data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The number of students in James' graduating class is greater than 100 but fewer than 200 and is 1 less than a multiple of 4, 2 less than a multiple of 5, and 3 less than a multiple of 6. How many students are in James' graduating class?"
}
] | math | {
"style": "rule",
"ground_truth": "183"
} | {
"index": 33400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of students in Teresa's graduating class is more than 50 and fewer than 100 and is 1 less than a multiple of 3, 2 less than a multiple of 4, and 3 less than a multiple of 5. How many students are in Teresa's graduating class?"
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 33401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of students studying in the 5th-6th grades of the school is expressed as a three-digit number. From the digits of this number (without repetitions), 6 different two-digit numbers can be formed, the sum of which is twice the number of students in the 5th-6th grades. H... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"index": 33402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 33403,
"solution": "1. **Define the variables:**\n Let $a$ be the first term, $n$ be the number of terms, and $d$ be the common difference in the arithmetic progression (A.P.). The last term of the A.P. can be expressed as $a + (n-1)d$.\n\n2. **Set up the equation for the difference between the last an... |
DeepScaleR | [
{
"role": "user",
"content": "The number of terms in the expansion of $(x+y+z)^4$ is 15."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 33404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of terms in the expansion of $[(a+3b)^{2}(a-3b)^{2}]^{2}$ when simplified is:"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33405,
"solution": "1. **Simplify the given expression**: \n Start by applying the property of exponents to the expression $[(a+3b)^2(a-3b)^2]^2$:\n \\[\n [(a+3b)^2(a-3b)^2]^2 = [(a+3b)(a-3b)]^4\n \\]\n This simplification uses the property that $(x^m)^n = x^{mn}$ and the fact that $(a+3b)(a-3b... |
DeepScaleR | [
{
"role": "user",
"content": "The number of the distinct solutions to the equation $|x-|2x+1||=3$ is"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33406,
"solution": "To solve the equation $|x - |2x + 1|| = 3$, we need to consider the different cases that arise from the absolute values.\n\n#### Step 1: Analyze the inner absolute value\nThe expression $|2x + 1|$ can be simplified by considering two cases:\n- **Case 1:** $2x + 1 \\geq 0 \\Rightarrow ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of trailing zeros in 2006! is to be calculated."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 33407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of trees in a park must be more than 80 and fewer than 150. The number of trees is 2 more than a multiple of 4, 3 more than a multiple of 5, and 4 more than a multiple of 6. How many trees are in the park?"
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 33408,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of triangles with all sides being positive integers and the longest side being 11 can be expressed as a combination problem, calculate the number of such triangles."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 33409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of triangles with vertices' coordinates $(x, y)$ that satisfy $1 \\leqslant x \\leqslant 4, 1 \\leqslant y \\leqslant 4$, and where $x$ and $y$ are integers is $\\qquad$ ."
}
] | math | {
"style": "rule",
"ground_truth": "516"
} | {
"index": 33410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations\n$ab+bc=44$\n$ac+bc=23$\nis"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33411,
"solution": "1. **Factor the second equation**: Given $ac + bc = 23$, we can factor out $c$:\n \\[\n c(a + b) = 23\n \\]\n Since $23$ is a prime number, the possible values for $c$ are $1$ and $23$.\n\n2. **Analyze the possible values of $c$**:\n - If $c = 23$, then $a + b = \\frac{23}{2... |
DeepScaleR | [
{
"role": "user",
"content": "The number of two-digit numbers that can be formed using the digits 0, 1, 2, 3, 4 without repeating any digit must be calculated."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of unordered pairs of edges of a given rectangular cuboid that determine a plane."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 33413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of values of $x$ satisfying the equation\n\\[\\frac {2x^2 - 10x}{x^2 - 5x} = x - 3\\]is:"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 33414,
"solution": "1. **Identify the domain of the equation**: The given equation is\n \\[\n \\frac{2x^2 - 10x}{x^2 - 5x} = x - 3.\n \\]\n We first note that the denominator $x^2 - 5x$ must not be zero to avoid division by zero. Factoring out $x$, we get:\n \\[\n x(x - 5) \\neq 0.\n \\]\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of white balls and red balls in a jar is in the ratio of $3:2$. If there are 9 white balls, how many red balls are there?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 33415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 33416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,2,\\ldots,9$ are arranged so that the $1$ st term is not $1$ and the $9$ th term is not $9$ . Calculate the probability that the third term is $3$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{43}{399}"
