data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is an angle $\\theta$ in the range $0^\\circ < \\theta < 45^\\circ$ which satisfies\n\\[\\tan \\theta + \\tan 2 \\theta + \\tan 3 \\theta = 0.\\]Calculate $\\tan \\theta$ for this angle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"index": 35501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$ , $B$ and $C$ , respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$ . On each turn, Ming chooses a two-line intersection inside $ABC... | math | {
"style": "rule",
"ground_truth": "45853"
} | {
"index": 35502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate s... | math | {
"style": "rule",
"ground_truth": "n^2 - n + 1"
} | {
"index": 35503,
"solution": "\nTo solve the problem involving cable car companies \\( A \\) and \\( B \\), we must determine the smallest integer \\( k \\) such that there are always two stations linked by both companies.\n\n### Understanding the Problem\n\n1. **Stations and Connections:**\n - We have \\( n^2 \... |
DeepScaleR | [
{
"role": "user",
"content": "There is an opaque bag containing 4 identical balls labeled with the numbers 1, 2, 3, and 4.\n(Ⅰ) If balls are drawn one by one without replacement twice, calculate the probability that the first ball drawn has an even number and the sum of the two balls’ numbers is divisible b... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 35504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown. Two large triangles a... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 35505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are consider... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 35506,
"solution": "If two of our big equilateral triangles have the same color for their center triangle and the same multiset of colors for their outer three triangles, we can carry one onto the other by a combination of rotation and reflection. Thus, to make two triangles distinct, they must differ ei... |
DeepScaleR | [
{
"role": "user",
"content": "There is an urn with $N$ slips of paper numbered from 1 to $N$. Two slips are drawn at random. What is the expected value of the ratio of the smaller number to the larger number drawn?"
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 35507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is infinite sequence of composite numbers $a_1,a_2,...,$ where $a_{n+1}=a_n-p_n+\\frac{a_n}{p_n}$ ; $p_n$ is smallest prime divisor of $a_n$ . It is known, that $37|a_n$ for every $n$ .\nFind possible values of $a_1$"
}
] | math | {
"style": "rule",
"ground_truth": "37^2"
} | {
"index": 35508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \\le k \\le 11$, has a remainder of $1$. \nWhat is the difference between the two smallest such integers?"
}
] | math | {
"style": "rule",
"ground_truth": "27720"
} | {
"index": 35509,
"solution": "1. **Identify the Problem Requirements:**\n We need to find integers greater than $1$ that, when divided by any integer $k$ such that $2 \\le k \\le 11$, leave a remainder of $1$. We are asked to find the difference between the two smallest such integers.\n\n2. **Set Up the Congruen... |
DeepScaleR | [
{
"role": "user",
"content": "There is one odd integer \\( N \\) between 400 and 600 that is divisible by both 5 and 11. What is the sum of the digits of \\( N \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 35510,
"solution": "If \\( N \\) is divisible by both 5 and 11, then \\( N \\) is divisible by \\( 5 \\times 11=55 \\). This is because 5 and 11 have no common divisor larger than 1. Therefore, we are looking for a multiple of 55 between 400 and 600 that is odd. One way to find such a multiple is to star... |
DeepScaleR | [
{
"role": "user",
"content": "There is one odd integer \\(N\\) between 400 and 600 that is divisible by both 5 and 11. What is the sum of the digits of \\(N\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 35511,
"solution": "If \\(N\\) is divisible by both 5 and 11, then \\(N\\) is divisible by \\(5 \\times 11=55\\). This is because 5 and 11 have no common divisor larger than 1. Therefore, we are looking for a multiple of 55 between 400 and 600 that is odd. One way to find such a multiple is to start with... |
DeepScaleR | [
{
"role": "user",
"content": "There is only one set of five prime numbers that form an arithmetic sequence with a common difference of 6. What is the sum of those five prime numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 35512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\\neq 0$, find the equation of the line. Enter your answer i... | math | {
"style": "rule",
"ground_truth": "y=10x-4"
} | {
"index": 35513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There was a bonus fund in a certain institution. It was planned to distribute the fund such that each employee of the institution would receive $50. However, it turned out that the last employee on the list would receive only $45. Then, in order to ensure fairness, it was decid... | math | {
"style": "rule",
"ground_truth": "950"
} | {
"index": 35514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There were 100 cards lying face up with the white side on the table. Each card has one white side and one black side. Kostya flipped 50 cards, then Tanya flipped 60 cards, and after that Olya flipped 70 cards. As a result, all 100 cards ended up lying with the black side up. Ho... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 35515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There were cards with numbers from 1 to 20 in a bag. Vlad pulled out 6 cards and asserted that all these cards can be divided into pairs such that the sums of the numbers in each pair were the same. Lena managed to sneak a peek at 5 of Vlad's cards: they had the numbers $2, 4, ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There were electronic clocks on the International Space Station, displaying time in the format HH:MM. Due to an electromagnetic storm, the device started malfunctioning, and each digit on the display either increased by 1 or decreased by 1. What was the actual time when the sto... | math | {
