data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Determine the least possible value of $f(1998),$ where $f:\\Bbb{N}\\to \\Bbb{N}$ is a function such that for all $m,n\\in {\\Bbb N}$, \n\n\\[f\\left( n^{2}f(m)\\right) =m\\left( f(n)\\right) ^{2}. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 7700,
"solution": "\nTo find the least possible value of \\( f(1998) \\), where \\( f: \\mathbb{N} \\to \\mathbb{N} \\) satisfies the functional equation\n\n\\[\nf\\left( n^{2}f(m)\\right) = m\\left( f(n)\\right) ^{2}\n\\]\n\nfor all \\( m, n \\in \\mathbb{N} \\), we begin by analyzing the given equation... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the least possible value of \\((x+2)(x+3)(x+4)(x+5) + 2024\\) where \\(x\\) is a real number."
}
] | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"index": 7701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.\n\n[hide=\"Note\"]Note. Here (and generally in MathLinks) natural numbers supposed to be positive.[/hide]"
}
] | math | {
"style": "rule",
"ground_truth": "$n=7920, 7921, 7922, 7923, 7924$"
} | {
"index": 7702,
"solution": "\nTo find the least possible value of the natural number \\( n \\) such that \\( n! \\) ends in exactly 1987 zeros, we need to determine the number of trailing zeros of a factorial. The number of trailing zeros of \\( n! \\) is given by the sum of the floor divisions of \\( n \\) by po... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the least real number $M$ such that the inequality \\[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \\leq M(a^{2}+b^{2}+c^{2})^{2}\\] holds for all real numbers $a$, $b$ and $c$."
}
] | math | {
"style": "rule",
"ground_truth": "M=\\frac 9{16\\sqrt 2}"
} | {
"index": 7703,
"solution": "\nTo find the least real number \\( M \\) such that the inequality \n\n\\[\n|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \\leq M(a^{2}+b^{2}+c^{2})^{2}\n\\]\n\nholds for all real numbers \\( a, b, \\) and \\( c \\), we proceed as follows:\n\n### Step 1: Expression Expansion\n\nFir... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the length of $BC$ in an acute triangle $ABC$ with $\\angle ABC = 45^{\\circ}$ , $OG = 1$ and $OG \\parallel BC$ . (As usual $O$ is the circumcenter and $G$ is the centroid.)"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the length of side $PQ$ in the right-angled triangle $PQR$, where $PR = 15$ units and $\\angle PQR = 45^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the length of side $x$ in the following diagram:\n\n[asy]\nimport olympiad;\n\ndraw((0,0)--(2,0)--(0,2*sqrt(2))--cycle);\ndraw((0,0)--(-2,0)--(0,2*sqrt(2))--cycle);\nlabel(\"10\",(-1, sqrt(2)),NW);\nlabel(\"$x$\",(sqrt(2),sqrt(2)),NE);\ndraw(\"$30^{\\circ}$\",(2.2,0),... | math | {
"style": "rule",
"ground_truth": "\\frac{20\\sqrt{3}}{3}"
} | {
"index": 7706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the length of the interval of solutions of the inequality $a \\le 3x + 6 \\le b$ where the length of the interval is $15$."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 7707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the longest side of the polygon formed by the system:\n$$\n\\begin{cases}\nx + y \\leq 4 \\\\\nx + 2y \\geq 4 \\\\\nx \\geq 0 \\\\\ny \\geq 0\n\\end{cases}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 7708,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximal size of a set of positive integers with the following properties:\n\n(1) The integers consist of digits from the set \\(\\{1,2,3,4,5,6\\}\\).\n\n(2) No digit occurs more than once in the same integer.\n\n(3) The digits in each integer are in increasing ord... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 7709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximal size of a set of positive integers with the following properties:\n\n1. The integers consist of digits from the set {1,2,3,4,5,6}.\n2. No digit occurs more than once in the same integer.\n3. The digits in each integer are in increasing order.\n4. Any two i... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 7710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximal size of a set of positive integers with the following properties: $1.$ The integers consist of digits from the set $\\{ 1,2,3,4,5,6\\}$ . $2.$ No digit occurs more than once in the same integer. $3.$ The digits in each integer are in increasing order. ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 7711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum and minimum values of the function $f(x)=x^3 - \\frac{3}{2}x^2 + 5$ on the interval $[-2, 2]$."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 7712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum difference between the \\(y\\)-coordinates of the intersection points of the graphs \\(y=5-x^2+2x^3\\) and \\(y=3+2x^2+2x^3\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{6}}{9}"
