data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The image of the point with coordinates $(-3,-1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(5,3)$. Find $m+b$."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The image of the point with coordinates $(1,1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(9,5)$. Find $m+b$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 32801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The image of the point with coordinates $(2, -4)$ under the reflection across the line $y = mx + b$ is the point with coordinates $(-4, 8)$. The line of reflection passes through the point $(0, 3)$. Find $m+b$."
}
] | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"index": 32802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The image shows a 3x3 grid where each cell contains one of the following characters: 华, 罗, 庚, 杯, 数, 学, 精, 英, and 赛. Each character represents a different number from 1 to 9, and these numbers satisfy the following conditions:\n1. The sum of the four numbers in each \"田\" (four ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 32803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The image shows a grid consisting of 25 small equilateral triangles. How many rhombuses can be formed from two adjacent small triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 32804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The imaginary part of the complex number $z = -4i + 3$ is $-4i$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 32805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The incircle $\\omega$ of triangle $ABC$ is tangent to $\\overline{BC}$ at $X$. Let $Y \\neq X$ be the other intersection of $\\overline{AX}$ with $\\omega$. Points $P$ and $Q$ lie on $\\overline{AB}$ and $\\overline{AC}$, respectively, so that $\\overline{PQ}$ is tangent to $\... | math | {
"style": "rule",
"ground_truth": "227"
} | {
"index": 32806,
"solution": "Let the incircle of $ABC$ be tangent to $AB$ and $AC$ at $M$ and $N$. By Brianchon's theorem on tangential hexagons $QNCBMP$ and $PYQCXB$, we know that $MN,CP,BQ$ and $XY$ are concurrent at a point $O$. Let $PQ \\cap BC = Z$. Then by La Hire's $A$ lies on the polar of $Z$ so $Z$ lies ... |
DeepScaleR | [
{
"role": "user",
"content": "The incircle of triangle \\( ABC \\) with center \\( O \\) touches the sides \\( AB \\), \\( BC \\), and \\( AC \\) at points \\( M \\), \\( N \\), and \\( K \\) respectively. It is given that angle \\( AOC \\) is four times larger than angle \\( MKN \\). Find angle \\( B \\)."... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 32807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle $\\alpha$ of the line $l: \\sqrt{3}x+3y+1=0$ is $\\tan^{-1}\\left( -\\frac{\\sqrt{3}}{3} \\right)$. Calculate the value of the angle $\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"index": 32808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line $\\sqrt {3}x-y+1=0$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{3}"
} | {
"index": 32809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line $\\sqrt{3}x + y - 1 = 0$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 32810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line $\\sqrt{3}x+y+2024=0$ is $\\tan^{-1}\\left(-\\frac{\\sqrt{3}}{1}\\right)$. Calculate the angle in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 32811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line $\\sqrt{3}x+y-1=0$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 32812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line $x+ \\sqrt {3}y+c=0$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"index": 32813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line $x- \\sqrt {3}y+3=0$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{6}"
} | {
"index": 32814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line $x-y+1=0$ can be calculated."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 32815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line $x-y+1=0$ is what value?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 32816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inclination angle of the line given by the parametric equations \\[\n\\begin{cases} \nx=1+t \\\\\ny=1-t\n\\end{cases}\n\\]\ncalculate the inclination angle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3\\pi }{4}"
} | {
"index": 32817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \\ldots\\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolv... | math | {