"style": "rule",
"ground_truth": "11:48"
} | {
"index": 35517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There were five teams entered in a competition. Each team consisted of either only boys or only girls. The number of team members was $9, 15, 17, 19$, and $21$. After one team of girls had been knocked out of the competition, the number of girls still competing was three times ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 35518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Determine the volume, in cubic light years, of the set of all possible locations for a base such that the sum of squares of the distances from the base to each of ... | math | {
"style": "rule",
"ground_truth": "\\frac{27 \\sqrt{6} \\pi}{8}"
} | {
"index": 35519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there wer... | math | {
"style": "rule",
"ground_truth": "28182"
} | {
"index": 35520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There were no more than 70 mushrooms in the basket, among which 52% were white. If you throw out the three smallest mushrooms, the white mushrooms will become half of the total.\n\nHow many mushrooms are in the basket?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 35521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35522,
"solution": "1. **Define the variable**: Let $x$ be the number of hours Theresa needs to work in the sixth week to achieve an average of 10 hours per week over 6 weeks.\n\n2. **Set up the equation for the average**: The average number of hours worked per week is given by the total hours worked div... |
DeepScaleR | [
{
"role": "user",
"content": "These two spinners are divided into thirds and quarters, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? Express your answer as a common fraction.\n\n[asy]\n\nsize(5cm,5cm);\n... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 35523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "They paid 100 rubles for a book and still need to pay as much as they would need to pay if they had paid as much as they still need to pay. How much does the book cost?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 35524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirteen blue and six green hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of green tiles with the same size and shape as the others, what will be the difference between the total number of green tiles and the total number... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 35525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty clever students from 6th, 7th, 8th, 9th, and 10th grades were tasked with creating forty problems for an olympiad. Any two students from the same grade came up with the same number of problems, while any two students from different grades came up with a different number ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 35526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. Ho... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 35527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty identical toothpicks were used to create the figure below. There are over 25 triangles in the figure. What is the fewest number of toothpicks that could be removed so that no triangles remain?\n\n[asy]\ndraw((0,0)--(8,0), linewidth(1));\ndraw(2dir(60)--(2dir(60)+(6,0)), ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 35528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty people are divided into three groups (I, II, and III) with 10 people in each group. How many different compositions of groups can there be?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{30!}{(10!)^3}"
} | {
"index": 35529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\\frac{p}{q},$ where $p$ and $q$ are relat... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"index": 35530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty teams play a tournament where every team plays every other team exactly once. Each game results either in a win or a loss with a $50\\%$ chance for either outcome. Calculate the probability that all teams win a unique number of games. Express your answer as $\\frac{m}{n}... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"index": 35531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty teams play in a league where each team plays every other team exactly once, and every game results in a win or loss with no ties. Each game is independent with a $50\\%$ chance of either team winning. Determine the probability that no two teams end up with the same numbe... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"index": 35532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty tiles are numbered 1 through 30 and are placed into box $C$. Thirty other tiles numbered 21 through 50 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 20 and the tile from box $D$ is eithe... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{45}"
} | {
"index": 35533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty-nine students from seven classes came up with 60 problems, with students of the same class coming up with the same number of problems (not equal to zero), and students from different classes coming up with a different number of problems. How many students came up with on... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 35534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty-nine students from seven classes invented 60 problems, with the students from each class inventing the same number of problems (which is not zero), and the students from different classes inventing different numbers of problems. How many students invented one problem eac... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 35535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence?"
