prompt listlengths 1 1 | source stringclasses 1
value | id int64 0 17.4k | data_source stringclasses 1
value | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
[
{
"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 polyhedron. What is the vol... | math_dapo | 100 | math_dapo | math | {
"ground_truth": "864",
"style": "rule"
} | {
"index": 100,
"split": "train"
} |
[
{
"content": "Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:\n\n\\[\n\\begin{align*}\n700 \\cdot 712 \\cdot 718 + 320,\\\\\n701 \... | math_dapo | 101 | math_dapo | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 101,
"split": "train"
} |
[
{
"content": "Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles. Please reason step by step, keep your thinki... | math_dapo | 102 | math_dapo | math | {
"ground_truth": "18",
"style": "rule"
} | {
"index": 102,
"split": "train"
} |
[
{
"content": "The prime numbers $a$, $b$, and $c$ satisfy the equation $a + b^2 = 4c^2$. Determine the sum of all possible values of $a + b + c$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 103 | math_dapo | math | {
"ground_truth": "31",
"style": "rule"
} | {
"index": 103,
"split": "train"
} |
[
{
"content": "In a bag, there are $1007$ black and $1007$ white balls, each randomly numbered from $1$ to $2014$. At each step, one ball is drawn and placed on the table. Additionally, if desired, two balls of different colors can be selected from the table and placed in a separate bag. This action earns points... | math_dapo | 104 | math_dapo | math | {
"ground_truth": "1007",
"style": "rule"
} | {
"index": 104,
"split": "train"
} |
[
{
"content": "Find all prime numbers $p$ such that $(p-3)^p + p^2$ is a perfect square. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 105 | math_dapo | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 105,
"split": "train"
} |
[
{
"content": "In the country Máxico, there are two islands: the island \"Mayor\" and the island \"Menor\". The island \"Mayor\" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island \"Menor\" has only one state with $31$ cities. \"Aeropapantla\" and \"Aerocenzontle\" are the airlines that of... | math_dapo | 106 | math_dapo | math | {
"ground_truth": "65",
"style": "rule"
} | {
"index": 106,
"split": "train"
} |
[
{
"content": "Several cells are marked in a $100 \\times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells, and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will ce... | math_dapo | 107 | math_dapo | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 107,
"split": "train"
} |
[
{
"content": "Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of energy to hop one step north or one step south, and $1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $100$ Joules of energy, and hops till he... | math_dapo | 108 | math_dapo | math | {
"ground_truth": "10201",
"style": "rule"
} | {
"index": 108,
"split": "train"
} |
[
{
"content": "In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \\in \\{0,1,\\dots,a\\}$ is chosen uniformly at random. Similarly,... | math_dapo | 109 | math_dapo | math | {
"ground_truth": "102",
"style": "rule"
} | {
"index": 109,
"split": "train"
} |
[
{
"content": "30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matches are **unusual** m... | math_dapo | 110 | math_dapo | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 110,
"split": "train"
} |
[
{
"content": "In the arrow-shaped polygon (refer to the provided figure), the angles at vertices $A$, $C$, $D$, $E$, and $F$ are right angles. The side lengths are given as follows: $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. Determine the area of the polygon that is closest to one of the following... | math_dapo | 111 | math_dapo | math | {
"ground_truth": "300",
"style": "rule"
} | {
"index": 111,
"split": "train"
} |
[
{
"content": "Let $u$, $v$, and $w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 112 | math_dapo | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 112,
"split": "train"
} |
[
{
"content": "For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 113 | math_dapo | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 113,
"split": "train"
} |
[
{
"content": "Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:\n\\[\nx_1(x_1+1) = A \\\\\nx_2(x_2+1) = A \\\\\nx_1^4 + 3x_1^3 + 5x_1 = x_2^4 + 3x_2^3 + 5x_2.\n\\] Please reason step by step, keep your thinking concise, and... | math_dapo | 114 | math_dapo | math | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": 114,
"split": "train"
} |
[
{
"content": "Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers satisfying the equation $x^2 - y^2 = n$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 115 | math_dapo | math | {
"ground_truth": "45",
"style": "rule"
} | {
"index": 115,
"split": "train"
} |
[
{
"content": "A function $f$ is defined on integers such that:\n\n- $f(n) = n + 3$ if $n$ is odd.\n- $f(n) = \\frac{n}{2}$ if $n$ is even.\n\nIf $k$ is an odd integer, determine the values for which $f(f(f(k))) = k$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxe... | math_dapo | 116 | math_dapo | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 116,
"split": "train"
} |
[
{
