data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "For every four points $P_{1},P_{2},P_{3},P_{4}$ on the plane, find the minimum value of $\\frac{\\sum_{1\\le\\ i<j\\le\\ 4}P_{i}P_{j}}{\\min_{1\\le\\ i<j\\le\\ 4}(P_{i}P_{j})}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{2}"
} | {
"index": 11700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For every integer $n \\ge 1$ , the function $f_n : \\left\\{ 0, 1, \\cdots, n \\right\\} \\to \\mathbb R$ is defined recursively by $f_n(0) = 0$ , $f_n(1) = 1$ and \\[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \\] for each $1 \\le k < n$ . Let $S_N = f_{N+1}(1) + f_{N+2}(2)... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 11701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For every integer $k$ with $k > 0$, let $R(k)$ be the probability that\n\\[\n\\left[\\frac{n}{k}\\right] + \\left[\\frac{200 - n}{k}\\right] = \\left[\\frac{200}{k}\\right]\n\\]\nfor an integer $n$ randomly chosen from the interval $1 \\leq n \\leq 199$. What is the minimum pos... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 11702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For every integer $n\\ge2$, let $\\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\\text{pow}(144)=\\text{pow}(2^4\\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides\n\n$\\prod_{n=2}^{5300}\\text{pow}(n)$?"
}
] | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 11703,
"solution": "To solve this problem, we need to determine how many times the prime factor $67$ appears in the product $\\prod_{n=2}^{5300}\\text{pow}(n)$, where $\\text{pow}(n)$ is defined as the largest power of the largest prime that divides $n$.\n\n1. **Identify the largest prime factor of 2010*... |
DeepScaleR | [
{
"role": "user",
"content": "For every integers $ a,b,c$ whose greatest common divisor is $n$ , if\n\\[ \\begin{array}{l} {x \\plus{} 2y \\plus{} 3z \\equal{} a} \n{2x \\plus{} y \\minus{} 2z \\equal{} b} \n{3x \\plus{} y \\plus{} 5z \\equal{} c} \\end{array}\n\\]\nhas a solution in integers, what is th... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 11704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For every non-empty subset of the natural number set $N^*$, we define the \"alternating sum\" as follows: arrange the elements of the subset in descending order, then start with the largest number and alternately add and subtract each number. For example, the alternating sum of... | math | {
"style": "rule",
"ground_truth": "448"
} | {
"index": 11705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For every positive integer $k$ , let $\\mathbf{T}_k = (k(k+1), 0)$ , and define $\\mathcal{H}_k$ as the homothety centered at $\\mathbf{T}_k$ with ratio $\\tfrac{1}{2}$ if $k$ is odd and $\\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that ... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 11706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For every positive integer \\( n \\), define \\( a_{n} \\) as the last digit of the sum of the digits of the number formed by writing \"2005\" \\( n \\) times consecutively. For example, \\(\\mathrm{a}_{1}=7\\) and \\(\\mathrm{a}_{2}=4\\).\n\na) What are the positive integers \... | math | {
"style": "rule",
"ground_truth": "9025"
} | {
"index": 11707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For every positive real number $x$, let\n\\[g(x) = \\lim_{r \\to 0} ((x+1)^{r+1} - x^{r+1})^{\\frac{1}{r}}.\\]\nFind $\\lim_{x \\to \\infty} \\frac{g(x)}{x}$."