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 35536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "This century will mark the 200th anniversary of the birth of the famous Russian mathematician Pafnuty Lvovich Chebyshev, a native of Kaluga province. The sum of the digits in the hundreds and thousands places of the year he was born is 3 times the sum of the digits in the units... | math | {
"style": "rule",
"ground_truth": "1821"
} | {
"index": 35537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "This month, I spent 26 days exercising for 20 minutes or more, 24 days exercising 40 minutes or more, and 4 days of exercising 2 hours exactly. I never exercise for less than 20 minutes or for more than 2 hours. What is the minimum number of hours I could have exercised this mo... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 35538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "This pattern is made from toothpicks. If the pattern is continued by adding two toothpicks to the previous stage, how many toothpicks are used to create the figure for the $15^{th}$ stage?\n\n[asy]draw((0,0)--(7.5,13)--(-7.5,13)--cycle);\ndraw((0,0)--(-15,0)--(-7.5,13)--cycle);... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 35539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "This puzzle features a unique kind of problem where only one digit is known. It appears to have a single solution and, surprisingly, filling in the missing digits is not very difficult. Given that a divisor multiplied by 7 results in a three-digit number, we conclude that the f... | math | {
"style": "rule",
"ground_truth": "124"
} | {
"index": 35540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "This was a highly dangerous car rally. It began with a small and very narrow bridge, where one out of five cars would fall into the water. Then followed a terrifying sharp turn, where three out of ten cars would go off the road. Next, there was a dark and winding tunnel where o... | math | {
"style": "rule",
"ground_truth": "69.76"
} | {
"index": 35541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "This year is 2017, and the sum of the digits of the year is 10. Find the sum of all the years in this century whose digits sum to 10."
}
] | math | {
"style": "rule",
"ground_truth": "18396"
} | {
"index": 35542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Thomas has constant speeds for both running and walking. When a down-escalator is moving, Thomas can run down it in 15 seconds or walk down it in 30 seconds. One day, when the escalator was broken (and stationary), it took Thomas 20 seconds to run down it. How long, in seconds,... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 35543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three $1 \times 1 \times 1$ cubes are joined side by side. What is the surface area of the resulting prism?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 35544,
"solution": "When three $1 \times 1 \times 1$ cubes are joined together as in the diagram, the resulting prism is $3 \times 1 \times 1$. This prism has four rectangular faces that are $3 \times 1$ and two rectangular faces that are $1 \times 1$. Therefore, the surface area is $4 \times(3 \times 1)... |
DeepScaleR | [
{
"role": "user",
"content": "Three $1 \\times 1 \\times 1$ cubes are joined face to face in a single row and placed on a table, and have a total of 11 exposed $1 \\times 1$ faces. Determine the number of exposed $1 \\times 1$ faces when sixty $1 \\times 1 \\times 1$ cubes are joined face to face in a singl... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 35545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three $1 \\times 1 \\times 1$ cubes are joined face to face in a single row and placed on a table. The cubes have a total of 11 exposed $1 \\times 1$ faces. If sixty $1 \\times 1 \\times 1$ cubes are joined face to face in a single row and placed on a table, how many $1 \\times... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 35546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three $\\Delta$'s and a $\\diamondsuit$ will balance nine $\\bullet$'s. One $\\Delta$ will balance a $\\diamondsuit$ and a $\\bullet$.\n\nHow many $\\bullet$'s will balance the two $\\diamondsuit$'s in this balance?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 35547,
"solution": "Let's denote the symbols with variables for easier manipulation:\n- Let $\\Delta = a$\n- Let $\\diamondsuit = b$\n- Let $\\bullet = c$\n\nWe are given two balance equations:\n1. \\(3a + b = 9c\\)\n2. \\(a = b + c\\)\n\nWe need to find how many $\\bullet$'s (i.e., how many $c$'s) will ... |
DeepScaleR | [
{
"role": "user",
"content": "Three $\\text{A's}$, three $\\text{B's}$, and three $\\text{C's}$ are placed in the nine spaces so that each row and column contains one of each letter. If $\\text{A}$ is placed in the upper left corner, how many arrangements are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 35548,
"solution": "1. **Fixing A in the upper left corner**: We start by placing an A in the upper left corner of the grid. The grid now looks like this:\n \\[\n \\begin{array}{|c|c|c|}\n \\hline\n A & & \\\\\n \\hline\n & & \\\\\n \\hline\n & & \\\\\n \\hline\n \\end... |
DeepScaleR | [
{
"role": "user",
"content": "Three 12 cm $\\times$12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedr... | math | {
"style": "rule",
"ground_truth": "864"
} | {
"index": 35549,
"solution": "Note that gluing two of the given polyhedra together along a hexagonal face (rotated $60^\\circ$ from each other) yields a cube, so the volume is $\\frac12 \\cdot 12^3 = 864$, so our answer is $\\boxed{864}$.\n Image: \n Sidenote: What if this simple and beautiful way was not realized... |
DeepScaleR | [
{
"role": "user",
"content": "Three Graces each had the same number of fruits and met 9 Muses. Each Grace gave an equal number of fruits to each Muse. After that, each Muse and each Grace had the same number of fruits. How many fruits did each Grace have before meeting the Muses?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three Triangles: Within triangle \\(ABC\\), a random point \\(M\\) is chosen. What is the probability that the area of one of the triangles \\(ABM, BCM,\\) and \\(CAM\\) will be greater than the sum of the areas of the other two?"