"content": "Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$. Please reason step by step, k... | math_dapo | 117 | math_dapo | math | {
"ground_truth": "1594",
"style": "rule"
} | {
"index": 117,
"split": "train"
} |
[
{
"content": "10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\\frac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from t... | math_dapo | 118 | math_dapo | math | {
"ground_truth": "828",
"style": "rule"
} | {
"index": 118,
"split": "train"
} |
[
{
"content": "On a semicircle with diameter $|AB|=d$, there are points $C$ and $D$ such that $|BC|=|CD|=a$ and $|DA|=b$, where $a$, $b$, and $d$ are different positive integers. Find the minimum possible value of $d$. Please reason step by step, keep your thinking concise, and put your final answer within \\box... | math_dapo | 119 | math_dapo | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 119,
"split": "train"
} |
[
{
"content": "Suppose in the plane, there are 10 pairwise nonparallel lines intersecting one another. What is the maximum possible number of polygons (with finite areas) that can be formed? Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
... | math_dapo | 120 | math_dapo | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 120,
"split": "train"
} |
[
{
"content": "The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How man... | math_dapo | 121 | math_dapo | math | {
"ground_truth": "108",
"style": "rule"
} | {
"index": 121,
"split": "train"
} |
[
{
"content": "Let $\\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\\Gamma$ with diameter $\\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\\Gamma$ at a point $X$. The len... | math_dapo | 122 | math_dapo | math | {
"ground_truth": "76",
"style": "rule"
} | {
"index": 122,
"split": "train"
} |
[
{
"content": "A herder has forgotten the number of cows she has and does not want to count all of them. She remembers these four facts about the number of cows:\n\n- It has $3$ digits.\n- It is a palindrome.\n- The middle digit is a multiple of $4$.\n- It is divisible by $11$.\n\nWhat is the sum of all possible... | math_dapo | 123 | math_dapo | math | {
"ground_truth": "726",
"style": "rule"
} | {
"index": 123,
"split": "train"
} |
[
{
"content": "All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string? Please reason step by step, keep your thinking conci... | math_dapo | 124 | math_dapo | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 124,
"split": "train"
} |
[
{
"content": "Let $f(n)$ be a function that fulfills the following properties:\n\n- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.\n- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.\n- If $a$ is a divisor of $b$, then: $f\\left(\\frac{b}{a}\\right) = |f(b) - f(a)|$.\n\... | math_dapo | 125 | math_dapo | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 125,
"split": "train"
} |
[
{
"content": "Find all integers $n > 1$ such that every prime that divides $n^6 - 1$ also divides $n^5 - n^3 - n^2 + 1$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 126 | math_dapo | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 126,
"split": "train"
} |
[
{
"content": "Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:\n- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers o... | math_dapo | 127 | math_dapo | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 127,
"split": "train"
} |
[
{
"content": "Let $n$ be a positive integer. Find as many zeros as possible for the last digits of the following expression: $1^n + 2^n + 3^n + 4^n$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 128 | math_dapo | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 128,
"split": "train"
} |
[
{
"content": "Find all positive integers $n$ such that $n = 100 \\times d(n)$, where $d(n)$ represents the number of positive divisors of $n$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 129 | math_dapo | math | {
"ground_truth": "2000",
"style": "rule"
} | {
"index": 129,
"split": "train"
} |
[
{
"content": "In $\\triangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of the segment $BD$? Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"... | math_dapo | 130 | math_dapo | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 130,
"split": "train"
} |
[
{
"content": "Find the number of ordered triplets $(a, b, c)$ of positive integers such that $a < b < c$ and $abc = 2008$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 131 | math_dapo | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 131,
"split": "train"
} |
[
{
"content": "Solve in the real numbers the equation $3^{x+1} = (x-1)(x-3)$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 132 | math_dapo | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 132,
"split": "train"
} |
[
{
"content": "Let $f(x) = x^3 + x + 1$. Suppose $g$ is a cubic polynomial such that $g(0) = -1$, and the roots of $g$ are the squares of the roots of $f$. Find $g(9)$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 133 | math_dapo | math | {
"ground_truth": "899",
"style": "rule"
} | {
"index": 133,
"split": "train"
} |
[
{
"content": "A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row... | math_dapo | 134 | math_dapo | math | {
"ground_truth": "2019",
"style": "rule"