} | {
"index": 33417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?\n\n(G. Galperin)"
}
] | math | {
"style": "rule",
"ground_truth": "1990"
} | {
"index": 33418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:\n\n 1. The largest isn't first, but it is in one of the first three places. \n 2. The smallest isn't last, but it is in one of the last three places. \n 3. The median isn't first or... | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"index": 33419,
"solution": "1. **Identify the largest and smallest numbers**: The largest number in the set $\\{-2, 4, 6, 9, 12\\}$ is $12$, and the smallest is $-2$.\n\n2. **Apply rule 1**: The largest number, $12$, cannot be first but must be in one of the first three places. Thus, $12$ can be in the second or... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a r... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33420,
"solution": "To solve this problem, we need to determine how many arrangements of the numbers $1, 2, 3, 4, 5$ in a circle are \"bad,\" meaning that it is not possible to find a subset of numbers that appear consecutively on the circle that sum to every integer from $1$ to $15$.\n\n#### Step 1: Cou... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1, 2, 3, 4, 5, 6, 7,$ and $8$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where $8$ and $1$ are considered to be consecutive, are written on faces that sha... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 33421,
"solution": "[asy] import three; draw((0,0,0)--(0,0,1)--(0,1,1)--(1,1,1)--(1,0,1)--(1,0,0)--(1,1,0)--(0,1,0)--(0,0,0)); draw((1,1,0)--(1,1,1)); draw((0,1,0)--(0,1,1)); draw((0,0,0)--(1,0,0)); draw((0,0,1)--(1,0,1)); for(int i = 0; i < 2; ++i) { for(int j = 0; j < 2; ++j) { for(int k = 0; k < 2; ++... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1, 2, 3, \\dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. O... | math | {
"style": "rule",
"ground_truth": "0, 2, 4, 6, \\ldots, 1022"
} | {
"index": 33422,
"solution": "\nWe begin with the numbers \\(1, 2, 3, \\ldots, 1024\\) written on a blackboard. The task involves repeatedly dividing these numbers into pairs, taking the non-negative difference of each pair, and then continuing this process until only one number remains after ten such operations. ... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1, 2, \\dots, 16$ are randomly placed into the squares of a $4 \\times 4$ grid. Each square gets one number, and each of the numbers is used once. Find the probability that the sum of the numbers in each row and each column is even."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{36}{20922789888000}"
} | {
"index": 33423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,2 \\cdots 11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{33}"
} | {
"index": 33424,
"solution": "Suppose the middle number is $k$. Then there are $k-1$ ways to pick the number smaller than $k$ to its left and $\\binom{11-k}{4}$ ways to pick the 4 numbers larger than $k$ to its right. Hence there is a total of $\\sum_{k=2}^{7}(k-1) \\cdot\\binom{11-k}{4}$ ways for there to be exac... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,2, \\ldots, 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k<10$, there exists an integer $k^{\\prime}>k$ such that there is at most one number between $k$ and $k^{\\prime}$ in the circle. If $p$ can be expresse... | math | {
"style": "rule",
"ground_truth": "1390"
} | {
"index": 33425,
"solution": "Let $n=10$ and call two numbers close if there is at most one number between them and an circular permutation focused if only $n$ is greater than all numbers close to it. Let $A_{n}$ be the number of focused circular permutations of $\\{1,2, \\ldots, n\\}$. If $n \\geq 5$, then there ... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,2, \\ldots, 10$ are written in a circle. There are four people, and each person randomly selects five consecutive integers (e.g. $1,2,3,4,5$, or $8,9,10,1,2$). If the probability that there exists some number that was not selected by any of the four people is $p$... | math | {
"style": "rule",
"ground_truth": "3690"
} | {
"index": 33426,
"solution": "The unselected numbers must be consecutive. Suppose that $\\{1,2, \\ldots, k\\}$ are the unselected numbers for some $k$. In this case, 1 cannot be selected, so there are 5 possible sets of consecutive numbers the people could have chosen. This leads to $5^{4}$ possibilities. Moreover... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,2, \\ldots, 20$ are put into a hat. Claire draws two numbers from the hat uniformly at random, $a<b$, and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, $c<d$. Let $N$ denote the number of integers $n$ that sat... | math | {
"style": "rule",
"ground_truth": "\\frac{181}{361}"
} | {
"index": 33427,
"solution": "The number of integers that satisfy exactly one of the two inequalities is equal to the number of integers that satisfy the first one, plus the number of integers that satisfy the second one, minus twice the number of integers that satisfy both. Parity-wise, this is just the number of... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,2, \\ldots, 2016$ are grouped into pairs in such a way that the product of the numbers in each pair does not exceed a certain natural number $N$. What is the smallest possible value of $N$ for which this is possible?"