"style": "rule",
"ground_truth": "063"
} | {
"index": 32818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing geometric sequence $x_{0},x_{1},x_{2},\\ldots$ consists entirely of integral powers of $3.$ Given that\n$\\sum_{n=0}^{7}\\log_{3}(x_{n}) = 308$ and $56 \\leq \\log_{3}\\left ( \\sum_{n=0}^{7}x_{n}\\right ) \\leq 57,$\n find $\\log_{3}(x_{14}).$"
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 32819,
"solution": "All these integral powers of $3$ are all different, thus in base $3$ the sum of these powers would consist of $1$s and $0$s. Thus the largest value $x_7$ must be $3^{56}$ in order to preserve the givens. Then we find by the given that $x_7x_6x_5\\dots x_0 = 3^{308}$, and we know that ... |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence $1, 3, 4, 9, 10, 12, 13, \\dots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $150^{\\text{th}}$ term of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "2280"
} | {
"index": 32820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence $1,3,4,9,10,12,13\\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Determine the $150^{\\mbox{th}}$ term of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "2280"
} | {
"index": 32821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence $1,3,4,9,10,12,13\\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\\mbox{th}}$ term of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "981"
} | {
"index": 32822,
"solution": "Notice that the first term of the sequence is $1$, the second is $3$, the fourth is $9$, and so on. Thus the $64th$ term of the sequence is $729$. Now out of $64$ terms which are of the form $729$ + $'''S'''$, $32$ of them include $243$ and $32$ do not. The smallest term that includes... |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence $1,3,4,9,10,12,13\\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $50^{\\mbox{th}}$ term of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "327"
} | {
"index": 32823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence $2,3,5,6,7,10,11,\\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "528"
} | {
"index": 32824,
"solution": "Because there aren't that many perfect squares or cubes, let's look for the smallest perfect square greater than $500$. This happens to be $23^2=529$. Notice that there are $23$ squares and $8$ cubes less than or equal to $529$, but $1$ and $2^6$ are both squares and cubes. Thus, ther... |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence $3, 15, 24, 48, \\ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?"
}
] | math | {
"style": "rule",
"ground_truth": "935"
} | {
"index": 32825,
"solution": "1. **Identify the form of the sequence**: The sequence consists of positive multiples of 3 that are one less than a perfect square. This can be represented as $n^2 - 1$, where $n^2 - 1$ is a multiple of 3.\n\n2. **Determine conditions for $n^2 - 1$ to be a multiple of 3**: \n - We h... |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence $3, 15, 24, 48, \\ldots\\,$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 32826,
"solution": "One less than a perfect square can be represented by $n^2 - 1 = (n+1)(n-1)$. Either $n+1$ or $n-1$ must be divisible by 3. This is true when $n \\equiv -1,\\ 1 \\equiv 2,\\ 1 \\pmod{3}$. Since 1994 is even, $n$ must $\\equiv 1 \\pmod{3}$. It will be the $\\frac{1994}{2} = 997$th such ... |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence \\( T = 2, 3, 5, 6, 7, 8, 10, 11, \\ldots \\) consists of all positive integers which are not perfect squares. What is the 2012th term of \\( T \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "2057"
} | {
"index": 32827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence \\(1, 3, 4, 9, 10, 12, 13, \\cdots\\) consists of some positive integers that are either powers of 3 or sums of distinct powers of 3. Find the value of the 2014th term."
}
] | math | {
"style": "rule",
"ground_truth": "88329"
} | {
"index": 32828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence consists of all those positive integers which are either powers of 2, powers of 3, or sums of distinct powers of 2 and 3. Find the $50^{\\rm th}$ term of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 32829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence of positive integers $a_1, a_2, a_3, \\dots$ follows the rule:\n\\[a_{n+2} = a_{n+1} + a_n\\]\nfor all $n \\geq 1$. If $a_6 = 50$, find $a_7$."