} | {
"index": 7713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum integer $ n $ such that for each positive integer $ k \\le \\frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 7714,
"solution": "\nWe need to determine the maximum integer \\( n \\) such that for each positive integer \\( k \\leq \\frac{n}{2} \\), there are two positive divisors of \\( n \\) with difference \\( k \\).\n\nTo solve this, we begin by considering the divisors of \\( n \\). Let the divisors be \\( d_... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum integer value of the expression \n\\[\\frac{3x^2 + 9x + 28}{3x^2 + 9x + 7}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 7715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum number of bishops that we can place in a $8 \\times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop. \n\nNote: A bishop threatens another one, if both are placed in different cells, in... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7716,
"solution": "To solve this problem, we need to determine the maximum number of bishops that can be placed on an \\(8 \\times 8\\) chessboard such that each bishop is threatened by at most one other bishop. The key here is to understand the attacking capability of bishops on a chessboard.\n\nBishops... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum number of different sets consisting of three terms that form arithmetic progressions and can be chosen from a sequence of real numbers \\( a_1, a_2, \\ldots, a_{101} \\), where\n\\[ \na_1 < a_2 < a_3 < \\cdots < a_{101} .\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"index": 7717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum number of elements in the set \\( S \\) that satisfy the following conditions:\n1. Each element in \\( S \\) is a positive integer not exceeding 100;\n2. For any two different elements \\( a \\) and \\( b \\) in \\( S \\), there exists another element \\( ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 7718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum possible value of \\[\\frac{\\left(x^2+5x+12\\right)\\left(x^2+5x-12\\right)\\left(x^2-5x+12\\right)\\left(-x^2+5x+12\\right)}{x^4}\\] over all non-zero real numbers $x$ .\n\n*2019 CCA Math Bonanza Lightning Round #3.4*"
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 7719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum possible value of the expression\n\n$$\n27abc + a\\sqrt{a^2 + 2bc} + b\\sqrt{b^2 + 2ca} + c\\sqrt{c^2 + 2ab}\n$$\n\nwhere \\(a, b, c\\) are positive real numbers such that \\(a + b + c = \\frac{1}{\\sqrt{3}}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3 \\sqrt{3}}"
} | {
"index": 7720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$."
}
] | math | {
"style": "rule",
"ground_truth": "\\boxed{987^2+1597^2}"
} | {
"index": 7721,
"solution": "\nWe are tasked with finding the maximum value of \\( m^2 + n^2 \\), where \\( m \\) and \\( n \\) are integers within the range \\( 1, 2, \\ldots, 1981 \\), satisfying the equation:\n\n\\[\n(n^2 - mn - m^2)^2 = 1.\n\\]\n\n### Step 1: Analyze the Equation\n\nThe equation given is a Pel... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the maximum value of the sum\n\\[S = \\sum_{n=1}^\\infty \\frac{n}{2^n} (a_1 a_2 \\cdots a_n)^{1/n}\\]\nover all sequences $a_1, a_2, a_3, \\cdots$ of nonnegative real numbers satisfying\n\\[\\sum_{k=1}^\\infty a_k = 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 7722,
"solution": "The answer is $2/3$. \n\nBy AM-GM, we have\n\\begin{align*}\n2^{n+1}(a_1\\cdots a_n)^{1/n} &= \\left((4a_1)(4^2a_2)\\cdots (4^na_n)\\right)^{1/n}\\\\\n& \\leq \\frac{\\sum_{k=1}^n (4^k a_k)}{n}.\n\\end{align*}\nThus\n\\begin{align*}\n2S &\\leq \\sum_{n=1}^\\infty \\frac{\\sum_{k=1}^n (... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the median and the mean of the set \\(\\{m, m+2, m+4, m+11, m+18\\}\\) where the median of the set is \\(16\\)."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 7723,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum number of digits to the right of the decimal point required to express the fraction $\\frac{987654321}{2^{30} \\cdot 5^5}$ as a decimal."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 7724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum of the expression\n\n$$\n\\frac{2}{|a-b|}+\\frac{2}{|b-c|}+\\frac{2}{|c-a|}+\\frac{5}{\\sqrt{ab+bc+ca}}\n$$\n\nunder the conditions that \\(ab + bc + ca > 0\\), \\(a + b + c = 1\\), and \\(a, b, c\\) are distinct."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{6}"