}
] | math | {
"style": "rule",
"ground_truth": "1017072"
} | {
"index": 33428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"index": 33429,
"solution": "\nTo solve this problem, we need to ensure that the sum of the numbers in each row and each column is odd. We can achieve this by having either three odd numbers or one odd and two even numbers in each row and column. Let's analyze the possible configurations and calculate the probabi... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1,2,\\ldots,64$ are written in the squares of an $8\\times 8$ chessboard, one number to each square. Then $2\\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find,... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33430,
"solution": "\nTo solve this problem, we need to maximize the number of \\(2 \\times 2\\) tiles that can be placed on a \\(8 \\times 8\\) chessboard, such that the sum of the numbers in each tile is less than 100. The numbers \\(1, 2, \\ldots, 64\\) are written on the chessboard, with each square ... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1-10$ are written in a circle randomly. Find the expected number of numbers which are at least 2 larger than an adjacent number."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{3}"
} | {
"index": 33431,
"solution": "For $1 \\leq i \\leq 10$, let $X_{i}$ be the random variable that is 1 if the $i$ in the circle is at least 2 larger than one of its neighbors, and 0 otherwise. The random variable representing number of numbers that are at least 2 larger than one of their neighbors is then just $X_{1... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1000^{2}, 1001^{2}, 1002^{2}, \\ldots$ have their last two digits discarded. How many of the first terms of the resulting sequence form an arithmetic progression?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 33432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 33433,
"solution": "Suppose that the two identical digits are both $1$. Since the thousands digit must be $1$, only one of the other three digits can be $1$. This means the possible forms for the number are\n$11xy,\\qquad 1x1y,\\qquad1xy1$\n Because the number must have exactly two identical digits, $x\\... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $1978^{n}$ and $1978^{m}$ have the same last three digits. Find the positive integers $n$ and $m$ such that $m+n$ is minimized, given that $n > m \\geq 1$."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 33434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $2^{0}, 2^{1}, \\cdots, 2^{15}, 2^{16}=65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one from the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number... | math | {
"style": "rule",
"ground_truth": "131069"
} | {
"index": 33435,
"solution": "If we reverse the order of the numbers in the final subtraction we perform, then the final number will be negated. Thus, the possible final numbers come in pairs with opposite signs. Therefore, the largest possible number is the negative of the smallest possible number. To get the sma... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 33436,
"solution": "1. **Calculate the total sum of the numbers given their average**: \n The average (arithmetic mean) of the numbers $3, 5, 7, a,$ and $b$ is given as $15$. The formula for the average of a set of numbers is:\n \\[\n \\text{Average} = \\frac{\\text{Sum of all numbers}}{\\text{Numb... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $4x, 2x-3, 4x-3$ are three consecutive terms in an arithmetic sequence. What is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 33437,
"solution": "Since $4x, 2x-3, 4x-3$ form an arithmetic sequence, then the differences between consecutive terms are equal, or $(2x-3)-4x=(4x-3)-(2x-3)$. Thus, $-2x-3=2x$ or $4x=-3$ and so $x=-\\frac{3}{4}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $5,6,10,17$, and 21 are rearranged so that the sum of the first three numbers is equal to the sum of the last three numbers. Which number is in the middle of this rearrangement?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33438,
"solution": "When a list of 5 numbers $a, b, c, d, e$ has the property that $a+b+c=c+d+e$, it is also true that $a+b=d+e$. With the given list of 5 numbers, it is likely easier to find two pairs with no overlap and with equal sum than to find two triples with one overlap and equal sum. After some ... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $\\log(a^3b^7)$, $\\log(a^5b^{12})$, and $\\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\\text{th}}$ term of the sequence is $\\log{b^n}$. What is $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 33439,
"solution": "\n1. **Define the terms in terms of logarithms**:\n Let $A = \\log(a)$ and $B = \\log(b)$. The given terms of the sequence can be rewritten using logarithmic properties:\n \\[\n \\begin{align*}\n \\log(a^3b^7) &= 3A + 7B, \\\\\n \\log(a^5b^{12}) &= 5A + 12B, \\\\\n \\log(a... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $\\sqrt{2u-1}$, $\\sqrt{2u+1}$ and $2\\sqrt{u}$ are the side lengths of a triangle. How many degrees are in the measure of the largest angle?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 33440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $\\sqrt{3v-2}$, $\\sqrt{3v+1}$, and $2\\sqrt{v}$ are the side lengths of a triangle. What is the measure of the largest angle?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 33441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $a, b, c, d$ belong to the interval $[-6.5, 6.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 33442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$."