}
] | math | {
"style": "rule",
"ground_truth": "e"
} | {
"index": 11708,
"solution": "The limit is $e$.\n\n\\textbf{First solution.}\nBy l'H\\^opital's Rule, we have\n\\begin{align*}\n&\\lim_{r\\to 0} \\frac{\\log((x+1)^{r+1}-x^{r+1})}{r} \\\\\n&\\quad = \\lim_{r\\to 0} \\frac{d}{dr} \\log((x+1)^{r+1}-x^{r+1}) \\\\\n&\\quad = \\lim_{r\\to 0} \\frac{(x+1)^{r+1}\\log(x+1... |
DeepScaleR | [
{
"role": "user",
"content": "For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always"
}... | math | {
"style": "rule",
"ground_truth": "-6[-W]"
} | {
"index": 11709,
"solution": "To solve this problem, we need to determine the cost of mailing a letter weighing $W$ ounces, where the rate is six cents for every ounce or portion thereof. This means that even a small fraction of an ounce over a whole number results in charging for an additional ounce.\n\n1. **Unde... |
DeepScaleR | [
{
"role": "user",
"content": "For every real number $x$, let $\\lfloor x\\rfloor$ denote the greatest integer not exceeding $x$, and let $f(x) = \\lfloor x\\rfloor(2014^{x-\\lfloor x\\rfloor}-1)$. The set of all numbers $x$ such that $1\\leq x<2014$ and $f(x)\\leq 1$ is a union of disjoint intervals. What i... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 11710,
"solution": "1. **Define the function and interval**: Given the function \\( f(x) = \\lfloor x \\rfloor (2014^{x - \\lfloor x \\rfloor} - 1) \\), we need to find the set of \\( x \\) such that \\( 1 \\leq x < 2014 \\) and \\( f(x) \\leq 1 \\).\n\n2. **Simplify the function**: Let \\( \\lfloor x \\... |
DeepScaleR | [
{
"role": "user",
"content": "For every real number $x$, what is the value of the expression $(x+1)^{2} - x^{2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "2x + 1"
} | {
"index": 11711,
"solution": "Expanding and simplifying, $(x+1)^{2} - x^{2} = (x^{2} + 2x + 1) - x^{2} = 2x + 1$."
} |
DeepScaleR | [
{
"role": "user",
"content": "For every subset $T$ of $U = \\{ 1,2,3,\\ldots,18 \\}$, let $s(T)$ be the sum of the elements of $T$, with $s(\\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\\frac{m}{n}$, where $m$ and $n... | math | {
"style": "rule",
"ground_truth": "683"
} | {
"index": 11712,
"solution": "Consider the elements of $U$ modulo $3.$\n Ignore the $0$'s because we're gonna multiply $\\binom{6}{0}+..+\\binom{6}{6}=2^6$ at the end. Let $a$ be the $1's$ and $b$ be the $2's$. The key here is that $2 \\equiv -1 \\pmod{3}$ so the difference between the number of $a$ and $b$ is a m... |
DeepScaleR | [
{
"role": "user",
"content": "For finite sets $A$ and $B$ , call a function $f: A \\rightarrow B$ an \\emph{antibijection} if there does not exist a set $S \\subseteq A \\cap B$ such that $S$ has at least two elements and, for all $s \\in S$ , there exists exactly one element $s'$ of $S$ such ... | math | {
"style": "rule",
"ground_truth": "1363641"
} | {
"index": 11713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For functions $f(x)$ and $g(x)$, let $m\\in \\{x|f(x)=0\\}$, $n\\in \\{x|g(x)=0\\}$. If there exist $m$ and $n$ such that $|m-n|\\leqslant 1$, then $f(x)$ and $g(x)$ are called \"zero-point related functions\". If the functions $f(x)=e^{x-2}+\\ln(x-1)-1$ and $g(x)=x(\\ln x-ax)-... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 11714,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$ . On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for e... | math | {
"style": "rule",
"ground_truth": "4910"
} | {
"index": 11715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For his birthday, Bert gets a box that holds $125$ jellybeans when filled to capacity. A few weeks later, Carrie gets a larger box full of jellybeans. Her box is twice as high, twice as wide and twice as long as Bert's. Approximately, how many jellybeans did Carrie get?"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 11716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For his birthday, Piglet baked a big cake weighing 10 kg and invited 100 guests. Among them was Winnie-the-Pooh, who has a weakness for sweets. The birthday celebrant announced the cake-cutting rule: the first guest cuts themselves a piece of cake equal to \\(1\\%\\) of the rem... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many $n$ in $\\{1, 2, 3, ..., 100 \\}$ is the tens digit of $n^2$ odd?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11718,
"solution": "1. **Express $n$ in terms of its digits**: Let $n = 10a + b$, where $a$ is the tens digit and $b$ is the units digit of $n$. Then $n^2 = (10a + b)^2$.\n\n2. **Expand $n^2$**: \n \\[\n n^2 = (10a + b)^2 = 100a^2 + 20ab + b^2\n \\]\n - The term $100a^2$ contributes only to the h... |
DeepScaleR | [
{
"role": "user",
"content": "For how many $n$ with $1 \\leq n \\leq 100$ can a unit square be divided into $n$ congruent figures?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 11719,
"solution": "We can divide the square into congruent rectangles for all $n$, so the answer is 100."