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 32830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence of positive integers $a_1, a_2, a_3, \\dots$ is defined by the rule\n\\[a_{n + 2} = a_{n + 1} + a_n\\]\nfor all $n \\ge 1.$ If $a_7 = 210$, then find $a_8.$"
}
] | math | {
"style": "rule",
"ground_truth": "340"
} | {
"index": 32831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence of positive integers $a_1,$ $a_2,$ $a_3,$ $\\dots$ has the property that\n\\[a_{n + 2} = a_{n + 1} + a_n\\]for all $n \\ge 1.$ If $a_7 = 120,$ then find $a_8.$"
}
] | math | {
"style": "rule",
"ground_truth": "194"
} | {
"index": 32832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence of positive integers $b_1, b_2, b_3, \\ldots$ follows the rule:\n\\[ b_{n + 2} = b_{n + 1} + b_n \\]\nfor all $n \\geq 1$. If $b_5 = 55$, find $b_6$."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 32833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\\dots$ follows the rule\n\\[b_{n + 2} = b_{n + 1} + b_n\\]for all $n \\ge 1.$ If $b_9 = 544,$ then find $b_{10}.$"
}
] | math | {
"style": "rule",
"ground_truth": "883"
} | {
"index": 32834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\\dots$ has the property that\n\\[b_{n + 2} = b_{n + 1} + b_n\\]for all $n \\ge 1.$ If $b_6 = 60,$ then find $b_7.$"
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"index": 32835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The infinite sequence $S=\\{s_1,s_2,s_3,\\ldots\\}$ is defined by $s_1=7$ and $s_n=7^{s_{n-1}}$ for each integer $n>1$. What is the remainder when $s_{100}$ is divided by $5$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 32836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inhabitants of the Isle of Concatenate use an extended alphabet of 25 letters (A through Y). Each word in their language has a maximum length of 5 letters, and every word must include the letter A at least once. How many such words are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "1863701"
} | {
"index": 32837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inhabitants of the island of Jumble use the standard Kobish alphabet ($20$ letters, A through T). Each word in their language is $4$ letters or less, and for some reason, they insist that all words contain the letter A at least once. How many words are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "30860"
} | {
"index": 32838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inscribed circle of triangle $ABC$ is tangent to $\\overline{AB}$ at $P,$ and its radius is $21$. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "345"
} | {
"index": 32839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inscribed circle of triangle $ABC$ is tangent to $\\overline{AB}$ at $P_{},$ and its radius is $21$. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "345"
} | {
"index": 32840,
"solution": "Let $Q$ be the tangency point on $\\overline{AC}$, and $R$ on $\\overline{BC}$. By the Two Tangent Theorem, $AP = AQ = 23$, $BP = BR = 27$, and $CQ = CR = x$. Using $rs = A$, where $s = \\frac{27 \\cdot 2 + 23 \\cdot 2 + x \\cdot 2}{2} = 50 + x$, we get $(21)(50 + x) = A$. By Heron's ... |
DeepScaleR | [
{
"role": "user",
"content": "The inscribed circle of triangle $DEF$ is tangent to $\\overline{DE}$ at point $P$ and its radius is $13$. Given that $DP = 17$ and $PE = 31$, and the tangent from vertex $F$ to the circle is $20$, determine the perimeter of triangle $DEF$."
}
] | math | {
"style": "rule",
"ground_truth": "136"
} | {
"index": 32841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The instantaneous rate of change of carbon-14 content is $-\\frac{\\ln2}{20}$ (becquerel/year) given that at $t=5730$. Using the formula $M(t) = M_0 \\cdot 2^{-\\frac{t}{5730}}$, determine $M(2865)$."
}
] | math | {
"style": "rule",
"ground_truth": "573\\sqrt{2}/2"
} | {
"index": 32842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The instructor of a summer math camp brought several shirts, several pairs of trousers, several pairs of shoes, and two jackets for the entire summer. In each lesson, he wore trousers, a shirt, and shoes, and he wore a jacket only on some lessons. On any two lessons, at least o... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 32843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$ . Let $k=a-1$ . If the $k$ -th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$ , find the highest possible value of $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"index": 32844,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer $N$ is the smallest positive integer that is a multiple of 2024, has more than 100 positive divisors (including 1 and $N$), and has fewer than 110 positive divisors (including 1 and $N$). What is the sum of the digits of $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 32845,
"solution": "Throughout this solution, we use the fact that if $N$ is a positive integer with $N>1$ and $N$ has prime factorization $p_{1}^{a_{1}} p_{2}^{a_{2}} \\cdots p_{m}^{a_{m}}$ for some distinct prime numbers $p_{1}, p_{2}, \\ldots, p_{m}$ and positive integers $a_{1}, a_{2}, \\ldots, a_{m}... |
DeepScaleR | [
{
"role": "user",
"content": "The integer $m$ is between $30$ and $80$ and is a multiple of $6$. When $m$ is divided by $8,$ the remainder is $2$. Similarly, when $m$ is divided by $5,$ the remainder is $2$. What is the value of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 32846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\\frac{m}{18}$."