}
] | math | {
"style": "rule",
"ground_truth": "306"
} | {
"index": 33443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $a,$ $b,$ $c,$ $d$ are equal to 1, 2, 3, 4, in some order. Find the largest possible value of\n\\[ab + bc + cd + da.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 33444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $a_1,$ $a_2,$ $a_3,$ $b_1,$ $b_2,$ $b_3,$ $c_1,$ $c_2,$ $c_3$ are equal to the numbers $1,$ $2,$ $3,$ $\\dots,$ $9$ in some order. Find the smallest possible value of\n\\[a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "214"
} | {
"index": 33445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $a_{1}, a_{2}, \\ldots, a_{100}$ are a permutation of the numbers $1,2, \\ldots, 100$. Let $S_{1}=a_{1}$, $S_{2}=a_{1}+a_{2}, \\ldots, S_{100}=a_{1}+a_{2}+\\ldots+a_{100}$. What maximum number of perfect squares can be among the numbers $S_{1}, S_{2}, \\ldots, S_{10... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 33446,
"solution": "We add initial term \\(S_{0}=0\\) to the sequence \\(S_{1}, S_{2}, \\ldots, S_{100}\\) and consider all the terms \\(S_{n_{0}}<S_{n_{1}}<\\ldots\\) that are perfect squares: \\(S_{n_{k}}=m_{k}^{2}\\) (in particular, \\(n_{0}=m_{0}=0\\)). Since \\(S_{100}=5050<72^{2}\\), all the number... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 42, $x$ is twice $y$. What is the value of $y$ when $x=-8$?"
}
] | math | {
"style": "rule",
"ground_truth": "-49"
} | {
"index": 33447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 1 through 25 are written on 25 cards with one number on each card. Sara picks one of the 25 cards at random. What is the probability that the number on her card will be a multiple of 2 or 5? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 33448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 1 to 2031 are written on a blackboard.\n1. Select any two numbers on the blackboard, find the absolute value of their difference, and erase these two numbers.\n2. Then select another number on the blackboard, find the absolute value of its difference from the previo... | math | {
"style": "rule",
"ground_truth": "2030"
} | {
"index": 33449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no ... | math | {
"style": "rule",
"ground_truth": "1598"
} | {
"index": 33450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 1, 2, ..., 2016 are written on a board. You are allowed to erase any two numbers and write their arithmetic mean instead. How must you proceed so that the number 1000 remains on the board?"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 33451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 1, 3, 6, 10, $\\ldots$, are called triangular numbers, as shown geometrically here. What is the $20^{\\text{th}}$ triangular number?\n\n[asy]\n\ndot((0,0));\nlabel(\"1\",(0,-1.5));\n\ndot((3,0));\ndot((4,0));\ndot((3,1));\nlabel(\"3\",(3.5,-1.5));\n\ndot((7,0));\nd... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 33452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 2, 3, 5, 7, 11, 13 are arranged in a multiplication table, with three along the top and the other three down the left. The multiplication table is completed and the sum of the nine entries is tabulated. What is the largest possible sum of the nine entries?\n\n\\[\... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 33453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 2, 3, 5, 7, 11, 13, 17, 19 are arranged in a multiplication table, with four along the top and the other four down the left. The multiplication table is completed and the sum of the sixteen entries is tabulated. What is the largest possible sum of the sixteen entrie... | math | {
"style": "rule",
"ground_truth": "1482"
} | {
"index": 33454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 2, 4, 6, and 8 are a set of four consecutive even numbers. Suppose the sum of five consecutive even numbers is 320. What is the smallest of the five numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 33455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 407 and 370 equal the sum of the cubes of their digits. For example, \\( 4^3 = 64 \\), \\( 0^3 = 0 \\), and \\( 7^3 = 343 \\). Adding 64, 0, and 343 gives you 407. Similarly, the cube of 3 (27), added to the cube of 7 (343), gives 370.\n\nCould you find a number, no... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"index": 33456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers 60, 221, and 229 are the legs and hypotenuse of a right triangle. Find the multiplicative inverse to 450 modulo 3599. (Express your answer as an integer $n$ with $0\\leq n<3599$.)"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 33457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( 2^{2021} \\) and \\( 5^{2021} \\) are written out one after the other. How many digits are written in total?"