} | {
"index": 7725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum of the following function defined in the interval $45^{\\circ}<x<90^{\\circ}$:\n\n$$\ny=\\tan x+\\frac{\\tan x}{\\sin \\left(2 x-90^{\\circ}\\right)}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"index": 7726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum\n\\[\n\\frac{a}{3b} + \\frac{b}{6c} + \\frac{c}{9a},\n\\]\nwhere \\( a, b, \\) and \\( c \\) are positive real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3\\sqrt[3]{2}}"
} | {
"index": 7727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum\n\\[\n\\frac{a}{3b} + \\frac{b}{6c} + \\frac{c}{9a},\n\\]\nwhere \\(a, b, c\\) are positive real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\cdot \\frac{1}{\\sqrt[3]{162}}"
} | {
"index": 7728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum\n\\[\n\\frac{a}{3b} + \\frac{b}{6c} + \\frac{c}{9a},\n\\]\nwhere \\(a\\), \\(b\\), and \\(c\\) are positive real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt[3]{162}}"
} | {
"index": 7729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum\n\\[\\frac{a}{2b} + \\frac{b}{4c} + \\frac{c}{8a},\\]where $a,$ $b,$ and $c$ are positive real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 7730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum\n\\[\\frac{a}{3b} + \\frac{b}{5c} + \\frac{c}{7a},\\]\nwhere $a,$ $b,$ and $c$ are positive real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt[3]{105}}"
} | {
"index": 7731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum \\[\\frac{a}{3b} + \\frac{b}{5c} + \\frac{c}{6a},\\] where \\( a, b, \\) and \\( c \\) are positive real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt[3]{90}}"
} | {
"index": 7732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\\equiv 0\\pmod{p} $ ."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 7733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the minimum value of the function $$y = \\frac {4x^{2}+2x+5}{x^{2}+x+1}$$ for \\(x > 1\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16 - 2\\sqrt{7}}{3}"
} | {
"index": 7734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the monotonicity of the function $f(x) = \\frac{x}{x^2 + 1}$ on the interval $(1, +\\infty)$, and find the maximum and minimum values of the function when $x \\in [2, 3]$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 7735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number $ABCC$ (written in decimal system) given that\n\n$$\nABCC = (DD - E) \\cdot 100 + DD \\cdot E\n$$\n\nwhere $A, B, C, D,$ and $E$ are distinct digits."
}
] | math | {
"style": "rule",
"ground_truth": "1966"
} | {
"index": 7736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of $8$ -tuples $(\\epsilon_1, \\epsilon_2,...,\\epsilon_8)$ such that $\\epsilon_1, \\epsilon_2, ..., 8 \\in \\{1,-1\\}$ and $\\epsilon_1 + 2\\epsilon_2 + 3\\epsilon_3 +...+ 8\\epsilon_8$ is a multiple of $3$ ."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 7737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of 0-1 binary sequences of ten 0's and ten 1's which do not contain three 0's together."
}
] | math | {
"style": "rule",
"ground_truth": "24068"
} | {
"index": 7738,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of 6-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 without repetition and ensuring that 1, 3, 5 are not adjacent to one another."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of 6-digit numbers composed of the digits 0, 1, 2, 3, 4, 5 without any repetition and with alternating even and odd digits."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of 8-tuples of nonnegative integers $\\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\\right)$ satisfying $0 \\leq a_{k} \\leq k$, for each $k=1,2,3,4$, and $a_{1}+a_{2}+a_{3}+a_{4}+2 b_{1}+3 b_{2}+4 b_{3}+5 b_{4}=19$."
}
] | math | {
"style": "rule",
"ground_truth": "1540"
} | {
"index": 7741,
"solution": "For each $k=1,2,3,4$, note that set of pairs $\\left(a_{k}, b_{k}\\right)$ with $0 \\leq a_{k} \\leq k$ maps bijectively to the set of nonnegative integers through the map $\\left(a_{k}, b_{k}\\right) \\mapsto a_{k}+(k+1) b_{k}$, as $a_{k}$ is simply the remainder of $a_{k}+(k+1) b_{k}... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \\in \\{1, 2, 3, \\ldots, 1000\\}$ and which are divisible by 121."
