data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number."
}
] | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"index": 10200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of different monic quadratic polynomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two different roots which are powers of 5 with natural exponents, and their coefficients do not exceed in absolute value $125^{... | math | {
"style": "rule",
"ground_truth": "5112"
} | {
"index": 10201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of different numbers of the form $\\left\\lfloor\\frac{i^2}{2015} \\right\\rfloor$ , with $i = 1,2, ..., 2015$ ."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 10202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of different patterns that can be created by shading exactly three of the nine small triangles, no two of which can share a side, considering patterns that can be matched by rotations or by reflections as the same."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of different possible rational roots of the polynomial:\n\\[6x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 10 = 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 10204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of different recommendation plans for the high school given that 3 male and 2 female students are selected as candidates, where both Russian and Japanese exams must include male participants, and 2 spots are available for Russian, 2 for Japanese, and 1 for Spani... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 10205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of digit of $\\sum_{n=0}^{99} 3^n$ .\nYou may use $\\log_{10} 3=0.4771$ .\n\n2012 Tokyo Institute of Technology entrance exam, problem 2-A"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 10206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of digits in the decimal representation of $2^{41}$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 10207,
"solution": "Noticing that $2^{10}=1024 \\approx 1000$ allows for a good estimate. Alternatively, the number of decimal digits of $n$ is given by $\\left\\lfloor\\log _{10}(n)\\right\\rfloor+1$. Using $\\log _{10}(2) \\approx 0.31$ also gives the correct answer. The exact value of $2^{41}$ is 2199... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of distinct arrangements in a row of all natural numbers from 1 to 10 such that the sum of any three consecutive numbers is divisible by 3."
}
] | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"index": 10208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of distinct numbers in the list\n\\[\\left\\lfloor \\frac{1^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac{2^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac{3^2}{1000} \\right\\rfloor, \\ \\dots, \\ \\left\\lfloor \\frac{1000^2}{1000} \\right\\rfloor.\\]"... | math | {
"style": "rule",
"ground_truth": "751"
} | {
"index": 10209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \\( (i+1) \\) does not immediately follow \\( i \\) for all \\( i \\) that runs from 1 to 7."
}
] | math | {
"style": "rule",
"ground_truth": "16687"
} | {
"index": 10210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer."
}
] | math | {
"style": "rule",
"ground_truth": "5600"
} | {
"index": 10211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of eight-digit numbers where the product of the digits equals 3375. The answer must be presented as an integer."
}
] | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 10212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of eight-digit numbers whose product of digits equals 1400. The answer must be presented as an integer."
}
] | math | {
"style": "rule",
"ground_truth": "5880"
} | {
"index": 10213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of eight-digit positive integers that are multiples of 9 and have all distinct digits."
}
] | math | {
"style": "rule",
"ground_truth": "181440"
} | {
"index": 10214,
"solution": "Note that $0+1+\\cdots+9=45$. Consider the two unused digits, which must then add up to 9. If it's 0 and 9, there are $8 \\cdot 7!$ ways to finish; otherwise, each of the other four pairs gives $7 \\cdot 7!$ ways to finish, since 0 cannot be the first digit. This gives a total of $36 ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of even digits in the base-7 representation of $403_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of five-digit palindromes."
}
] | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 10216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of five-digit positive integers, $n$, that satisfy the following conditions:\n (a) the number $n$ is divisible by $5,$\n (b) the first and last digits of $n$ are equal, and\n (c) the sum of the digits of $n$ is divisible by $5.$"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 10217,
"solution": "The number takes a form of $5\\text{x,y,z}5$, in which $5|(x+y+z)$. Let $x$ and $y$ be arbitrary digits. For each pair of $x,y$, there are exactly two values of $z$ that satisfy the condition of $5|(x+y+z)$. Therefore, the answer is $10\\times10\\times2=\\boxed{200}$\n ~Shreyas S"
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of four-digit numbers in which all digits are different, the first digit is divisible by 2, and the sum of the first and last digits is divisible by 3."
}
] | math | {
"style": "rule",
"ground_truth": "672"
} | {
"index": 10218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of four-digit numbers with distinct digits, formed using the digits 0, 1, 2, ..., 9, such that the absolute difference between the units and hundreds digit is 8."