}
] | math | {
"style": "rule",
"ground_truth": "555"
} | {
"index": 32847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer $n > 9$ is a root of the quadratic equation $x^2 - ax + b=0$. In this equation, the representation of $a$ in the base-$n$ system is $19$. Determine the base-$n$ representation of $b$."
}
] | math | {
"style": "rule",
"ground_truth": "90_n"
} | {
"index": 32848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\\frac{n}{15}$."
}
] | math | {
"style": "rule",
"ground_truth": "592"
} | {
"index": 32849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer $n$ is the smallest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\\frac{n}{15}$."
}
] | math | {
"style": "rule",
"ground_truth": "592"
} | {
"index": 32850,
"solution": "Any multiple of 15 is a multiple of 5 and a multiple of 3.\n Any multiple of 5 ends in 0 or 5; since $n$ only contains the digits 0 and 8, the units digit of $n$ must be 0.\n The sum of the digits of any multiple of 3 must be divisible by 3. If $n$ has $a$ digits equal to 8, the sum o... |
DeepScaleR | [
{
"role": "user",
"content": "The integer $x$ has 12 positive factors. The numbers 12 and 15 are factors of $x$. What is $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 32851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer $y$ has 24 positive factors. The numbers 20 and 35 are factors of $y$. What is the smallest possible value of $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "1120"
} | {
"index": 32852,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer 119 is a multiple of which number?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 32853,
"solution": "The ones digit of 119 is not even, so 119 is not a multiple of 2. The ones digit of 119 is not 0 or 5, so 119 is not a multiple of 5. Since $120=3 \\times 40$, then 119 is 1 less than a multiple of 3 so is not itself a multiple of 3. Since $110=11 \\times 10$ and $121=11 \\times 11$, ... |
DeepScaleR | [
{
"role": "user",
"content": "The integer 2014 is between which powers of 10?"
}
] | math | {
"style": "rule",
"ground_truth": "10^{3} \\text{ and } 10^{4}"
} | {
"index": 32854,
"solution": "Since \\( 10^{0}=1,10^{1}=10,10^{2}=100,10^{3}=1000,10^{4}=10000 \\), and \\( 10^{5}=100000 \\), then 2014 is between \\( 10^{3} \\) and \\( 10^{4} \\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer 2019 can be formed by placing two consecutive two-digit positive integers, 19 and 20, in decreasing order. What is the sum of all four-digit positive integers greater than 2019 that can be formed in this way?"
}
] | math | {
"style": "rule",
"ground_truth": "478661"
} | {
"index": 32855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer 48178 includes the block of digits 178. How many integers between 10000 and 100000 include the block of digits 178?"
}
] | math | {
"style": "rule",
"ground_truth": "280"
} | {
"index": 32856,
"solution": "Since 100000 does not include the block of digits 178, each integer between 10000 and 100000 that includes the block of digits 178 has five digits. Such an integer can be of the form $178 x y$ or of the form $x 178 y$ or of the form $x y 178$ for some digits $x$ and $y$. The leading d... |
DeepScaleR | [
{
"role": "user",
"content": "The integer 636405 may be written as the product of three 2-digit positive integers. What is the sum of these three integers?"
}
] | math | {
"style": "rule",
"ground_truth": "259"
} | {
"index": 32857,
"solution": "We begin by factoring the given integer into prime factors. Since 636405 ends in a 5, it is divisible by 5, so $636405=5 \\times 127281$. Since the sum of the digits of 127281 is a multiple of 3, then it is a multiple of 3, so $636405=5 \\times 3 \\times 42427$. The new quotient (4242... |
DeepScaleR | [
{
"role": "user",
"content": "The integer 843301 is prime. The primorial of a prime number $p$, denoted $p \\#$, is defined to be the product of all prime numbers less than or equal to $p$. Determine the number of digits in $843301 \\#$. Your score will be $$\\max \\left\\{\\left\\lfloor 60\\left(\\frac{1}{... | math | {
"style": "rule",
"ground_truth": "365851"
} | {
"index": 32858,
"solution": "Remark: 843301\\#-1 is the largest known prime number of the form $p \\#-1$, where $p$ is prime."
