data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "For an arithmetic sequence $a_1,$ $a_2,$ $a_3,$ $\\dots,$ let\n\\[S_n = a_1 + a_2 + a_3 + \\dots + a_n,\\]and let\n\\[T_n = S_1 + S_2 + S_3 + \\dots + S_n.\\]If you are told the value of $S_{2019},$ then you can uniquely determine the value of $T_n$ for some integer $n.$ What ... | math | {
"style": "rule",
"ground_truth": "3028"
} | {
"index": 11500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For an arithmetic sequence $b_1, b_2, b_3, \\dots,$ let\n\\[S_n = b_1 + b_2 + b_3 + \\dots + b_n,\\]and let\n\\[T_n = S_1 + S_2 + S_3 + \\dots + S_n.\\]Given the value of $S_{2023},$ then you can uniquely determine the value of $T_n$ for some integer $n.$ What is this integer $... | math | {
"style": "rule",
"ground_truth": "3034"
} | {
"index": 11501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For an arithmetic sequence $b_1,$ $b_2,$ $b_3,$ $\\dots,$ let\n\\[P_n = b_1 + b_2 + b_3 + \\dots + b_n,\\]and let\n\\[Q_n = P_1 + P_2 + P_3 + \\dots + P_n.\\]If you are told the value of $P_{2023},$ then you can uniquely determine the value of $Q_n$ for some integer $n.$ What ... | math | {
"style": "rule",
"ground_truth": "3034"
} | {
"index": 11502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For an arithmetic sequence \\(a_1, a_2, a_3, \\dots\\), let\n\\[ S_n = a_1 + a_2 + a_3 + \\dots + a_n, \\]\nand let\n\\[ T_n = S_1 + S_2 + S_3 + \\dots + S_n. \\]\nIf you know the value of \\( S_{2023}, \\) then you can uniquely determine the value of \\( T_n \\) for some integ... | math | {
"style": "rule",
"ground_truth": "3034"
} | {
"index": 11503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, it is known that (a_2 - 1)^3 + 2014(a_2 - 1) = sin \\frac{2011\\pi}{3} and (a_{2013} - 1)^3 + 2014(a_{2013} - 1) = cos \\frac{2011\\pi}{6}. Determine the value of S_{2014}."
}
] | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"index": 11504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2}=1$ a point $P$ on the ellipse makes lines connecting it to the two foci $F1$ and $F2$ perpendicular to each other. Then, the area of $\\triangle PF1F2$ is ________."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For an even integer positive integer $n$ Kevin has a tape of length $4 n$ with marks at $-2 n,-2 n+1, \\ldots, 2 n-1,2 n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2, \\ldots, n-1, n$, and places a stone on each of these points. We call a stone 'stuck' if it is ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{n-1}"
} | {
"index": 11506,
"solution": "After we have selected the positions of the initial $n$ stones, we number their positions: $a_{1}<a_{2}<\\ldots<a_{n}$. The conditions on how we move the stones imply that the expected value of $\\left(a_{i}-a_{j}\\right)$ after $t$ minutes is still equal to $a_{i}-a_{j}$. In addition... |
DeepScaleR | [
{
"role": "user",
"content": "For an integer $m\\geq 1$, we consider partitions of a $2^m\\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectan... | math | {
"style": "rule",
"ground_truth": "2^{m+2}(m+1)"
} | {
"index": 11507,
"solution": "\nTo determine the smallest possible sum of rectangle perimeters when a \\(2^m \\times 2^m\\) chessboard is partitioned into rectangles such that each of the \\(2^m\\) cells along one diagonal is a separate rectangle, we begin by analyzing the conditions and the required configuration... |
DeepScaleR | [
{
"role": "user",
"content": "For an integer $n \\geq 0$, let $f(n)$ be the smallest possible value of $|x+y|$, where $x$ and $y$ are integers such that $3 x-2 y=n$. Evaluate $f(0)+f(1)+f(2)+\\cdots+f(2013)$."