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many $n=2,3,4,\\ldots,109,110$ is the base-$n$ number $432143_n$ a multiple of $11$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many $n=2,3,4,\\ldots,99,100$ is the base-$n$ number $215216_n$ a multiple of $5$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many $n=2,3,4,\\ldots,99,100$ is the base-$n$ number $235236_n$ a multiple of $7$?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 11722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many (not necessarily positive) integer values of $n$ is the value of $4000 \\cdot \\left(\\frac{2}{5}\\right)^n$ an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 11723,
"solution": "1. **Understanding the Expression**: We start by analyzing the expression $4000 \\cdot \\left(\\frac{2}{5}\\right)^n$. This expression can be rewritten as:\n \\[\n 4000 \\cdot \\left(\\frac{2}{5}\\right)^n = 4000 \\cdot 2^n \\cdot 5^{-n}\n \\]\n This shows that the expression ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many bases between two and nine inclusive does the representation of $576_{10}$ have a final digit of 1?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 11724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many different digits $n$ is the three-digit number $14n$ divisible by $n$?\n\nNote: $14n$ refers to a three-digit number with the unit digit of $n,$ not the product of $14$ and $n.$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 11725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many different negative values of $x$ is $\\sqrt{x +150}$ a positive integer?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exactly 2004 of other lines?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many digits $C$ is the positive three-digit number $1C3$ a multiple of 3?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many even integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 4?"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 11729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many even positive integers $n$ less than or equal to 800 is $$(\\sin t - i\\cos t)^n = \\sin nt - i\\cos nt$$ true for all real $t$?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 11730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many four-digit whole numbers does the sum of the digits equal $30$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $m$ ,\n(i) $1\\le m \\le 5000$ (ii) $[\\sqrt{m}] =[\\sqrt{m+125}]$ Note: $[x]$ is the greatest integer function"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 11732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 11733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 8a = 0$$ have integer solutions for $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 11734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 11735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $b$ does there exist a polynomial function with integer coefficients such that $f(2)=2010$ and $f(b)=8$?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 11736,
"solution": "We can take $f(x)=-\\frac{2002}{d}(x-b)+2010$ for all divisors $d$ of -2002. To see that we can't get any others, note that $b-2$ must divide $f(b)-f(2)$, so $b-2$ divides -2002 (this is because $b-2$ divides $b^{n}-2^{n}$ and hence any sum of numbers of the form $b^{n}-2^{n}$)."
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\\frac{n}{1260}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 11737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\\frac{n}{1400}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "142"
} | {
"index": 11738,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\\frac{n}{2520}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 11739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 120 inclusive does the decimal representation of $\\frac{n}{120}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 11740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11741,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\\frac{n}{2940}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 11742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 349 inclusive does the decimal representation of $\\frac{n}{350}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 11743,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 474 inclusive does the decimal representation of $\\frac{n}{475}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 11744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 500 inclusive does the decimal representation of $\\frac{n}{2520}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 11745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 990 inclusive does the decimal representation of $\\frac{n}{1000}$ terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"index": 11746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 11747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of $x$ is $x^2 < 7x$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 11748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integer values of \\( n \\) between 1 and 999 inclusive does the decimal representation of \\( \\frac{n}{1000} \\) terminate?"
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 11749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $a$ with $|a| \\leq 2005$ , does the system\r $x^2=y+a$ \r $y^2=x+a$ \r\nhave integer solutions?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 11750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $1 \\leq k \\leq 2013$ does the decimal representation of $k^{k}$ end with a 1?"
}
] | math | {
"style": "rule",
"ground_truth": "202"
} | {
"index": 11751,
"solution": "We claim that this is only possible if $k$ has a units digit of 1. Clearly, it is true in these cases. Additionally, $k^{k}$ cannot have a units digit of 1 when $k$ has a units digit of $2,4,5,6$, or 8. If $k$ has a units digit of 3 or 7, then $k^{k}$ has a units digit of 1 if and onl... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $N$ between $1$ and $1990$ is the improper fraction $\\frac{N^2+7}{N+4}$ not in lowest terms?"