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 33458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( 62, 63, 64, 65, 66, 67, 68, 69, \\) and \\( 70 \\) are divided by, in some order, the numbers \\( 1, 2, 3, 4, 5, 6, 7, 8, \\) and \\( 9 \\), resulting in nine integers. The sum of these nine integers is \\( S \\). What are the possible values of \\( S \\)?"
}
... | math | {
"style": "rule",
"ground_truth": "187"
} | {
"index": 33459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( a, b, c, \\) and \\( d \\) are distinct positive integers chosen from 1 to 10 inclusive. What is the least possible value \\(\\frac{a}{b}+\\frac{c}{d}\\) could have?\n\nA) \\(\\frac{2}{10}\\)\n\nB) \\(\\frac{3}{19}\\)\n\nC) \\(\\frac{14}{45}\\)\n\nD) \\(\\frac{2... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{45}"
} | {
"index": 33460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-13.5, 13.5]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\)."
}
] | math | {
"style": "rule",
"ground_truth": "756"
} | {
"index": 33461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-4 ; 4]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\)."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 33462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-5, 5]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\)."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 33463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-7, 7]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\)."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 33464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-7.5, 7.5]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\)."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 33465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( p_1, p_2, p_3, q_1, q_2, q_3, r_1, r_2, r_3 \\) are equal to the numbers \\( 1, 2, 3, \\dots, 9 \\) in some order. Find the smallest possible value of\n\\[\nP = p_1 p_2 p_3 + q_1 q_2 q_3 + r_1 r_2 r_3.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "214"
} | {
"index": 33466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( x \\) and \\( y \\) are such that the equalities \\( \\operatorname{ctg} x - \\operatorname{ctg} y = 2 \\) and \\( 5 \\sin (2x - 2y) = \\sin 2x \\sin 2y \\) hold. Find \\( \\operatorname{tg} x \\operatorname{tg} y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{6}{5}"
} | {
"index": 33467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( x \\) and \\( y \\) satisfy the equation:\n\n$$\n\\sqrt{x y}+\\sqrt{(1-x)(1-y)}=\\sqrt{7 x(1-y)}+\\frac{\\sqrt{y(1-x)}}{\\sqrt{7}}\n$$\n\nFind the maximum value of the expression \\( x + 7y \\). Justify your answer."
}
] | math | {
"style": "rule",
"ground_truth": "57/8"
} | {
"index": 33468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\( x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3 \\) are equal to the numbers \\( 1, 2, 3, \\ldots, 9 \\) in some order. Find the smallest possible value of\n\\[ x_1 x_2 x_3 + y_1 y_2 y_3 + z_1 z_2 z_3. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "214"
} | {
"index": 33469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(1, 2, 3, \\ldots, 10\\) are written in some order around a circle. Peter computed the sums of all 10 triples of neighboring numbers and wrote the smallest of these sums on the board. What is the maximum possible number that could have been written on the board?"
... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 33470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(1, 2, 3, \\ldots, 400\\) are written on 400 cards. Two players, \\(A\\) and \\(B\\), play the following game:\n\n1. In the first step, \\(A\\) takes 200 cards for themselves.\n2. \\(B\\) then takes 100 cards from both the remaining 200 cards and the 200 cards tha... | math | {
"style": "rule",
"ground_truth": "20000"
} | {
"index": 33471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(1, 2, 3, \\ldots, 7\\) are randomly divided into two non-empty subsets. What is the probability that the sum of the numbers in the two subsets is equal? If the probability is expressed as \\(\\frac{p}{q}\\) in its lowest terms, find \\(p + q\\)."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 33472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(1000^{2}, 1001^{2}, 1002^{2}, \\ldots\\) have their last three digits discarded. How many of the first terms in the resulting sequence form an arithmetic progression?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 33473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(2, 3, 12, 14, 15, 20, 21\\) may be divided into two sets so that the product of the numbers in each set is the same.\nWhat is this product?"