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer \\( n \\) has a total of 10 divisors. These divisors are arranged in ascending order, and the 8th divisor is \\( \\frac{n}{3} \\). Find the maximum value of the integer \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 32859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer \\( n \\) is the smallest positive number that satisfies the following conditions:\n1. \\( n \\) is a multiple of 75.\n2. \\( n \\) has exactly 75 positive divisors (including 1 and itself).\n\nFind the value of \\( \\frac{n}{75} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 32860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer numbers from $1$ to $2002$ are written in a blackboard in increasing order $1,2,\\ldots, 2001,2002$. After that, somebody erases the numbers in the $ (3k+1)-th$ places i.e. $(1,4,7,\\dots)$. After that, the same person erases the numbers in the $(3k+1)-th$ positions... | math | {
"style": "rule",
"ground_truth": "2,6,10"
} | {
"index": 32861,
"solution": "\nLet's analyze the process of elimination step by step, starting from the list of integers from 1 to 2002:\n\n### Step 1:\nInitially, the numbers \\(1, 2, \\ldots, 2002\\) are written on the board. In this first step, numbers at positions \\(1, 4, 7, \\ldots\\) (i.e., \\( (3k+1) \\)-... |
DeepScaleR | [
{
"role": "user",
"content": "The integer parts of two finite decimals are 7 and 10, respectively. How many possible values are there for the integer part of the product of these two finite decimals?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 32862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \\leq y \\leq 8$. Determine the number of such integer points $(x, y)$."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \\leq y \\leq 8$. How many such integer points $(x, y)$ are there?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer sequence \\(a_1, a_2, a_3, \\dots\\) is defined as follows: \\(a_1 = 1\\). For \\(n \\geq 1\\), \\(a_{n+1}\\) is the smallest integer greater than \\(a_n\\) such that for all \\(i, j, k \\in \\{1, 2, \\dots, n+1\\}\\), the condition \\(a_i + a_j \\neq 3a_k\\) is sat... | math | {
"style": "rule",
"ground_truth": "66016"
} | {
"index": 32865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integer solution for the inequality $|2x-m|\\leq1$ with respect to $x$ is uniquely $3$ ($m$ is an integer).\n(I) Find the value of the integer $m$;\n(II) Given that $a, b, c \\in R$, if $4a^4+4b^4+4c^4=m$, find the maximum value of $a^2+b^2+c^2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"index": 32866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers $a$ , $b$ , $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\\leq 2022$ and $a+b+c+d = ac + bd = 0$ . Determine the largest possible value of $d$ ,"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 32867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers $1,2, \\ldots, 64$ are written in the squares of a $8 \\times 8$ chess board, such that for each $1 \\leq i<64$, the numbers $i$ and $i+1$ are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?"
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 32868,
"solution": "Our answer is $26+52+54+56+58+60+62+64$. One possible configuration: WLOG, we seek to maximize the sum of the numbers on the main diagonal (top left to bottom right). If we color the squares in a checker-board pattern and use the fact that $a$ and $a+1$ lie on different colored square... |
DeepScaleR | [
{
"role": "user",
"content": "The integers $1,2,4,5,6,9,10,11,13$ are to be placed in the circles and squares below with one number in each shape. Each integer must be used exactly once and the integer in each circle must be equal to the sum of the integers in the two neighbouring squares. If the integer $x... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32869,
"solution": "From the given information, if $a$ and $b$ are in two consecutive squares, then $a+b$ goes in the circle between them. Since all of the numbers that we can use are positive, then $a+b$ is larger than both $a$ and $b$. This means that the largest integer in the list, which is 13, canno... |
DeepScaleR | [
{
"role": "user",
"content": "The integers $G$ and $H$ are chosen such that\n\\[\\frac{G}{x+5}+\\frac{H}{x^2-4x}=\\frac{x^2-2x+10}{x^3+x^2-20x}\\]for all real values of $x$ except $-5$, $0$, and $4$. Find $H/G$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers $a, b$ and $c$ satisfy the equations $a+5=b$, $5+b=c$, and $b+c=a$. What is the value of $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 32871,
"solution": "Since $a+5=b$, then $a=b-5$. Since $a=b-5$ and $c=5+b$ and $b+c=a$, then $b+(5+b)=b-5$, $2b+5=b-5$, $b=-10$. (If $b=-10$, then $a=b-5=-15$ and $c=5+b=-5$ and $b+c=(-10)+(-5)=(-15)=a$, as required.)"