}
] | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"index": 35551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three aluminum cans can be recycled to make a new can. How many new cans can eventually be made from 243 aluminum cans? (Remember that the first new cans that are made can then be recycled into even newer cans!) Do not include the original 243 cans in your count."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 35552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three ants begin on three different vertices of a tetrahedron. Every second, they choose one of the three edges connecting to the vertex they are on with equal probability and travel to the other vertex on that edge. They all stop when any two ants reach the same vertex at the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 35553,
"solution": "At every second, each ant can travel to any of the three vertices they are not currently on. Given that, at one second, the three ants are on different vertices, the probability of them all going to the same vertex is $\\frac{1}{27}$ and the probability of them all going to different ... |
DeepScaleR | [
{
"role": "user",
"content": "Three balls are drawn simultaneously from the urn (as described in Problem 4). Find the probability that all the drawn balls are blue (event $B$)."
}
] | math | {
"style": "rule",
"ground_truth": "1/12"
} | {
"index": 35554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three balls are lying on a table, each pair touching each other. The radii of the balls form a geometric progression with a common ratio \\( q \\neq 1 \\). The radius of the middle ball is 2012. Find the ratio of the sum of the squares of the sides of the triangle formed by the... | math | {
"style": "rule",
"ground_truth": "4024"
} | {
"index": 35555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up ... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 35556,
"solution": "To solve this problem, we need to calculate the probability that three balls tossed into bins end up evenly spaced in distinct bins. We start by considering the possible configurations for the balls and their corresponding probabilities.\n\n#### Step 1: Define the problem\nWe are give... |
DeepScaleR | [
{
"role": "user",
"content": "Three balls are randomly placed into three boxes. Let the random variable $\\xi$ denote the maximum number of balls in any one box. Determine the mathematical expectation $E(\\xi)$ of $\\xi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{9}"
} | {
"index": 35557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three balls labeled 1, 2, and 3 are placed in a jar. A ball is drawn from the jar, its number is recorded, and it is then returned to the jar. This process is repeated three times, with each ball having an equal chance of being drawn in each trial. If the sum of the recorded nu... | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{7}$"
} | {
"index": 35558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three balls marked $1,2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numb... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"index": 35559,
"solution": "1. **Understanding the Problem:**\n - We have three balls numbered $1, 2, 3$.\n - Each ball is equally likely to be drawn.\n - The drawing process is repeated three times with replacement.\n - We need to find the probability that the ball numbered $2$ was drawn all three times... |
DeepScaleR | [
{
"role": "user",
"content": "Three balls numbered 1, 2, and 3 are placed in a bag. A ball is drawn, the number recorded, and then the ball is returned. This process is repeated three times. Calculate the probability that the sum of the three recorded numbers is less than 8."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{27}"
} | {
"index": 35560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three boxes each contain an equal number of hockey pucks. Each puck is either black or gold. All 40 of the black pucks and exactly $\\frac{1}{7}$ of the gold pucks are contained in one of the three boxes. Determine the total number of gold hockey pucks."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 35561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three boys \\( B_{1}, B_{2}, B_{3} \\) and three girls \\( G_{1}, G_{2}, G_{3} \\) are to be seated in a row according to the following rules:\n1) A boy will not sit next to another boy and a girl will not sit next to another girl,\n2) Boy \\( B_{1} \\) must sit next to girl \\... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 35562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three boys and three girls are lined up for a photo. Boy A is next to boy B, and exactly two girls are next to each other. Calculate the total number of different ways they can be arranged."