}
] | math | {
"style": "rule",
"ground_truth": "154"
} | {
"index": 10219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of four-digit numbers, composed of the digits 1, 2, 3, 4, 5, 6, 7 (each digit can be used no more than once), that are divisible by 15."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 10220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of four-element subsets of $\\{1,2,3,4,\\dots, 20\\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\\{3,5,13,19\\}$ and $\\{6,10,20,18\\}$ are two such subset... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 10221,
"solution": "Let's say our four elements in our subset are $a,b,c,d$. We have two cases. Note that the order of the elements / the element letters themselves don't matter since they are all on equal grounds at the start.\n$\\textrm{Case } 1 \\textrm{:}$ $a+b = 16$ and $c+d = 24$.\n List out possib... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(xy) + f(xz) - f(x) f(yz) \\ge 1\\]for all real numbers $x,$ $y,$ and $z.$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(x + f(y)) = x + y\\]for all real numbers $x$ and $y.$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(x + y) f(x - y) = (f(x) + f(y))^2 - 4x^2 f(y)\\]for all real numbers $x$ and $y.$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 10224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions $f$ from $\\{0, 1, 2, 3, 4, 5, 6\\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and \\[|x - y| \\leq |f(x) - f(y)| \\leq 3|x - y|\\] for all $x$ and $y$ in $\\{0, 1, 2, 3, 4, 5, 6\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "185"
} | {
"index": 10225,
"solution": "Because $f(n)$ and $f(n+1)$ can differ by at most 3 for $n=0,1,2,3,4,5$, $f$ can decrease at most once. This is because if $f$ decreases $2$ times (or more) then we get a contradiction: $12 = f(6) \\le 3\\cdot 4 + (-1)\\cdot 2 = 10$.\n If $f$ never decreases, then $f(n+1)-f(n)\\in \\{... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions $f(n),$ taking the integers to the integers, such that\n\\[f(a + b) + f(ab) = f(a) f(b) + 1\\]for all integers $a$ and $b.$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 10226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions $f(x)$ from $\\{1, 2, 3, 4, 5\\}$ to $\\{1, 2, 3, 4, 5\\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\\{1, 2, 3, 4, 5\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "756"
} | {
"index": 10227,
"solution": "Just to visualize solution 1. If we list all possible $(x,f(x))$, from ${1,2,3,4,5}$ to ${1,2,3,4,5}$ in a specific order, we get $5 \\cdot 5 = 25$ different $(x,f(x))$ 's. Namely:\n\\[(1,1) (1,2) (1,3) (1,4) (1,5)\\] \\[(2,1) (2,2) (2,3) (2,4) (2,5)\\] \\[(3,1) (3,2) (3,3) (3,4) (3,5... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions of the form $f(x) = ax^2 + bx + c$ such that\n\\[f(x) f(-x) = f(x^2).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that\n\\[f(x) f(-x) = f(x^3).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of functions of the form \\( f(x) = ax^3 + bx^2 + cx + d \\) such that\n\\[ f(x)f(-x) = f(x^3). \\]"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integer points that satisfy the system of inequalities:\n\\[\n\\begin{cases}\ny \\leqslant 3x \\\\\ny \\geqslant \\frac{1}{3}x \\\\\nx + y \\leqslant 100\n\\end{cases}\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2551"
} | {
"index": 10231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\\log(kx)=2\\log(x+2)$ has exactly one real solution."
}
] | math | {
"style": "rule",
"ground_truth": "501"
} | {
"index": 10232,
"solution": "[asy] Label f; f.p=fontsize(5); xaxis(-3,3,Ticks(f,1.0)); yaxis(-3,26,Ticks(f,1.0)); real f(real x){return (x+2)^2;} real g(real x){return x*-1;} real h(real x){return x*-2;} real i(real x){return x*-3;} real j(real x){return x*8;} draw(graph(f,-2,3),green); draw(graph(g,-2,2),red); d... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $ c$ such that $ \\minus{}2007 \\leq c \\leq 2007$ and there exists an integer $ x$ such that $ x^2 \\plus{} c$ is a multiple of $ 2^{2007}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "670"
} | {
"index": 10233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $a$ with $1\\le a\\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation\n\\[x^2(x^2+2z) - y^2(y^2+2z)=a.\\]\n\n*Ray Li.*\n\n<details><summary>Clarifications</summary>[list=1][*] $x,y,z$ are not necessarily distin... | math | {
"style": "rule",
"ground_truth": "1257"
} | {
"index": 10234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $n$ with $1\\le n\\le 100$ for which $n-\\phi(n)$ is prime. Here $\\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$ .\n\n*Proposed by Mehtaab Sawhney*"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 10235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $c$ such that the equation \\[\\left||20|x|-x^2|-c\\right|=21\\]has $12$ distinct real solutions."