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 11752,
"solution": "1. **Identify the condition for $\\frac{N^2+7}{N+4}$ not being in lowest terms**: We need to find the number of integers $N$ such that $\\gcd(N^2+7, N+4) > 1$.\n\n2. **Express $N$ in terms of a divisor $m$ of $N+4**: Assume $m$ is a divisor of $N+4$. Then, $N+4 \\equiv 0 \\pmod{m}$, w... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $a$ with $1 \\leq a \\leq 10$ is $a^{2014}+a^{2015}$ divisible by 5?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 11753,
"solution": "First, we factor $a^{2014}+a^{2015}$ as $a^{2014}(1+a)$. If $a=5$ or $a=10$, then the factor $a^{2014}$ is a multiple of 5, so the original expression is divisible by 5. If $a=4$ or $a=9$, then the factor $(1+a)$ is a multiple of 5, so the original expression is divisible by 5. If $a=... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $a(1 \\leq a \\leq 200)$ is the number $a^{a}$ a square?"
}
] | math | {
"style": "rule",
"ground_truth": "107"
} | {
"index": 11754,
"solution": "107 If $a$ is even, we have $a^{a}=\\left(a^{a / 2}\\right)^{2}$. If $a$ is odd, $a^{a}=\\left(a^{(a-1) / 2}\\right)^{2} \\cdot a$, which is a square precisely when $a$ is. Thus we have 100 even values of $a$ and 7 odd square values $\\left(1^{2}, 3^{2}, \\ldots, 13^{2}\\right)$ for a... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $m$, with $1 \\leq m \\leq 30$, is it possible to find a value of $n$ so that $n!$ ends with exactly $m$ zeros?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 11755,
"solution": "If $n!$ ends with exactly $m$ zeroes, then $n!$ is divisibe by $10^{m}$ but not divisible by $10^{m+1}$. In this case, we can write $n!=10^{m} \\cdot q$ where $q$ is not divisible by 10. Since $2<5$, when $n \\geq 2$, the product $n!=1 \\cdot 2 \\cdot 3 \\cdots \\cdots(n-1) \\cdot n$ ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between $1$ and $50$, inclusive, is $\\frac{(n^2-1)!}{(n!)^n}$ an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 11756,
"solution": "1. **Understanding the Expression**: We start by analyzing the expression \\[\\frac{(n^2-1)!}{(n!)^n}.\\] We need to determine for how many integers $n$ between $1$ and $50$, inclusive, this expression is an integer.\n\n2. **Rewriting the Expression**: We can rewrite the expression as... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 100 is the greatest common divisor of 15 and $n$ equal to 3?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 11757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 11 (inclusive) is $\\frac{n}{12}$ a repeating decimal?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 11758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 150 is the greatest common divisor of 18 and $n$ equal to 6?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 11759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 150 is the greatest common divisor of 21 and $n$ equal to 3?"
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 11760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 20 (inclusive) is $\\frac{n}{18}$ a repeating decimal?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 11761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 20 (inclusive) is $\\frac{n}{42}$ a repeating decimal?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 2005, inclusive, is $2 \\cdot 6 \\cdot 10 \\cdots(4 n-2)$ divisible by $n!$?"
}
] | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"index": 11763,
"solution": "Note that $$\\begin{aligned} 2 \\cdot 6 \\cdot 10 \\cdots(4 n-2) & =2^{n} \\cdot 1 \\cdot 3 \\cdot 5 \\cdots(2 n-1) \\\\ & =2^{n} \\cdot \\frac{1 \\cdot 2 \\cdot 3 \\cdots 2 n}{2 \\cdot 4 \\cdot 6 \\cdots 2 n} \\\\ & =\\frac{1 \\cdot 2 \\cdot 3 \\cdots 2 n}{1 \\cdot 2 \\cdot 3 \\cdots... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 500 inclusive is $$(\\sin t + i\\cos t)^n = \\sin(2nt) + i\\cos(2nt)$$ true for all real $t$?"
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ is $\\frac n{20-n}$ the square of an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 11765,
"solution": "To solve the problem, we need to find the integers $n$ for which $\\frac{n}{20-n}$ is a perfect square.\n\n1. **Examine the range of $n$:**\n - If $n < 0$ or $n > 20$, the fraction $\\frac{n}{20-n}$ is negative, and thus cannot be a perfect square.\n - If $n = 20$, the fraction is... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ is it true that $\\sqrt{n} \\le \\sqrt{4n - 6} < \\sqrt{2n + 5}$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 11766,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ with $1 \\le n \\le 2012$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right) \n\\]equal to zero?"
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"index": 11767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ with $1 \\le n \\le 2016$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 2 + e^{4 \\pi i k / n} \\right)^n - 1 \\right) \n\\]\nequal to zero?"