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 35563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three boys and two girls are to stand in a row according to the following requirements. How many different arrangements are there? (Answer with numbers) \n(Ⅰ) The two girls stand next to each other; \n(Ⅱ) Girls cannot stand at the ends; \n(Ⅲ) Girls are arranged from left to ... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 35564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three boys played a \"Word\" game in which they each wrote down ten words. For each word a boy wrote, he scored three points if neither of the other boys had the same word; he scored one point if only one of the other boys had the same word. No points were awarded for words whi... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 35565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 35566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three candles can burn for 30, 40, and 50 minutes, respectively (but are not ignited simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How long are exactly two candles burning simultaneo... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 35567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade, the second card is a 10, and the third card is a queen?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{11050}"
} | {
"index": 35568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a 4, the second card is a $\\clubsuit$, and the third card is a 2?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{663}"
} | {
"index": 35569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Jack, the second card is a Queen, and the third card is a King? Assuming the dealing is done without replacement."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{16575}"
} | {
"index": 35570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\\clubsuit$, the second card is a $\\heartsuit$, and the third card is a king?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{2550}"
} | {
"index": 35571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a $\\spadesuit$, and the third card is a 3?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{11050}"
} | {
"index": 35572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a diamond, and the third card is a Jack?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{650}"
} | {
"index": 35573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cards are dealt successively without replacement from a standard deck of 52 cards. What is the probability that the first card is a $\\heartsuit$, the second card is a King, and the third card is a $\\spadesuit$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{2550}"
} | {
"index": 35574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that\n\n(a) the numbers are all different,\n(b) they sum to $13$, and\n(c) they are in increasing order, left to right.\nFirst, Casey looks at the number on... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 35575,
"solution": "1. **Initial Possibilities**: Given the conditions (a) all numbers are different, (b) they sum to $13$, and (c) they are in increasing order, we list all possible sets of numbers:\n - $(1,2,10)$\n - $(1,3,9)$\n - $(1,4,8)$\n - $(1,5,7)$\n - $(2,3,8)$\n - $(2,4,7)$\n - $(... |
DeepScaleR | [
{
"role": "user",
"content": "Three cars leave city $A$ at the same time and travel along a closed path composed of three straight segments $AB, BC$, and $CA$. The speeds of the first car on these segments are 12, 10, and 15 kilometers per hour, respectively. The speeds of the second car are 15, 15, and 10 ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 35576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cars start simultaneously from City A, heading towards City B along the same highway. The second car travels 4 kilometers per hour less than the first car and 6 kilometers per hour more than the third car. The second car arrives at City B 3 minutes later than the first ca... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 35577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three chiefs of Indian tribes are sitting by a fire with three identical pipes. They are holding a war council and smoking. The first chief can finish a whole pipe in ten minutes, the second in thirty minutes, and the third in an hour. How should the chiefs exchange the pipes a... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 35578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three children need to cover a distance of 84 kilometers using two bicycles. Walking, they cover 5 kilometers per hour, while bicycling they cover 20 kilometers per hour. How long will it take for all three to reach the destination if only one child can ride a bicycle at a time... | math | {
"style": "rule",
"ground_truth": "8.4"
} | {
"index": 35579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three circles are drawn around vertices \\( A, B, \\) and \\( C \\) of a regular hexagon \\( ABCDEF \\) with side length 2 units, such that the circles touch each other externally. What is the radius of the smallest circle?"