} | {
"index": 134,
"split": "train"
} |
[
{
"content": "Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get? Please reason step by step, keep your thinking concise, and put your final answe... | math_dapo | 135 | math_dapo | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 135,
"split": "train"
} |
[
{
"content": "Willy Wonka has $n$ distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is $p(n)$, then we have the following table:\n\n\\[\n\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nn & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline\np(n) & 1... | math_dapo | 136 | math_dapo | math | {
"ground_truth": "17007",
"style": "rule"
} | {
"index": 136,
"split": "train"
} |
[
{
"content": "In rectangle $ABCD$, diagonal $AC$ is intersected by the angle bisector from $B$ at point $B'$ and by the angle bisector from $D$ at point $D'$. Similarly, diagonal $BD$ is intersected by the angle bisector from $A$ at point $A'$ and by the angle bisector from $C$ at point $C'$. The area of quadri... | math_dapo | 137 | math_dapo | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 137,
"split": "train"
} |
[
{
"content": "Find the smallest positive $m$ such that there exist at least 11 even and 11 odd positive integers $n$ for which $\\frac{n^3 + m}{n + 2}$ is an integer. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 138 | math_dapo | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 138,
"split": "train"
} |
[
{
"content": "What is the smallest positive integer $n$ such that $2013^n$ ends in $001$? In other words, find the smallest $n$ for which the rightmost three digits of $2013^n$ are $001$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}... | math_dapo | 139 | math_dapo | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 139,
"split": "train"
} |
[
{
"content": "A $8 \\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maxim... | math_dapo | 140 | math_dapo | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 140,
"split": "train"
} |
[
{
"content": "Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k \\geq 2$ such that $3^n = x^k + y^k$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "us... | math_dapo | 141 | math_dapo | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 141,
"split": "train"
} |
[
{
"content": "Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For ... | math_dapo | 142 | math_dapo | math | {
"ground_truth": "45",
"style": "rule"
} | {
"index": 142,
"split": "train"
} |
[
{
"content": "There are $25$ towns in a country. Find the smallest $k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:\n1) From each town, there are exactly $k$ direct routes to $k$ other towns.\n2) If two towns are not connected by a d... | math_dapo | 143 | math_dapo | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 143,
"split": "train"
} |
[
{
"content": "Let $x, y$ be positive integers such that:\n\\[\nx^4 = (x-1)(y^3 - 23) - 1\n\\]\nFind the maximum possible value of $x + y$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 144 | math_dapo | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 144,
"split": "train"
} |
[
{
"content": "If the four sides of a quadrilateral are $2$, $3$, $6$, and $x$, find the sum of all possible integral values for $x$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 145 | math_dapo | math | {
"ground_truth": "54",
"style": "rule"
} | {
"index": 145,
"split": "train"
} |
[
{
"content": "Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\\frac{n}{100}$ probability of $1$ meter forward, to meter $n+1... | math_dapo | 146 | math_dapo | math | {
"ground_truth": "2500",
"style": "rule"
} | {
"index": 146,
"split": "train"
} |
[
{
"content": "Blue rolls a fair $n$-sided die numbered with integers from $1$ to $n$, and then flips a coin. The coin is weighted to land heads either $\\frac{1}{3}$ or $\\frac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\\frac{1}{15}$, find $n$. Please reason ste... | math_dapo | 147 | math_dapo | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 147,
"split": "train"
} |
[
{
"content": "A positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$? Please reason step by step, kee... | math_dapo | 148 | math_dapo | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 148,
"split": "train"
} |
[
{
"content": "Let $m,n$ be natural numbers such that $m + 3n - 5 = 2\\text{LCM}(m,n) - 11\\text{GCD}(m,n)$. Find the maximum possible value of $m+n$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 149 | math_dapo | math | {
"ground_truth": "70",
"style": "rule"
} | {
"index": 149,
"split": "train"
} |
[
{
"content": "25 checkers are placed on the 25 leftmost squares of a $1 \\times N$ board. A checker can either move to the empty adjacent square to its right or jump over an adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find the minimal $N$ such that all the checke... | math_dapo | 150 | math_dapo | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": 150,
"split": "train"
} |
[
{
"content": "Suppose $1$, $2$, and $3$ are the roots of the equation $x^4 + ax^2 + bx = c$. Find the value of $c$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 151 | math_dapo | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 151,