}
] | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"index": 7742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of all positive integers which cannot be written in the form $80k + 3m$ , where $k,m \\in N = \\{0,1,2,...,\\}$"
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 7743,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of angles $\\theta$ between 0 and $2 \\pi$, other than integer multiples of $\\pi / 2$, such that the quantities $\\sin \\theta, \\cos \\theta$, and $\\tan \\theta$ form a geometric sequence in some order."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7744,
"solution": "If $\\sin \\theta, \\cos \\theta$, and $\\tan \\theta$ are in a geometric progression, then the product of two must equal the square of the third. Using this criterion, we have 3 cases. - Case 1: $\\sin \\theta \\cdot \\tan \\theta=\\cos ^{2} \\theta$. This implies that $\\left(\\sin ^... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of angles between 0 and $2 \\pi,$ other than integer multiples of $\\frac{\\pi}{2},$ such that $\\sin \\theta,$ $\\cos \\theta$, and $\\tan \\theta$ form a geometric sequence in some order."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of arrangements of the letters a, b, c, d, e in a sequence such that neither a nor b is adjacent to c."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of decreasing sequences of positive integers \\(b_1 \\geq b_2 \\geq b_3 \\geq \\cdots \\geq b_7 \\leq 1500\\) such that \\(b_i - i\\) is divisible by 3 for \\(1 \\leq i \\le 7\\). Express the number of such sequences as \\({m \\choose n}\\) for some integer... | math | {
"style": "rule",
"ground_truth": "506"
} | {
"index": 7747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of different arrangements for assigning 6 repatriated international students to work in Jinan and Qingdao, given that at least 2 students must be assigned to Jinan and at least 3 students must be assigned to Qingdao."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 7748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of different ways to schedule volleyball, basketball, and table tennis competitions in 4 different gyms, given that each sport must be held in only one gym and that no more than 2 sports can take place in the same gym."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of digits in the product of $84,123,457,789,321,005$ and $56,789,234,567,891$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 7750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of digits in the value of $2^{12} \\times 5^8 $."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of digits in the value of $2^{15} \\times 5^{10} \\times 3^2$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of digits in the value of $2^{15} \\times 5^{11}$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of digits in the value of $2^{15} \\times 5^{12} - 10^5$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7754,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of distinct odd numbers that can be formed by rearranging the digits of the number \"34396\"."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of distinct terms in the simplified expansion of $[(x+5y)^3(x-5y)^3]^{3}$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of four-digit integers $n$ such that $n$ and $2n$ are both palindromes."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7757,
"solution": "Let $n=\\underline{a} \\underline{b} \\underline{b} \\underline{a}$. If $a, b \\leq 4$ then there are no carries in the multiplication $n \\times 2$, and $2n=(2a)(2b)(2b)(2a)$ is a palindrome. We shall show conversely that if $n$ and $2n$ are palindromes, then necessarily $a, b \\leq 4... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of functions $f: \\{1, 2, 3\\} \\rightarrow \\{1, 2, 3\\}$ satisfying the property $f(f(x)) = f(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of increasing sequences of positive integers $a_1 \\le a_2 \\le a_3 \\le \\cdots \\le a_8 \\le 1023$ such that $a_i-i$ is even for $1\\le i \\le 8$. The answer can be expressed as $\\binom{m}{n}$ for some $m > n$. Compute the remainder when $m$ is divided b... | math | {
"style": "rule",
"ground_truth": "515"
} | {
"index": 7759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of integers $2 \\leq n \\leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 7760,
"solution": "Only $n \\equiv 1(\\bmod 210)$ work. Proof: we require $\\operatorname{gcd}(n, 210)=1$. Note that $\\forall p \\leq 7$ the order of $n$ $(\\bmod p)$ divides $p-1$, hence is relatively prime to any $p \\leq 7$. So $n^{n} \\equiv 1(\\bmod p) \\Longleftrightarrow n \\equiv 1(\\bmod p)$ fo... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of integers $D$ such that whenever $a$ and $b$ are both real numbers with $-1 / 4<a, b<1 / 4$, then $\\left|a^{2}-D b^{2}\\right|<1$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 7761,
"solution": "We have $$-1<a^{2}-D b^{2}<1 \\Rightarrow \\frac{a^{2}-1}{b^{2}}<D<\\frac{a^{2}+1}{b^{2}}$$ We have $\\frac{a^{2}-1}{b^{2}}$ is maximal at $-15=\\frac{.25^{2}-1}{.25^{2}}$ and $\\frac{a^{2}+1}{b^{2}}$ is minimal at $\\frac{0^{2}+1}{.25^{2}}=16$. However, since we cannot have $a, b= \\p... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of intersection points of 10 lines, given that only two of them are parallel and exactly three of these lines intersect at one point."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 7762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 7763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure."