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{3}"
} | {
"index": 35580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three circles have the same center O. Point X divides segment OP in the ratio 1:3, with X being closer to O. Calculate the ratio of the area of the circle with radius OX to the area of the circle with radius OP. Next, find the area ratio of a third circle with radius 2*OX to th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 35581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.\n\n[asy]\ndraw(Circle((0,-0.58),2.15),linewidth(0.7));\ndraw(Circ... | math | {
"style": "rule",
"ground_truth": "\\frac{3+2\\sqrt{3}}{3}"
} | {
"index": 35582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three circles with radii 2, 3, and 10 units are placed inside a larger circle such that all circles are touching one another. Determine the value of the radius of the larger circle."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 35583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three circles with radii 2, 3, and 3 touch each other. What is the area of the triangle formed by joining the centers of these circles?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circl... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 35585,
"solution": "Notice that any line that passes through the bottom circle's center cuts it in half, so all we really care about are the top two circles. Suppose $\\ell$ is the desired line. Draw lines $\\ell_1$ and $\\ell_2$ both parallel to $\\ell$ such that $\\ell_1$ passes through $(14,92)$ and $... |
DeepScaleR | [
{
"role": "user",
"content": "Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest ... | math | {
"style": "rule",
"ground_truth": "408"
} | {
"index": 35586,
"solution": "Let $A,B,C$ be the fraction of bananas taken by the first, second, and third monkeys respectively. Then we have the system of equations \\[\\frac{3}{4}A+\\frac{3}{8}B+\\frac{11}{24}C=\\frac{1}{2}\\] \\[\\frac{1}{8}A+\\frac{1}{4}B+\\frac{11}{24}C=\\frac{1}{3}\\] \\[\\frac{1}{8}A+\\frac... |
DeepScaleR | [
{
"role": "user",
"content": "Three clients are at the hairdresser, each paying their bill at the cash register.\n\n- The first client pays the same amount that is in the register and takes 10 reais as change.\n- The second client performs the same operation as the first.\n- The third client performs the sa... | math | {
"style": "rule",
"ground_truth": "8.75"
} | {
"index": 35587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \\tfrac{b}{c} \\sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ ... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 35588,
"solution": "Reinterpret the problem in the following manner. Equilateral triangle $ABC$ has a point $X$ on the interior such that $AX = 5,$ $BX = 4,$ and $CX = 3.$ A $60^\\circ$ counter-clockwise rotation about vertex $A$ maps $X$ to $X'$ and $B$ to $C.$\n[asy]import cse5; size(200); defaultpen(l... |
DeepScaleR | [
{
"role": "user",
"content": "Three concentric circles have radii $5$ meters, $15$ meters, and $25$ meters respectively. Calculate the total distance a beetle travels, which starts at a point $P$ on the outer circle, moves inward along a radius to the middle circle, traces a one-third arc of the middle circ... | math | {
"style": "rule",
"ground_truth": "15\\pi + 70"
} | {
"index": 35589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three concentric circles have radii of 1, 2, and 3 units, respectively. Points are chosen on each of these circles such that they are the vertices of an equilateral triangle. What can be the side length of this equilateral triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"index": 35590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three concentric circles with radii 5 meters, 10 meters, and 15 meters, form the paths along which an ant travels moving from one point to another symmetrically. The ant starts at a point on the smallest circle, moves radially outward to the third circle, follows a path on each... | math | {
"style": "rule",
"ground_truth": "\\frac{65\\pi}{3} + 20"
} | {
"index": 35591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cones are placed on a table on their bases, touching each other. The radii of their bases are 1, 12, and 12, and the apex angles of the cones are $-4 \\operatorname{arctg} \\frac{1}{3}$, $4 \\operatorname{arctg} \\frac{2}{3}$, and $4 \\operatorname{arctg} \\frac{2}{3}$ re... | math | {
"style": "rule",
"ground_truth": "40/21"
} | {
"index": 35592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 35593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 35594,
"solution": "1. **Identify the diameters of the circles**: Given that the circle centered at $Q$ has a diameter of $4$, and since the circles are congruent, the circles centered at $P$ and $R$ also have diameters of $4$.\n\n2. **Determine the height of the rectangle**: Consider a diameter of circl... |
DeepScaleR | [
{
"role": "user",
"content": "Three congruent cones, each with a radius of 8 cm and a height of 8 cm, are enclosed within a cylinder. The base of each cone is consecutively stacked and forms a part of the cylinder’s interior base, while the height of the cylinder is 24 cm. Calculate the volume of the cylind... | math | {
"style": "rule",
"ground_truth": "1024\\pi"
} | {
"index": 35595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three congruent isosceles triangles $DAO$, $AOB$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12$. These triangles are arranged to form trapezoid $ABCD$, as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$.\n\n[asy]\npair A, B, C, D, O, P;\nA= (6, 8);\n... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=12$ and $AB=DO=OC=16$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$. Points $X$ and $Y$ are the midpoints of $AD$ and $BC$, re... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=13$ and $AB=DO=OC=15$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ such that $OP$ is perpendicular to $AB$.\n\nPoint $X$ is the midpoint of $AD$ and point $Y$ is t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three congruent isosceles triangles are constructed inside an equilateral triangle with a side length of $\\sqrt{2}$. Each base of the isosceles triangle is placed on one side of the equilateral triangle. If the total area of the isosceles triangles equals $\\frac{1}{2}$ the ar... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 35599,
"solution": ""
} |
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