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 11768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ with $1 \\le n \\le 2023$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right) \n\\]equal to zero, where $n$ needs to be an even multiple of $5$?"
}
] | math | {
"style": "rule",
"ground_truth": "202"
} | {
"index": 11769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ with $1 \\le n \\le 2023$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right)^2 \n\\]equal to zero?"
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 11770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$, for $1 \\leq n \\leq 1000$, is the number $\\frac{1}{2}\\binom{2 n}{n}$ even?"
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"index": 11771,
"solution": "In fact, the expression $\\binom{2 n}{n}$ is always even, and it is not a multiple of four if and only if $n$ is a power of 2, and there are 10 powers of 2 between 1 and 1000. Let $f(N)$ denote the number of factors of 2 in $N$. Thus, $$f(n!)=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $x$ does a triangle with side lengths $10, 24$ and $x$ have all its angles acute?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 11772,
"solution": "To determine the number of integers $x$ for which a triangle with sides $10, 24,$ and $x$ has all acute angles, we need to consider both the triangle inequality and the condition for all angles to be acute.\n\n#### Step 1: Apply the Triangle Inequality\nThe triangle inequality states ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $x$ is the number $x^4 - 53x^2 + 150$ negative?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $x$ is the number $x^4-51x^2+50$ negative?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11774,
"solution": "1. **Substitute and Simplify**: Let $x^2 = u$. Then the expression $x^4 - 51x^2 + 50$ becomes $u^2 - 51u + 50$.\n\n2. **Factorization**: Factor the quadratic expression:\n \\[\n u^2 - 51u + 50 = (u - 1)(u - 50)\n \\]\n This factorization is possible because the roots of the qu... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius $10$ centered at $(5, 5)$?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 11775,
"solution": "1. **Identify the equation of the circle**: The circle is centered at $(5, 5)$ with a radius of $10$. The equation of a circle with center $(h, k)$ and radius $r$ is given by:\n \\[\n (x-h)^2 + (y-k)^2 = r^2\n \\]\n Substituting $h = 5$, $k = 5$, and $r = 10$, we get:\n \\[\... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers \\( n \\) between 1 and 100 is the greatest common divisor (gcd) of 21 and \\( n \\) equal to 7?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers \\( n \\) between 1 and 15 (inclusive) is \\(\\frac{n}{18}\\) a repeating decimal?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers \\( n \\) between 1 and 20 (inclusive) is \\( \\frac{n}{18} \\) a repeating decimal?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 11778,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers \\( n \\) is \\(\\frac{2n^3 - 12n^2 - 2n + 12}{n^2 + 5n - 6}\\) equal to an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 11779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers \\( n \\), with \\( 2 \\leq n \\leq 80 \\), is \\( \\frac{(n-1)(n)(n+1)}{8} \\) equal to an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 11780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers \\(n\\) with \\(1 \\le n \\le 2020\\) is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right)\n\\]\nequal to zero?"
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 11781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many non-negative real values of $x$ is $\\sqrt{144-\\sqrt[3]{x}}$ an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 11782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many non-negative real values of $x$ is $\\sqrt{169-\\sqrt[4]{x}}$ an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 11783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 11784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many odd integers $k$ between 0 and 100 does the equation $2^{4m^{2}}+2^{m^{2}-n^{2}+4}=2^{k+4}+2^{3m^{2}+n^{2}+k}$ have exactly two pairs of positive integers $(m, n)$ that are solutions?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 11785,
"solution": "Step 1: Using parity and properties of powers of 2 to simplify the equation. We note that if $2^{x}=2^{y}$ for some real numbers $x$ and $y$, then $x=y$. We examine equations of the form $2^{a}+2^{b}=2^{c}+2^{d}$ where $a, b, c$, and $d$ are integers. We may assume without loss of gen... |
DeepScaleR | [
{
"role": "user",
"content": "For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?\n\na square\na rectangle that is not a square\na rhombus that is not a square\na parallelogram that ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11786,
"solution": "To solve this problem, we need to determine which of the given types of quadrilaterals have a point that is equidistant from all four vertices. This point is the circumcenter of the quadrilateral, and the quadrilateral must be cyclic (i.e., it can be inscribed in a circle) for such a ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many of the given drawings can the six dots be labelled to represent the links between suspects?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11787,
"solution": "Two of the five drawings can be labelled to represent the given data."