"split": "train"
} |
[
{
"content": "Find all positive integers $n$ for which $2^n + 2021n$ is a perfect square. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 152 | math_dapo | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 152,
"split": "train"
} |
[
{
"content": "How many different ways are there to write 2004 as a sum of one or more positive integers which are all \"approximately equal\" to each other? Two numbers are called approximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of... | math_dapo | 153 | math_dapo | math | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": 153,
"split": "train"
} |
[
{
"content": "Find the smallest positive integer $n$ such that $n^4 + (n+1)^4$ is composite. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 154 | math_dapo | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 154,
"split": "train"
} |
[
{
"content": "A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. Determine, with proof, the maximum number of distinct license plates that the state can use. Please reason step by step, keep your thinking co... | math_dapo | 155 | math_dapo | math | {
"ground_truth": "100000",
"style": "rule"
} | {
"index": 155,
"split": "train"
} |
[
{
"content": "In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam. Please reason step by step, keep you... | math_dapo | 156 | math_dapo | math | {
"ground_truth": "2000",
"style": "rule"
} | {
"index": 156,
"split": "train"
} |
[
{
"content": "Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a perfect square. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 157 | math_dapo | math | {
"ground_truth": "179",
"style": "rule"
} | {
"index": 157,
"split": "train"
} |
[
{
"content": "It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 158 | math_dapo | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 158,
"split": "train"
} |
[
{
"content": "Each pair of vertices of a regular $67$-gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected and how they were painted, ... | math_dapo | 159 | math_dapo | math | {
"ground_truth": "2011",
"style": "rule"
} | {
"index": 159,
"split": "train"
} |
[
{
"content": "Determine the value of $k$ for the quartic equation:\n\n$$x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$$\n\nGiven that the product of two of its four roots is $-32$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 160 | math_dapo | math | {
"ground_truth": "86",
"style": "rule"
} | {
"index": 160,
"split": "train"
} |
[
{
"content": "There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines i... | math_dapo | 161 | math_dapo | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": 161,
"split": "train"
} |
[
{
"content": "There is a pile of eggs. Joan counted the eggs, but her count was off by $1$ in the $1$'s place. Tom counted the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs an... | math_dapo | 162 | math_dapo | math | {
"ground_truth": "439",
"style": "rule"
} | {
"index": 162,
"split": "train"
} |
[
{
"content": "Determine the least integer $k$ for which the following scenario could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number o... | math_dapo | 163 | math_dapo | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 163,
"split": "train"
} |
[
{
"content": "Find the smallest positive integer $n$, such that $3^k + n^k + (3n)^k + 2014^k$ is a perfect square for all natural numbers $k$, but not a perfect cube for all natural numbers $k$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "us... | math_dapo | 164 | math_dapo | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 164,
"split": "train"
} |
[
{
"content": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n + 76$ equal squares. Find the value of $n$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 165 | math_dapo | math | {
"ground_truth": "324",
"style": "rule"
} | {
"index": 165,
"split": "train"
} |
[
{
"content": "In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$. Please reason step by step, keep your thinkin... | math_dapo | 166 | math_dapo | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 166,
"split": "train"
} |
[
{
"content": "Let $ABCDEFGHIJ$ be a regular 10-sided polygon with all its vertices on a circle centered at $O$ with a radius of 5. The diagonals $AD$ and $BE$ intersect at $P$, and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the length of the segment $PQ$. Please reason step by step, keep your think... | math_dapo | 167 | math_dapo | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 167,
"split": "train"
} |
[
{
"content": "The difference between the maximal and the minimal diagonals of the regular $n$-gon equals its side ($n > 5$). Find $n$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 168 | math_dapo | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 168,
"split": "train"
} |
[
{
"content": "A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the... | math_dapo | 169 | math_dapo | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 169,
"split": "train"
} |
[
{
"content": "Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$, and $a+b$ are all distinct squares. What is the smallest possible value of $b$? Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 170 | math_dapo | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": 170,