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 10236,
"solution": "Graph $y=|20|x|-x^2|$ (If you are having trouble, look at the description in the next two lines and/or the diagram in Solution 1). Notice that we want this to be equal to $c-21$ and $c+21$.\n We see that from left to right, the graph first dips from very positive to $0$ at $x=-20$, th... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $n$ greater than 1 such that for any integer $a$, $n$ divides $a^{25} - a$."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 10237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $n$ such that $$ 1+\\left\\lfloor\\frac{100 n}{101}\\right\\rfloor=\\left\\lceil\\frac{99 n}{100}\\right\\rceil $$"
}
] | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"index": 10238,
"solution": "Consider $f(n)=\\left\\lceil\\frac{99 n}{100}\\right\\rceil-\\left\\lfloor\\frac{100 n}{101}\\right\\rfloor$. Note that $f(n+10100)=\\left\\lceil\\frac{99 n}{100}+99 \\cdot 101\\right\\rceil-\\left\\lfloor\\frac{100 n}{101}+100^{2}\\right\\rfloor=f(n)+99 \\cdot 101-100^{2}=f(n)-1$. Th... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $n$ such that \\[1+\\left\\lfloor\\dfrac{100n}{101}\\right\\rfloor=\\left\\lceil\\dfrac{99n}{100}\\right\\rceil.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"index": 10239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $n$ that satisfy\n\\[10 < n^2 < 99.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $n$ that satisfy:\n\\[50 < n^2 < 200.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 10241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $n$ with $1 \\leq n \\leq 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of 1001."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 10242,
"solution": "Note that $1001=7 \\cdot 11 \\cdot 13$, so the stated product must be a multiple of 7, as well as a multiple of 11, as well as a multiple of 13. There are 4 possible residues of $n$ modulo 11 for which the product is a multiple of 11; similarly, there are 4 possible residues of $n$ mo... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers $x$ such that the following three conditions all hold: - $x$ is a multiple of 5 - $121<x<1331$ - When $x$ is written as an integer in base 11 with no leading 0 s (i.e. no 0 s at the very left), its rightmost digit is strictly greater than its leftmos... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 10243,
"solution": "We will work in base 11, so let $x=\\overline{\\operatorname{def}}_{11}$ such that $d>0$. Then, based on the first two conditions, we aim to find multiples of 5 between $100_{11}$ and $1000_{11}$. We note that $$\\overline{d e f}_{11} \\equiv 11^{2} \\cdot d+11 \\cdot e+f \\equiv d+e+... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers \\( n \\) that satisfy\n\\[ 15 < n^2 < 120. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 10244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers \\( n \\) that satisfy\n\\[ 20 < n^2 < 200. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 10245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers between 1 and 2013 with the property that the sum of its digits equals 9."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 10246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of integers from 1 to 1000 inclusive that give the same remainder when divided by 11 and by 12."
}
] | math | {
"style": "rule",
"ground_truth": "87"
} | {
"index": 10247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of natural numbers \\( k \\) not exceeding 242400, such that \\( k^2 + 2k \\) is divisible by 303."
}
] | math | {
"style": "rule",
"ground_truth": "3200"
} | {
"index": 10248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of natural numbers \\( k \\) not exceeding 353500 such that \\( k^{2} + k \\) is divisible by 505."
}
] | math | {
"style": "rule",
"ground_truth": "2800"
} | {
"index": 10249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of natural numbers \\( k \\) that do not exceed 291000 and such that \\( k^{2} - 1 \\) is divisible by 291."