}
] | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"index": 33474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(2, 3, 4, 5, 6, 7, 8\\) are to be placed, one per square, in the diagram shown so that the sum of the four numbers in the horizontal row equals 21 and the sum of the four numbers in the vertical column also equals 21. In how many different ways can this be done?"
... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(2^{2021}\\) and \\(5^{2021}\\) are written out one after the other. How many digits were written in total?"
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 33476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-11.5, 11.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\)."
}
] | math | {
"style": "rule",
"ground_truth": "552"
} | {
"index": 33477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-12.5, 12.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\)."
}
] | math | {
"style": "rule",
"ground_truth": "650"
} | {
"index": 33478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-4.5, 4.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\)."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 33479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-5.5, 5.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\)."
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 33480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-6, 6]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\)."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 33481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-6.5 ; 6.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\)."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 33482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-7.5, 7.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\)."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 33483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-8.5, 8.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\)."
}
] | math | {
"style": "rule",
"ground_truth": "306"
} | {
"index": 33484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a, b,\\) and \\(c\\) (not necessarily integers) satisfy the conditions\n\n\\[\na + b + c = 0 \\quad \\text{and} \\quad \\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a} = 100\n\\]\n\nWhat is the value of \\(\\frac{b}{a} + \\frac{c}{b} + \\frac{a}{c}\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "-101"
} | {
"index": 33485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a\\) and \\(b\\) are such that \\(|a| \\neq |b|\\) and \\(\\frac{a+b}{a-b} + \\frac{a-b}{a+b} = 6\\). Find the value of the expression \\(\\frac{a^{3} + b^{3}}{a^{3} - b^{3}} + \\frac{a^{3} - b^{3}}{a^{3} + b^{3}}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18}{7}"
} | {
"index": 33486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers \\(a_{1}, a_{2}, \\ldots, a_{n}\\) are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. What is the largest possible \\(n\\) for which this is true?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers assigned to 100 athletes range from 1 to 100. If each athlete writes down the largest odd factor of their number on a blackboard, what is the sum of all the numbers written by the athletes?"
}
] | math | {
"style": "rule",
"ground_truth": "3344"
} | {
"index": 33488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 33489,
"solution": "1. **Identify the total sum of numbers on the cube**: Each number from $1$ to $8$ is placed on a vertex of the cube. The sum of these numbers is $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8$. Using the formula for the sum of the first $n$ natural numbers, $\\frac{n(n+1)}{2}$, where $n = 8$, we get:... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 1000 are written in a circle. Starting from the first one, every 15th number (i.e., the numbers 1, 16, 31, etc.) is crossed out, and during subsequent rounds, already crossed-out numbers are also taken into account. The crossing out continues until it turn... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"index": 33490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 150, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square nor a perfect cube? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"index": 33491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 150, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1. For example, $2^{... | math | {
"style": "rule",
"ground_truth": "\\frac{133}{150}"
} | {
"index": 33492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square, a perfect cube, nor a sixth power? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{183}{200}"
} | {
"index": 33493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 200, inclusive, are placed in a bag. A number is randomly selected from the bag. What is the probability that it is neither a perfect square, a perfect cube, nor a multiple of 7? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{39}{50}"
} | {
"index": 33494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 8 are arranged at the vertices of a cube in such a way that the sum of the numbers at any three vertices on the same face is at least 10. What is the minimum possible sum of the numbers on the vertices of one face?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 8 are placed at the vertices of a cube so that the sum of the numbers at any three vertices, which lie on the same face, is at least 10. What is the minimum possible sum of the numbers on one face?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 9 are arranged in the cells of a $3 \\times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal, it is 21. What is the sum of the numbers in the five shaded cells?\n\n![Table with shaded cells](https://cdn.mathpix.com/... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 33498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 9 are divided into three groups of three numbers, and then the numbers in each group are multiplied. $A$ is the largest of the three products. What is the smallest possible value of $A$?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 33499,
"solution": ""
} |
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