"split": "train"
} |
[
{
"content": "Determine the value of the parameter $m$ such that the equation $(m-2)x^2 + (m^2-4m+3)x - (6m^2-2) = 0$ has real solutions, and the sum of the cubes of these solutions is equal to zero. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role"... | math_dapo | 171 | math_dapo | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 171,
"split": "train"
} |
[
{
"content": "Find all positive integers $l$ for which the equation $$a^3+b^3+ab=(lab+1)(a+b)$$ has a solution over positive integers $a,b$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 172 | math_dapo | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 172,
"split": "train"
} |
[
{
"content": "Given a regular polygon with $n$ sides, it is known that there are 1200 ways to choose three of the vertices of the polygon such that they form the vertices of a right triangle. What is the value of $n$? Please reason step by step, keep your thinking concise, and put your final answer within \\box... | math_dapo | 173 | math_dapo | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": 173,
"split": "train"
} |
[
{
"content": "Kobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board as follows:\n\n1. From $P$ and $Q$ to $2P-Q$ and $2Q-P$, or \n2. ... | math_dapo | 174 | math_dapo | math | {
"ground_truth": "2179",
"style": "rule"
} | {
"index": 174,
"split": "train"
} |
[
{
"content": "In quadrilateral $ABCD$, it is given that $AB = DB$ and $AD = BC$. If the angles are $\nabla ABD = 36^{\\circ}$ and $\nabla BCD = 54^{\\circ}$, find the measure of angle $\nabla ADC$ in degrees. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
... | math_dapo | 175 | math_dapo | math | {
"ground_truth": "102",
"style": "rule"
} | {
"index": 175,
"split": "train"
} |
[
{
"content": "Find all positive integers $n$ such that $n$ has exactly 6 positive divisors $1 < d_{1} < d_{2} < d_{3} < d_{4} < n$ and $1 + n = 5(d_{1} + d_{2} + d_{3} + d_{4})$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 176 | math_dapo | math | {
"ground_truth": "1519",
"style": "rule"
} | {
"index": 176,
"split": "train"
} |
[
{
"content": "Find the smallest natural number $n$ such that for any coloration of the numbers $1, 2, \\dots, n$ with three different colors, there exist two numbers of the same color, whose difference is a perfect square. Please reason step by step, keep your thinking concise, and put your final answer within ... | math_dapo | 177 | math_dapo | math | {
"ground_truth": "25",
"style": "rule"
} | {
"index": 177,
"split": "train"
} |
[
{
"content": "For some real number $c$, the graphs of the equation $y = |x - 20| + |x + 18|$ and the line $y = x + c$ intersect at exactly one point. What is $c$? Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 178 | math_dapo | math | {
"ground_truth": "18",
"style": "rule"
} | {
"index": 178,
"split": "train"
} |
[
{
"content": "Find all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square number. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 179 | math_dapo | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 179,
"split": "train"
} |
[
{
"content": "Find all values of the parameter $a$ for which the sum of all real solutions of the equation $x^4 - 5x + a = 0$ is equal to $a$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 180 | math_dapo | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 180,
"split": "train"
} |
[
{
"content": "Draw a regular hexagon. Then, construct a square on each edge of the hexagon. Form equilateral triangles by connecting the vertices of neighboring squares. If this pattern is extended symmetrically to infinity, what is the ratio of the number of triangles to the number of squares? Please reason st... | math_dapo | 181 | math_dapo | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 181,
"split": "train"
} |
[
{
"content": "Let $n$ be a positive integer. If $S$ is a nonempty set of positive integers, then we say $S$ is $n$-complete if all elements of $S$ are divisors of $n$, and if $d_1$ and $d_2$ are any elements of $S$, then $n$ divides $d_1$ and gcd$(d_1, d_2)$ are in $S$. How many $2310$-complete sets are there? ... | math_dapo | 182 | math_dapo | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 182,
"split": "train"
} |
[
{
"content": "Let $p$ be a polynomial with integer coefficients such that $p(15) = 6$, $p(22) = 1196$, and $p(35) = 26$. Find an integer $n$ such that $p(n) = n + 82$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 183 | math_dapo | math | {
"ground_truth": "28",
"style": "rule"
} | {
"index": 183,
"split": "train"
} |
[
{
"content": "Find the smallest positive integer $N$ satisfying the following three properties:\n- $N$ leaves a remainder of $5$ when divided by $7$.\n- $N$ leaves a remainder of $6$ when divided by $8$.\n- $N$ leaves a remainder of $7$ when divided by $9$. Please reason step by step, keep your thinking concise... | math_dapo | 184 | math_dapo | math | {
"ground_truth": "502",
"style": "rule"
} | {
"index": 184,
"split": "train"
} |
[
{