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers $r$ and $k$ are randomly selected, where $-3 < r < 6$ and $1 < k < 8$. What is the probability that the division $r \\div k$ is an integer value? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 32872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers -5 and 6 are shown on a number line. What is the distance between them?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 32873,
"solution": "The distance between two numbers on the number line is equal to their positive difference. Here, this distance is $6-(-5)=11$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers 195 and 61 are expressed in base 4 and added. What is the resulting sum, expressed in base 4?"
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 32874,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers \\( m \\) and \\( n \\) satisfy the equation \\( 3^{m} \\times n = 7! + 8! + 9! \\). What is the smallest possible value for \\( n \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "560"
} | {
"index": 32875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers \\( n \\) and \\( m \\) satisfy the inequalities \\( 3n - m < 5 \\), \\( n + m > 26 \\), and \\( 3m - 2n < 46 \\). What can \\( 2n + m \\) be equal to? Indicate all possible values."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 32876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers \\( r \\) and \\( k \\) are randomly selected, where \\(-5 < r < 10\\) and \\(0 < k < 10\\). What is the probability that the division \\( r \\div k \\) results in \\( r \\) being a square number? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{63}"
} | {
"index": 32877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers \\(1,2,3,4,5,6,7,8,9,10\\) are written on a blackboard. Each day, a teacher chooses one of the integers uniformly at random and decreases it by 1. Let \\(X\\) be the expected value of the number of days which elapse before there are no longer positive integers on t... | math | {
"style": "rule",
"ground_truth": "120.75280458176904"
} | {
"index": 32878,
"solution": "Answer: 120.75280458176904"
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers \\(a, b,\\) and \\(c\\) satisfy the equations \\(a + 5 = b\\), \\(5 + b = c\\), and \\(b + c = a\\). What is the value of \\(b\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 32879,
"solution": "Since \\(a + 5 = b\\), then \\(a = b - 5\\). Substituting \\(a = b - 5\\) and \\(c = 5 + b\\) into \\(b + c = a\\), we obtain \\(b + (5 + b) = b - 5\\). Simplifying, we get \\(2b + 5 = b - 5\\), which gives \\(b = -10\\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers from 1 to \\( n \\), inclusive, are equally spaced in order around a circle. The diameter through the position of the integer 7 also goes through the position of 23. What is the value of \\( n \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 32880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The integers that can be expressed as a sum of three distinct numbers chosen from the set $\\{4,7,10,13, \\ldots,46\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 32881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The intercept of the line $5x - 2y - 10 = 0$ on the x-axis is $a$, and on the y-axis is $b$. Find the values of $a$ and $b$."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 32882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The intercept on the x-axis, the intercept on the y-axis, and the slope of the line 4x-5y-20=0 are respectively."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{5}"
} | {
"index": 32883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The interior angles of a convex polygon form an arithmetic sequence, with the smallest angle being $120^\\circ$ and the common difference being $5^\\circ$. Determine the number of sides $n$ for the polygon."