}
] | math | {
"style": "rule",
"ground_truth": "297"
} | {
"index": 7764,
"solution": "First, let us count the total number of rectangles in the grid without the hole in the middle. There are $\\binom{7}{2}=21$ ways to choose the two vertical boundaries of the rectangle, and there are 21 ways to choose the two horizontal boundaries of the rectangles. This makes $21^{2}=4... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of numbers between $1$ and $3000$ that are integer multiples of $5$ or $7$, but not $35$."
}
] | math | {
"style": "rule",
"ground_truth": "943"
} | {
"index": 7765,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of ordered pairs $(a,b)$ of integers such that $\\log_a b + 6\\log_b a=5, 2 \\leq a \\leq 2005,$ and $2 \\leq b \\leq 2005.$"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 7766,
"solution": "Let $k=\\log_a b$. Then our equation becomes $k+\\frac{6}{k}=5$. Multiplying through by $k$ and solving the quadratic gives us $k=2$ or $k=3$. Hence $a^2=b$ or $a^3=b$.\n For the first case $a^2=b$, $a$ can range from 2 to 44, a total of 43 values. For the second case $a^3=b$, $a$ can ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of ordered pairs $(m, n)$ that satisfy $m$ and $n \\in \\{-1,0,1,2,3\\}$, and the equation $mx^2 + 2x + n = 0$ has real solutions."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 7767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of ordered pairs of positive integers \\((a, b)\\) satisfying the equation\n\\[ 100(a + b) = ab - 100. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 7768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of pairs $(a,b)$ of real numbers such that $10, a, b, ab$ is an arithmetic progression."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 7769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of pairs \\((a, b)\\) of integers with \\(1 \\leq b < a \\leq 200\\) such that the sum \\((a+b) + (a-b) + ab + \\frac{a}{b}\\) is a square of a number."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 7770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of pairs of positive integers $x,y$ such that $x\\le y$ , $\\gcd (x,y)=5!$ and $\\text{lcm}(x,y)=50!$ ."
}
] | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"index": 7771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of palindromes between 1000 and 10000 that are multiples of 6."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of perfect cubic divisors in the product $1! \\cdot 2! \\cdot 3! \\cdot \\ldots \\cdot 6!$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of positive integers $n$ satisfying:\n\n\n\n- $n<10^6$\n- $n$ is divisible by 7\n- $n$ does not contain any of the digits 2,3,4,5,6,7,8."
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"index": 7774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of positive integers $a$ less than $12$ such that the congruence $ax\\equiv 1\\pmod{12}$ has a solution in $x$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of possible values for \\( m \\) such that the lengths of the sides of a triangle are \\( \\ln 20 \\), \\( \\ln 60 \\), and \\( \\ln m \\), and the triangle has a positive area."
}
] | math | {
"style": "rule",
"ground_truth": "1196"
} | {
"index": 7776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of prime dates in a non-leap year where the day and the month are both prime numbers, and the year has one fewer prime month than usual."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 7777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of quadratic polynomials $P(x)=p_{1} x^{2}+p_{2} x-p_{3}$, where $p_{1}, p_{2}, p_{3}$ are not necessarily distinct (positive) prime numbers less than 50, whose roots are distinct rational numbers."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 7778,
"solution": "The existence of distinct rational roots means that the given quadratic splits into linear factors. Then, since $p_{1}, p_{3}$ are both prime, we get that the following are the only possible factorizations: - $(p_{1} x-p_{3})(x+1) \\Rightarrow p_{2}=p_{1}-p_{3}$ - $(p_{1} x+p_{3})(x-1)... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of real number $a$ , such that for every $a$ , equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:\n(a) $x_0$ is an even integer;\n(b) $|x_0|<1000$ ."
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 7779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of real solutions to the equation:\n\\[\n\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{150}{x - 150} = x^2.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "151"
} | {
"index": 7780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of relatively prime dates in the month with the second fewest relatively prime dates."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 7781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of revolutions a wheel, with a fixed center and with an outside diameter of 8 feet, would require to cause a point on the rim to travel one mile."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{660}{\\pi}"
} | {
"index": 7782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of scalene triangles where all sides are integers and have a perimeter less than 20."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of six-digit palindromes."
}
] | math | {
"style": "rule",
"ground_truth": "9000"
} | {
"index": 7784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of six-letter words where the first and last letters are the same, and the second and fifth letters are also the same."