"content": "A time is chosen randomly and uniformly in a 24-hour day. The probability that at that time, the non-reflex angle between the hour hand and minute hand on a clock is less than $\\frac{360}{11}$ degrees is $\\frac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m + n$. Please reason step... | math_dapo | 185 | math_dapo | math | {
"ground_truth": "211",
"style": "rule"
} | {
"index": 185,
"split": "train"
} |
[
{
"content": "Find all integers $n$ such that $1 < n < 10^6$ and $n^3 - 1$ is divisible by $10^6 n - 1$. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 186 | math_dapo | math | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": 186,
"split": "train"
} |
[
{
"content": "Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every subsequent term is obtained by multiplying its preceding term by $a$ and adding $b$. \n\nFor example, if $a = 2$ and $b = 4$, the first three terms are $5, ... | math_dapo | 187 | math_dapo | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 187,
"split": "train"
} |
[
{
"content": "Find the sum of all integers $n$ such that $n^4 + n^3 + n^2 + n + 1$ is a perfect square. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 188 | math_dapo | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 188,
"split": "train"
} |
[
{
"content": "Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins al... | math_dapo | 189 | math_dapo | math | {
"ground_truth": "1116",
"style": "rule"
} | {
"index": 189,
"split": "train"
} |
[
{
"content": "Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression:\n\n$$ \\frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2} $$\n\nIf $T+t$ can be expressed in the form $\\frac{m}{n}$ where $m,n$ are nonzero integers with $\\text{GCD}(m,n)=1$, find the value of ... | math_dapo | 190 | math_dapo | math | {
"ground_truth": "25",
"style": "rule"
} | {
"index": 190,
"split": "train"
} |
[
{
"content": "Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of this row. Assume that each box must be placed so that it straddles two lower boxes. Including the arrangement with no boxes, ... | math_dapo | 191 | math_dapo | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": 191,
"split": "train"
} |
[
{
"content": "We are given weights ranging from 1 to 5771, i.e., 1, 2, 3, ..., 5770, 5771. These weights are to be partitioned into $n$ sets such that each set has an equal total weight. Determine the maximal value of $n$ for which this partitioning is possible. Please reason step by step, keep your thinking co... | math_dapo | 192 | math_dapo | math | {
"ground_truth": "2886",
"style": "rule"
} | {
"index": 192,
"split": "train"
} |
[
{
"content": "Determine all prime numbers $p$ such that $5^p + 4p^4$ is a perfect square, i.e., the square of an integer. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 193 | math_dapo | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 193,
"split": "train"
} |
[
{
"content": "Find the smallest positive integer $n$ such that $n$ is divisible by $100$ and has exactly $100$ divisors. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math_dapo | 194 | math_dapo | math | {
"ground_truth": "162000",
"style": "rule"
} | {
"index": 194,
"split": "train"
} |
[
{
"content": "A frog is located at the origin of the Cartesian coordinate system. Every second, the frog jumps either horizontally or vertically to one of the four adjacent points with integer coordinates. Determine the number of different points where the frog can be located after 2015 seconds. Please reason s... | math_dapo | 195 | math_dapo | math | {
"ground_truth": "4064256",
"style": "rule"
} | {
"index": 195,
"split": "train"
} |
[
{
"content": "Numbers $a$, $b$, and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, and $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax^2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the s... | math_dapo | 196 | math_dapo | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 196,
"split": "train"
} |
[
{
"content": "There are $64$ booths arranged around a circular table, each containing a chip. The chips and booths are numbered from $1$ to $64$. At the center of the table, there are $1996$ light bulbs, all initially turned off. \n\nEvery minute, the chips move simultaneously in a circular pattern as follows: ... | math_dapo | 197 | math_dapo | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 197,
"split": "train"
} |
[
{
"content": "Find the number of positive integer solutions $(a,b,c,d)$ to the equation $(a^2+b^2)(c^2-d^2)=2020$. Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different. Please reason step by step, keep your thinking concise, and put your final answer within \\boxed{}.",
"role": "user"
... | math_dapo | 198 | math_dapo | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 198,
"split": "train"
} |
[
{
"content": "Consider an isosceles triangle $ABC$ with sides $BC = 30$, $CA = AB = 20$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $M$ be the midpoint of $AD$. Let $PQ$ be a chord of the circumcircle of triangle $ABC$, such that $M$ lies on $PQ$ and $PQ$ is parallel to $BC$. Determine t... | math_dapo | 199 | math_dapo | math | {
"ground_truth": "25",
"style": "rule"
} | {
"index": 199,
"split": "train"
} |
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