}
] | math | {
"style": "rule",
"ground_truth": "n = 9"
} | {
"index": 32884,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8a^2}{a^2+1}"
} | {
"index": 32885,
"solution": "\n#### Step 1: Analyze the given equations\nThe equations given are:\n1. \\((x+ay)^2 = 4a^2\\)\n2. \\((ax-y)^2 = a^2\\)\n\nThese can be rewritten as:\n- For equation 1: \\(x+ay = \\pm 2a\\)\n- For equation 2: \\(ax-y = \\pm a\\)\n\n#### Step 2: Convert to standard line equations\nFrom... |
DeepScaleR | [
{
"role": "user",
"content": "The interior of a right, circular cone is 8 inches tall with a 2-inch radius at the opening. The interior of the cone is filled with ice cream, and the cone has a hemisphere of ice cream exactly covering the opening of the cone. What is the volume of ice cream? Express your ans... | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"index": 32886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\\angle DBA = \\angle DCB$ and $\\angle ADB = \\angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degree... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 32887,
"solution": "1. **Assigning angles in arithmetic progression**: Since the internal angles of quadrilateral $ABCD$ form an arithmetic progression, we denote them as $a$, $a+d$, $a+2d$, and $a+3d$. The sum of the internal angles of any quadrilateral is $360^\\circ$, so:\n \\[\n a + (a+d) + (a+2d... |
DeepScaleR | [
{
"role": "user",
"content": "The intersecting squares from left to right have sides of lengths 12, 9, 7, and 3, respectively. By how much is the sum of the black areas greater than the sum of the gray areas?"
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 32888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$ . Given that the distance between the centers of the two squares is $2$ , the perimeter of the rectangle can be expressed as $P$ . Find $10P$ ."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 32889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inverse of $f(x) = \\frac{2x-1}{x+5}$ may be written in the form $f^{-1}(x)=\\frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are real numbers. Find $a/c$."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 32890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inverse of $f(x) = \\frac{3x - 2}{x + 4}$ may be written in the form $f^{-1}(x)=\\frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are real numbers. Find $a/c$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 32891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The inverse of \\( f(x) = \\frac{3x-2}{x+4} \\) may be written in the form \\( f^{-1}(x) = \\frac{ax+b}{cx+d} \\), where \\( a, b, c, \\) and \\( d \\) are real numbers. Find \\( \\frac{a}{c} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 32892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\\frac13$ at pH $3.55$ , while its charge is $\\frac12$ at pH $9.6$ . Charge increases linearly with pH. What is the isoelectric point of glycine?"
}
] | math | {
"style": "rule",
"ground_truth": "5.97"
} | {
"index": 32893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\\angle ACB$ intersect $AB$ at $D$ and $E$. What is the area of $\\triangle CDE$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50-25\\sqrt{3}}{2}"
} | {
"index": 32894,
"solution": "1. **Identify the properties of triangle $ABC$:**\n Given that $ABC$ is an isosceles right triangle with a right angle at $C$ and an area of $12.5$, we can find the lengths of the legs. The area of a triangle is given by:\n \\[\n \\text{Area} = \\frac{1}{2} \\times \\text{base} ... |
DeepScaleR | [
{
"role": "user",
"content": "The isosceles trapezoid has base lengths of 24 units (bottom) and 12 units (top), and the non-parallel sides are each 12 units long. How long is the diagonal of the trapezoid?"
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"index": 32895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The isosceles trapezoid shown has side lengths as labeled. How long is segment AC? [asy]\nunitsize(1.5mm);\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\ndotfactor=3;\n\npair A=(0,0), B=(21,0), C=(15,8), D=(6,8);\npair[] dots={A,B,C,D};\n\ndraw(A--B--C--D--cycle);\ndot(dots);\nl... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 32896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The isosceles triangle and the square shown here have the same area in square units. What is the height of the triangle, $h$, in terms of the side length of the square, $s$?\n\n[asy]\ndraw((0,0)--(0,10)--(10,10)--(10,0)--cycle);\nfill((0,0)--(17,5)--(0,10)--cycle,white);\ndraw(... | math | {
"style": "rule",
"ground_truth": "2s"
} | {
"index": 32897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns hom... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 32898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The jury, when preparing versions of the district math olympiad problems for grades $7, 8, 9, 10, 11$, aims to ensure that each version for each grade contains exactly 7 problems, of which exactly 4 do not appear in any other version. What is the maximum number of problems that... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 32899,
"solution": ""
} |
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