}
] | math | {
"style": "rule",
"ground_truth": "456976"
} | {
"index": 7785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of six-letter words where the first and last two letters are the same (e.g., \"aabbaa\")."
}
] | math | {
"style": "rule",
"ground_truth": "456976"
} | {
"index": 7786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of solutions to\n\\[2\\sin^3 x - 5 \\sin^2 x + 2 \\sin x = 0\\]in the range $0 \\le x \\le 2 \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 7787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of solutions to the equation\n\\[\\tan (10 \\pi \\cos \\theta) = \\cot (10 \\pi \\sin \\theta)\\]\nwhere $\\theta \\in (0, 2 \\pi).$"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 7788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of solutions to the equation\n\\[\\tan (7 \\pi \\cos \\theta) = \\cot (3 \\pi \\sin \\theta)\\] where $\\theta \\in (0, 3\\pi).$"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 7789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of subsets $S$ of $\\{1,2, \\ldots, 1000\\}$ that satisfy the following conditions: - $S$ has 19 elements, and - the sum of the elements in any non-empty subset of $S$ is not divisible by 20 ."
}
] | math | {
"style": "rule",
"ground_truth": "8 \\cdot\\binom{50}{19}"
} | {
"index": 7790,
"solution": "First we prove that each subset must consist of elements that have the same residue mod 20. Let a subset consist of elements $a_{1}, \\ldots, a_{19}$, and consider two lists of partial sums $$\\begin{aligned} & a_{1}, a_{1}+a_{2}, a_{1}+a_{2}+a_{3}, \\ldots, a_{1}+a_{2}+\\cdots+a_{19} ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of subsets $S$ of $\\{1,2,3, \\ldots, 10\\}$ with the following property: there exist integers $a<b<c$ with $a \\in S, b \\notin S, c \\in S$."
}
] | math | {
"style": "rule",
"ground_truth": "968"
} | {
"index": 7791,
"solution": "968. There are $2^{10}=1024$ subsets of $\\{1,2, \\ldots, 10\\}$ altogether. Any subset without the specified property must be either the empty set or a block of consecutive integers. To specify a block of consecutive integers, we either have just one element (10 choices) or a pair of ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of three-element subsets of the set \\(\\{1, 2, 3, 4, \\ldots, 120\\}\\) for which the sum of the three elements is a multiple of 3."
}
] | math | {
"style": "rule",
"ground_truth": "93640"
} | {
"index": 7792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of times and the positions in which it appears $\\frac12$ in the following sequence of fractions: $$ \\frac11, \\frac21, \\frac12 , \\frac31 , \\frac22 , \\frac13 , \\frac41,\\frac32,\\frac23,\\frac14,..., \\frac{1}{1992} $$"
}
] | math | {
"style": "rule",
"ground_truth": "664"
} | {
"index": 7793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of triples $0 \\leq k, m, n \\leq 100$ of integers such that $$ 2^{m} n-2^{n} m=2^{k} $$"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 7794,
"solution": "First consider when $n \\geq m$, so let $n=m+d$ where $d \\geq 0$. Then we have $2^{m}\\left(m+d-2^{d} m\\right)=$ $2^{m}\\left(m\\left(1-2^{d}\\right)+d\\right)$, which is non-positive unless $m=0$. So our first set of solutions is $m=0, n=2^{j}$. Now, we can assume that $m>n$, so let... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of unordered triples of distinct points in the $4 \\times 4 \\times 4$ lattice grid $\\{0,1,2,3\\}^{3}$ that are collinear in $\\mathbb{R}^{3}$ (i.e. there exists a line passing through the three points)."
}
] | math | {
"style": "rule",
"ground_truth": "376"
} | {
"index": 7795,
"solution": "Define a main plane to be one of the $x y, y z, z x$ planes. Define a space diagonal to be a set of collinear points not parallel to a main plane. We classify the lines as follows: (a) Lines parallel to two axes (i.e. orthogonal to a main plane). Notice that given a plane of the form $... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of ways a student can schedule four mathematics courses — algebra, geometry, number theory, and statistics — on an 8-period day, given that no two mathematics courses can be scheduled in consecutive periods."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 7796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word \"BALLOONIST\"."
}
] | math | {
"style": "rule",
"ground_truth": "907200"
} | {
"index": 7797,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word \"PERCEPTION\"."
}
] | math | {
"style": "rule",
"ground_truth": "907,200"
} | {
"index": 7798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word \"SUCCESS\"."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 7799,
"solution": ""
} |
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