}
] | math | {
"style": "rule",
"ground_truth": "4000"
} | {
"index": 10250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of natural numbers \\( k \\), not exceeding 333300, such that \\( k^{2} - 2k \\) is exactly divisible by 303."
}
] | math | {
"style": "rule",
"ground_truth": "4400"
} | {
"index": 10251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of natural numbers not exceeding 2022 and not belonging to either the arithmetic progression \\(1, 3, 5, \\ldots\\) or the arithmetic progression \\(1, 4, 7, \\ldots\\)."
}
] | math | {
"style": "rule",
"ground_truth": "674"
} | {
"index": 10252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$ ."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 10253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of non-positive integers for which the values of the quadratic polynomial \\(2x^2 + 2021x + 2019\\) are non-positive."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 10254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of odd digits in the base-4 representation of $233_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 10255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered $64$-tuples $(x_0,x_1,\\dots,x_{63})$ such that $x_0,x_1,\\dots,x_{63}$ are distinct elements of $\\{1,2,\\dots,2017\\}$ and \\[ x_0 + x_1 + 2x_2 + 3x_3 + \\cdots + 63 x_{63} \\] is divisible by 2017."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{2016!}{1953!}- 63! \\cdot 2016$"
} | {
"index": 10256,
"solution": "The desired count is $\\frac{2016!}{1953!}- 63! \\cdot 2016$, which we compute using the principle of inclusion-exclusion. As in A2, we use the fact that 2017 is prime; this means that we can do linear algebra over the field \\mathbb{F}_{2017}. In particular, every nonzero homogeneous... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered 17-tuples $(a_1, a_2, a_3, \\dots, a_{17})$ of integers, such that the square of any number in the 17-tuple is equal to the sum of the other 16 numbers."
}
] | math | {
"style": "rule",
"ground_truth": "12378"
} | {
"index": 10257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered integer pairs \\((a, b)\\) such that the equation \\(x^{2} + a x + b = 167 y\\) has integer solutions \\((x, y)\\), where \\(1 \\leq a, b \\leq 2004\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2020032"
} | {
"index": 10258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered integer pairs \\((a, b)\\) such that the equation \\(x^{2} + a x + b = 167y\\) has integer solutions \\((x, y)\\), where \\(1 \\leq a, b \\leq 2004\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2020032"
} | {
"index": 10259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered integer tuples \\((k_1, k_2, k_3, k_4)\\) that satisfy \\(0 \\leq k_i \\leq 20\\) for \\(i = 1, 2, 3, 4\\), and \\(k_1 + k_3 = k_2 + k_4\\)."
}
] | math | {
"style": "rule",
"ground_truth": "6181"
} | {
"index": 10260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs $(A, B)$ such that the following conditions hold: $A$ and $B$ are disjoint subsets of $\\{1,2, \\ldots, 50\\}$, $|A|=|B|=25$, and the median of $B$ is 1 more than the median of $A$."
}
] | math | {
"style": "rule",
"ground_truth": "\\binom{24}{12}^{2}"
} | {
"index": 10261,
"solution": "The median of both sets, which we will call $a$ and $b$ respectively, are more than exactly 12 of the members in their own set. Since $a$ and $b$ are consecutive, they must also be higher than the lower half of the other set and lower than the higher half of the other set, meaning tha... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs $(a,b)$ of complex numbers such that\n\\[a^3 b^5 = a^7 b^2 = 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 10262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs $(a,b)$ of complex numbers such that\n\\[a^4 b^6 = a^8 b^3 = 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 10263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\\{1, 2, ..., 30\\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$."
}
] | math | {
"style": "rule",
"ground_truth": "295"
} | {
"index": 10264,
"solution": "This solution refers to the Remarks section.\n By the Euclidean Algorithm, we have \\[\\gcd\\left(2^m+1,2^m-1\\right)=\\gcd\\left(2,2^m-1\\right)=1.\\] We are given that $\\gcd\\left(2^m+1,2^n-1\\right)>1.$ Multiplying both sides by $\\gcd\\left(2^m-1,2^n-1\\right)$ gives \\begin{alig... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x \\le 2y \\le 60$ and $y \\le 2x \\le 60$."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 10265,
"solution": "$y\\le2x\\le60$\n Multiplying both sides by 2 yields:\n$2y\\le4x\\le120$\n Then the two inequalities can be merged to form the following inequality:\n$x\\le2y\\le4x\\le120$\n Additionally, we must ensure that $2y<60$\n Therefore we must find pairs $(x,y)$ that satisfy the inequality a... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs $(x,y)$ of real numbers such that\n\\[16^{x^2 + y} + 16^{x + y^2} = 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs \\((a, b)\\) of positive integers such that \\(a\\) and \\(b\\) both divide \\(20^{19}\\), but \\(ab\\) does not."
