a total of 176 pieces. If 16 pieces are taken out from the second box and put into the first box, the number of pieces in the first box is 31 more than m times the number of pieces in the second box (m is an integer greater than 1). Then, determine the... | math | {
"style": "rule",
"ground_truth": "131"
} | {
"index": 36181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two boys $A$ and $B$ start at the same time to ride from Port Jervis to Poughkeepsie, $60$ miles away. $A$ travels $4$ miles an hour slower than $B$. $B$ reaches Poughkeepsie and at once turns back meeting $A$ $12$ miles from Poughkeepsie. The rate of $A$ was:"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 36182,
"solution": "1. **Define Variables:**\n Let the speed of boy $A$ be $a$ mph, and the speed of boy $B$ be $b$ mph. Given that $A$ travels $4$ mph slower than $B$, we have:\n \\[ b = a + 4 \\]\n\n2. **Set Up Distance Equations:**\n - Boy $A$ travels until the meeting point, which is $12$ miles... |
DeepScaleR | [
{
"role": "user",
"content": "Two boys and three girls stand in a row for a photo. If boy A does not stand at either end, and exactly two of the three girls are adjacent, determine the number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 36183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 36184,
"solution": "1. **Define the problem setup**: Two boys start at the same point A on a circular track and move in opposite directions with speeds of 5 ft/s and 9 ft/s respectively. We need to find the number of times they meet excluding the start and finish.\n\n2. **Calculate relative speed**: Sinc... |
DeepScaleR | [
{
"role": "user",
"content": "Two brothers had tickets to a stadium located 10 km from their home. Initially, they planned to walk to the stadium. However, they changed their plan and decided to use a bicycle. They agreed that one would start on the bicycle and the other would walk simultaneously. After cov... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 36185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two bullets are placed in two consecutive chambers of a 6-chamber pistol. The cylinder is then spun. The pistol is fired but the first shot is a blank. Let \\( p \\) denote the probability that the second shot is also a blank if the cylinder is spun after the first shot and let... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 36186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two candidates participated in an election with \\(p+q\\) voters. Candidate \\(A\\) received \\(p\\) votes and candidate \\(B\\) received \\(q\\) votes, with \\(p > q\\). During the vote counting, only one vote is registered at a time on a board. Let \\(r\\) be the probability ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2019}"
} | {
"index": 36187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two candles of the same height are lighted at the same time. The first is consumed in $4$ hours and the second in $3$ hours.\nAssuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\frac{2}{5}"
} | {
"index": 36188,
"solution": "1. **Set up the equations for the heights of the candles**: Let the initial height of each candle be 1 unit. The first candle burns completely in 4 hours, so it burns at a rate of $\\frac{1}{4}$ units per hour. The second candle burns completely in 3 hours, so it burns at a rate of $\... |
DeepScaleR | [
{
"role": "user",
"content": "Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other?"
}
] | math | {
"style": "rule",
"ground_truth": "1:36"
} | {
"index": 36189,
"solution": "1. **Identify the rate of burning for each candle**: \n - The first candle burns out in $3$ hours, i.e., $180$ minutes.\n - The second candle burns out in $4$ hours, i.e., $240$ minutes.\n\n2. **Set up the function for the length of each candle's stub**:\n - Let $\\ell$ be the i... |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 12?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{663}"
} | {
"index": 36190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a heart and the second card is a 10?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{52}"
} | {
"index": 36191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are face cards (Jacks, Queens, or Kings) totaling to a numeric value of 20?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{221}"
} | {
"index": 36192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) and total to 15?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{221}"
} | {
"index": 36193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{884}"
} | {
"index": 36194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is an ace?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{52}"
} | {
"index": 36195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is either a 10 or a Jack?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{442}"
} | {
"index": 36196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are chosen consecutively without replacement from a standard 52-card deck. What is the probability that the first card is a face card (Jack, Queen, or King) and the second card is a number card (2 through 10) with the two cards totaling to 15?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{221}"
} | {
"index": 36197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards (13 hearts, 13 clubs, 13 spades, and 13 diamonds). What is the probability that the first card is a 6 and the second card is a Queen?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{663}"
} | {
"index": 36198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\\diamondsuit$ and the second card is an ace?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{52}"
} | {
"index": 36199,
"solution": ""
} |
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