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 11788,
"solution": "To determine the number of ordered pairs $(b,c)$ of positive integers such that neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions, we analyze the discriminants of these quadratic equations.\n\n1. **Analyze the discriminants:**\n - For $x^2 + bx + c = 0$, the dis... |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered pairs $(x,y)$ of integers is it true that $0 < x < y < 10^6$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "997"
} | {
"index": 11789,
"solution": "\\begin{eqnarray*} \\frac{x+y}{2} &=& \\sqrt{xy} + 2\\\\ x+y-4 &=& 2\\sqrt{xy}\\\\ y - 2\\sqrt{xy} + x &=& 4\\\\ \\sqrt{y} - \\sqrt{x} &=& \\pm 2\\end{eqnarray*}\n Because $y > x$, we only consider $+2$.\n For simplicity, we can count how many valid pairs of $(\\sqrt{x},\\sqrt{y})$ th... |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(a, b)$ such that $a \\le 50$ is it true that $x^2 - ax + b$ has integer roots?"
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"index": 11790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$?"
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"index": 11791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 11792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(x, y)$, with $x < y$, is the harmonic mean of $x$ and $y$ equal to $9^{15}$?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 11793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(x,y),$ with $y<x\\le 100,$ are both $\\frac xy$ and $\\frac{x+1}{y+1}$ integers?"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 11794,
"solution": "We know that $x \\equiv 0 \\mod y$ and $x+1 \\equiv 0 \\mod y+1$.\n Write $x$ and $ky$ for some integer $k$. Then, $ky+1 \\equiv 0\\mod y+1$. We can add $k$ to each side in order to factor out a $y+1$. So, $ky+k+1 \\equiv k \\mod y+1$ or $k(y+1)+1 \\equiv k \\mod y+1$. We know that $k... |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered triples $(a, b, c)$ of positive integers are the equations $abc+9=ab+bc+ca$ and $a+b+c=10$ satisfied?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 11795,
"solution": "Subtracting the first equation from the second, we obtain $1-a-b-c+ab+bc+ca-abc=(1-a)(1-b)(1-c)=0$. Since $a, b$, and $c$ are positive integers, at least one must equal 1. Note that $a=b=c=1$ is not a valid triple, so it suffices to consider the cases where exactly two or one of $a, b... |
DeepScaleR | [
{
"role": "user",
"content": "For how many ordered triplets $(a, b, c)$ of positive integers less than 10 is the product $a \\times b \\times c$ divisible by 20?"
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 11796,
"solution": "One number must be 5. The other two must have a product divisible by 4. Either both are even, or one is divisible by 4 and the other is odd. In the former case, there are $48=3 \\times 4 \\times 4$ possibilities: 3 positions for the 5, and any of 4 even numbers to fill the other two. ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many pairs $(m, n)$ with $m$ and $n$ integers satisfying $1 \\leq m \\leq 100$ and $101 \\leq n \\leq 205$ is $3^{m}+7^{n}$ divisible by 10?"
}
] | math | {
"style": "rule",
"ground_truth": "2625"
} | {
"index": 11797,
"solution": "The units digits of powers of 3 cycle $3,9,7,1$ and the units digits of powers of 7 cycle $7,9,3,1$. For $3^{m}+7^{n}$ to be divisible by 10, one of the following must be true: units digit of $3^{m}$ is 3 and $7^{n}$ is 7, or 9 and 1, or 7 and 3, or 1 and 9. The number of possible pai... |
DeepScaleR | [
{
"role": "user",
"content": "For how many pairs of consecutive integers in $\\{1000,1001,1002,\\ldots,2000\\}$ is no carrying required when the two integers are added?"
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 11798,
"solution": "Consider what carrying means: If carrying is needed to add two numbers with digits $abcd$ and $efgh$, then $h+d\\ge 10$ or $c+g\\ge 10$ or $b+f\\ge 10$. 6. Consider $c \\in \\{0, 1, 2, 3, 4\\}$. $1abc + 1ab(c+1)$ has no carry if $a, b \\in \\{0, 1, 2, 3, 4\\}$. This gives $5^3=125$ po... |
DeepScaleR | [
{
"role": "user",
"content": "For how many pairs of consecutive integers in $\\{3000,3001,3002,\\ldots,4000\\}$ is no borrowing required when the first integer is subtracted from the second?"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 11799,
"solution": ""
} |
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