}
] | math | {
"style": "rule",
"ground_truth": "444600"
} | {
"index": 10267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality\n\n\\[\n1 < a < b+2 < 10.\n\\]\n\n*Proposed by Lewis Chen*"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 10268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of integers $(a, b) \\in\\{1,2, \\ldots, 35\\}^{2}$ (not necessarily distinct) such that $a x+b$ is a \"quadratic residue modulo $x^{2}+1$ and 35 \", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the followi... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 10269,
"solution": "By the Chinese remainder theorem, we want the product of the answers modulo 5 and modulo 7 (i.e. when 35 is replaced by 5 and 7, respectively). First we do the modulo 7 case. Since $x^{2}+1$ is irreducible modulo 7 (or more conceptually, in $\\mathbb{F}_{7}[x]$ ), exactly half of the ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $a b$ is not."
}
] | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"index": 10270,
"solution": "First consider the case $a, b>0$. We have $720=2^{4} \\cdot 3^{2} \\cdot 5$, so the number of divisors of 720 is $5 * 3 * 2=30$. We consider the number of ways to select an ordered pair $(a, b)$ such that $a, b, a b$ all divide 720. Using the balls and urns method on each of the prime... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of integers $(a, b)$ such that the sequence\\[3, 4, 5, a, b, 30, 40, 50\\]is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression."
}
] | math | {
"style": "rule",
"ground_truth": "228"
} | {
"index": 10271,
"solution": "divide cases into $7\\leq a<20; 21\\leq a\\leq28$.(Notice that $a$ can't be equal to $6,20$, that's why I divide them into two parts. There are three cases that arithmetic sequence forms: $3,12,21,30;4,16,28,40;3,5,7,9$.(NOTICE that $5,20,35,50$ IS NOT A VALID SEQUENCE!) So when $7\\l... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of integers $(a,b)$ with $1 \\leq a \\leq 100$ and $b \\geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "2600"
} | {
"index": 10272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 10273,
"solution": "Solving for $m$ gives us $m = \\frac{503-3n}{5},$ so in order for $m$ to be an integer, we must have $3n \\equiv 503 \\mod 5 \\longrightarrow n \\equiv 1 \\mod 5.$ The smallest possible value of $n$ is obviously $1,$ and the greatest is $\\frac{503 - 5}{3} = 166,$ so the total number ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 10274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit."
}
] | math | {
"style": "rule",
"ground_truth": "738"
} | {
"index": 10275,
"solution": "There are $\\left\\lfloor\\frac{999}{10}\\right\\rfloor = 99$ numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities are when $a$ or $b$ have a 0 in the tens digit, and since the equation is symmetric, we will just count when $a$ has a 0 in the te... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$."
}
] | math | {
"style": "rule",
"ground_truth": "231"
} | {
"index": 10276,
"solution": "In this problem, we want to find the number of ordered pairs $(m, n)$ such that $m^2n = 20^{20}$. Let $x = m^2$. Therefore, we want two numbers, $x$ and $n$, such that their product is $20^{20}$ and $x$ is a perfect square. Note that there is exactly one valid $n$ for a unique $x$, wh... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \\leq 2020$ such that $3 x^{2}+10 x y+3 y^{2}$ is the power of some prime."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 10277,
"solution": "We can factor as $(3 x+y)(x+3 y)$. If $x \\geq y$, we need $\\frac{3 x+y}{x+3 y} \\in\\{1,2\\}$ to be an integer. So we get the case where $x=y$, in which we need both to be a power of 2, or the case $x=5 y$, in which case we need $y$ to be a power of 2. This gives us $11+9+9=29$ solu... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\\{1, 2, 3, 4, 5, 6, 7\\}$ and $3abc + 4abd + 5bcd$ is even. For example, $(2, 2, 5, 1)$ and $(3, 1, 4, 6)$ satisfy the conditions."
... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 10278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that\n\\begin{align*}\na^2 + b^2 + c^2 + d^2 &= 4, \\\\\n(a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16.\n\\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 10279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that\n\\[\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}^{-1} = \\renewcommand{\\arraystretch}{1.5} \\begin{pmatrix} \\frac{1}{a} & \\frac{1}{b} \\\\ \\frac{1}{c} & \\frac{1}{d} \\end{pmatrix} \\renewcomman... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 10280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered quadruples \\((a,b,c,d)\\) of nonnegative real numbers such that\n\\[\na^2 + b^2 + c^2 + d^2 = 9,\n\\]\n\\[\n(a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 81.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 10281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered quadruples of positive integers $(a, b, c, d)$ such that $a, b, c$, and $d$ are all (not necessarily distinct) factors of 30 and $abcd>900$."
}
] | math | {
"style": "rule",
"ground_truth": "1940"
} | {
"index": 10282,
"solution": "Since $abcd>900 \\Longleftrightarrow \\frac{30}{a} \\frac{30}{b} \\frac{30}{c} \\frac{30}{d}<900$, and there are $\\binom{4}{2}^{3}$ solutions to $abcd=2^{2} 3^{2} 5^{2}$, the answer is $\\frac{1}{2}\\left(8^{4}-\\binom{4}{2}^{3}\\right)=1940$ by symmetry."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered quintuples $(a,b,c,d,e)$ of nonnegative real numbers such that:\n\\begin{align*}\na^2 + b^2 + c^2 + d^2 + e^2 &= 5, \\\\\n(a + b + c + d + e)(a^3 + b^3 + c^3 + d^3 + e^3) &= 25.\n\\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 10283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive integers, $a$ is a factor of $b$, $a$ is a factor of $c$, and $a+b+c=100$."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 10284,
"solution": "Denote $x = \\frac{b}{a}$ and $y = \\frac{c}{a}$. The last condition reduces to $a(1 + x + y) = 100$. Therefore, $1 + x + y$ is equal to one of the 9 factors of $100 = 2^25^2$.\n Subtracting the one, we see that $x + y = \\{0,1,3,4,9,19,24,49,99\\}$. There are exactly $n - 1$ ways to ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered triples of divisors $(d_{1}, d_{2}, d_{3})$ of 360 such that $d_{1} d_{2} d_{3}$ is also a divisor of 360."
}
] | math | {
"style": "rule",
"ground_truth": "800"
} | {
"index": 10286,
"solution": "Since $360=2^{3} \\cdot 3^{2} \\cdot 5$, the only possible prime divisors of $d_{i}$ are 2,3 , and 5 , so we can write $d_{i}=2^{a_{i}} \\cdot 3^{b_{i}} \\cdot 5^{c_{i}}$, for nonnegative integers $a_{i}, b_{i}$, and $c_{i}$. Then, $d_{1} d_{2} d_{3} \\mid 360$ if and only if the foll... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered triples of integers $(a, b, c)$ with $1 \\leq a, b, c \\leq 100$ and $a^{2} b+b^{2} c+c^{2} a=a b^{2}+b c^{2}+c a^{2}$"
}
] | math | {
"style": "rule",
"ground_truth": "29800"
} | {
"index": 10287,
"solution": "This factors as $(a-b)(b-c)(c-a)=0$. By the inclusion-exclusion principle, we get $3 \\cdot 100^{2}-3 \\cdot 100+100=29800$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy $(ab+1)(bc+1)(ca+1)=84$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10288,
"solution": "The solutions are $(0,1,83)$ and $(1,2,3)$ up to permutation. First, we do the case where at least one of $a, b, c$ is 0. WLOG, say $a=0$. Then we have $1+bc=84 \\Longrightarrow bc=83$. As 83 is prime, the only solution is $(0,1,83)$ up to permutation. Otherwise, we claim that at leas... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered triples of positive integers $(a, b, c)$ such that $6a+10b+15c=3000$."
}
] | math | {
"style": "rule",
"ground_truth": "4851"
} | {
"index": 10289,
"solution": "Note that $6a$ must be a multiple of 5, so $a$ must be a multiple of 5. Similarly, $b$ must be a multiple of 3, and $c$ must be a multiple of 2. Set $a=5A, b=3B, c=2C$. Then the equation reduces to $A+B+C=100$. This has $\\binom{99}{2}=4851$ solutions."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that\n1. each of $T_1, T_2$ , and $T_3$ is a subset of $\\{1, 2, 3, 4\\}$ ,\n2. $T_1 \\subseteq T_2 \\cup T_3$ ,\n3. $T_2 \\subseteq T_1 \\cup T_3$ , and\n4. $T_3\\subseteq T_1 \\cup T_2$ ."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"index": 10290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs $(n,C)$ of positive integers such that $C\\leq 100$ and $n^2+n+C$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 10291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\\cdot 2 \\cdot 3\\cdot ... \\cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \\cdot 2^2 \\cdot 3^2\\cdot ... \\cdot 50^2$."
... | math | {
"style": "rule",
"ground_truth": "32768"
} | {
"index": 10292,
"solution": "\nTo solve this problem, we need to examine the conditions given for the pairs \\((a, b)\\) of positive integers:\n1. The greatest common divisor (GCD) of \\(a\\) and \\(b\\) is \\(1 \\cdot 2 \\cdot 3 \\cdot \\ldots \\cdot 50\\).\n2. The least common multiple (LCM) of \\(a\\) and \\(b... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 10293,
"solution": "We start by analyzing the given equation:\n\\[ m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1. \\]\n\nWe can rewrite the left-hand side as:\n\\[ m^3 + 6m^2 + 5m = m(m^2 + 6m + 5) = m(m+1)(m+5). \\]\nThis factorization is achieved by recognizing that $m^2 + 6m + 5$ can be factored as $(m+1)(... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs $(m,n)$ of positive integers with $1\\le m<n\\le 30$ such that there exists a real number $x$ satisfying \\[\\sin(mx)+\\sin(nx)=2.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 10294,
"solution": "We know that the range of $sin$ is between $-1$ and $1$.\n Thus, the only way for the sum to be $2$ is for $sin$ of $mx$ and $nx$ to both be $1$.\n The $sin$ of $(90+360k)$ is equal to 1.\n Assuming $mx$ and $nm$ are both positive, m and n could be $1,5,9,13,17,21,25,29$. There are $8... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs of integers $x, y$ with different parities such that $\\frac{1}{x}+\\frac{1}{y} = \\frac{1}{2520}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 10295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs of integers $(x, y)$ such that $x^{2}+2y^{2}<25$."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 10296,
"solution": "We do casework on $y$. If $y=0$, we have $x^{2}<25$, so we get 9 values of $x$. If $y=\\pm 1$, then $x^{2}<23$, so we still have 9 values of $x$. If $y=\\pm 2$, we have $x^{2}<17$, so we have 9 values of $x$. If $y=\\pm 3$, we have $x^{2}<7$, we get 5 values of $x$. Therefore, the fin... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs of integers \\((a, b)\\) with \\(1 \\leq a<b \\leq 57\\) such that \\(a^{2}\\) has a smaller remainder than \\(b^{2}\\) when divided by 57."
}
] | math | {
"style": "rule",
"ground_truth": "738"
} | {
"index": 10297,
"solution": "There are no such pairs when \\(b=57\\), so we may only consider pairs with \\(1 \\leq a<b \\leq 56\\). The key idea is that unless \\(a^{2} \\bmod 57=b^{2} \\bmod 57,(a, b)\\) can be paired with \\((57-b, 57-a)\\) and exactly one of them satisfies \\(a^{2} \\bmod 57<b^{2} \\bmod 57\\... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs of integers \\((x ; y)\\) that satisfy the equation \\(y^{2} - xy = 700000000\\)."
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 10298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs of natural numbers \\((x, y)\\) such that \\(1 \\leq x, y \\leq 1000\\) and \\(x^2 + y^2\\) is divisible by 5."
}
] | math | {
"style": "rule",
"ground_truth": "200000"
} | {
"index": 10299,
"solution": ""
} |
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