}
] | math | {
"style": "rule",
"ground_truth": "2416"
} | {
"index": 11508,
"solution": "First, we can use $3 x-2 y=n$ to get $x=\\frac{n+2 y}{3}$. Thus $|x+y|=\\left|\\frac{n+5 y}{3}\\right|$. Given a certain $n$, the only restriction on $y$ is that $3|n+2 y \\Longleftrightarrow 3| n+5 y$. Hence the set of possible $x+y$ equals the set of integers of the form $\\frac{n+5... |
DeepScaleR | [
{
"role": "user",
"content": "For an integer $n$, let $f_{9}(n)$ denote the number of positive integers $d \\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_{1}, b_{2}, \\ldots, b_{m}$ are real numbers such that $f_{9}(n)=\\sum_{j=1}^{m} b_{j} f_{9}(n-j)$ for all $n>m$. Find the smallest ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 11509,
"solution": "Let $M=9$. Consider the generating function $$F(x)=\\sum_{n \\geq 1} f_{M}(n) x^{n}=\\sum_{d=1}^{M} \\sum_{k \\geq 1} x^{d k}=\\sum_{d=1}^{M} \\frac{x^{d}}{1-x^{d}}$$ Observe that $f_{M}(n)=f_{M}\\left(n+M\\right.$ !) for all $n \\geq 1$ (in fact, all $n \\leq 0$ as well). Thus $f_{M}... |
DeepScaleR | [
{
"role": "user",
"content": "For an integer $n>2$, the tuple $(1, 2, \\ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\\{1, 2... | math | {
"style": "rule",
"ground_truth": "n \\geq 14"
} | {
"index": 11510,
"solution": "\nGiven the problem, we start with the sequence \\( (1, 2, \\ldots, n) \\) on a blackboard. The challenge is to determine for which integers \\( n > 2 \\), it is possible to obtain every permutation of \\( \\{1, 2, \\ldots, n\\} \\) by repeatedly swapping two numbers whose sum is a pe... |
DeepScaleR | [
{
"role": "user",
"content": "For an integer $x \\geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \\ldots$ define... | math | {
"style": "rule",
"ground_truth": "142"
} | {
"index": 11511,
"solution": "\nWe are given a sequence \\( x_0, x_1, x_2, \\ldots \\) defined by \\( x_0 = 1 \\) and the recursive formula\n\n\\[\nx_{n+1} = \\frac{x_n p(x_n)}{q(x_n)}\n\\]\n\nwhere \\( p(x) \\) is the least prime that does not divide \\( x \\) and \\( q(x) \\) is the product of all primes less th... |
DeepScaleR | [
{
"role": "user",
"content": "For an upcoming holiday, the weather forecast indicates a probability of $30\\%$ chance of rain on Monday and a $60\\%$ chance of rain on Tuesday. Moreover, once it starts raining, there is an additional $80\\%$ chance that the rain will continue into the next day without inter... | math | {
"style": "rule",
"ground_truth": "72\\%"
} | {
"index": 11512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any $n\\in\\mathbb N$ , denote by $a_n$ the sum $2+22+222+\\cdots+22\\ldots2$ , where the last summand consists of $n$ digits of $2$ . Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$ ."
}
] | math | {
"style": "rule",
"ground_truth": "222"
} | {
"index": 11513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \\in \\mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \\in \\mathbb{N}$, such that $k \\mid m^{h} - 1, m \\mid n^{\\frac{m^{h}-1}{k}} + 1$."
}
] | math | {
"style": "rule",
"ground_truth": "2^{r+1}"
} | {
"index": 11514,
"solution": "\nFor any \\( h = 2^{r} \\) (where \\( r \\) is a non-negative integer), we need to find all \\( k \\in \\mathbb{N} \\) which satisfy the following condition: There exists an odd natural number \\( m > 1 \\) and \\( n \\in \\mathbb{N} \\), such that \\( k \\mid m^{h} - 1 \\) and \\( m... |
DeepScaleR | [
{
"role": "user",
"content": "For any $x \\in (0, +\\infty)$, the inequality $(x-a+\\ln \\frac{x}{a})(-2x^2+ax+10) \\leq 0$ always holds. Then, the range of the real number $a$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 11515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any $x \\in \\mathbb{R}$, the function $f(x)$ represents the minimum value among the three function values $y_{1}=4x+1$, $y_{2}=x+2$, $y_{3}=-2x+4$. The maximum value of $f(x)$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 11516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any 2016 complex numbers \\( z_1, z_2, \\ldots, z_{2016} \\), it holds that\n\\[ \n\\sum_{k=1}^{2016} |z_k|^2 \\geq \\lambda \\min_{1 \\leq k \\leq 2016} \\{ |z_{k+1} - z_k|^2 \\}, \n\\]\nwhere \\( z_{2017} = z_1 \\). Find the maximum value of \\( \\lambda \\)."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 11517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any finite sequence of positive integers \\pi, let $S(\\pi)$ be the number of strictly increasing subsequences in \\pi with length 2 or more. For example, in the sequence $\\pi=\\{3,1,2,4\\}$, there are five increasing sub-sequences: $\\{3,4\\},\\{1,2\\},\\{1,4\\},\\{2,4\\}... | math | {
"style": "rule",
"ground_truth": "8287"
} | {
"index": 11518,
"solution": "For each subset of Joy's set of cards, we compute the number of orders of cards in which the cards in the subset are arranged in increasing order. When we sum over all subsets of Joy's cards, we will obtain the desired sum. Consider any subset of $k$ cards. The probability that they a... |
DeepScaleR | [
{
"role": "user",
"content": "For any finite set $S$, let $|S|$ denote the number of elements in $S$. Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\\{1,2,3,4,5\\}$ that satisfy \\[|A| \\cdot |B| = |A \\cap B| \\cdot |A \\cup B|\\]"
}
] | math | {
"style": "rule",
"ground_truth": "454"
} | {
"index": 11519,
"solution": "The answer is \\begin{align*} \\sum_{k=0}^{5}\\left[2\\binom{5}{k}2^{5-k}-\\binom{5}{k}\\right] &= 2\\sum_{k=0}^{5}\\binom{5}{k}2^{5-k}-\\sum_{k=0}^{5}\\binom{5}{k} \\\\ &=2(2+1)^5-(1+1)^5 \\\\ &=2(243)-32 \\\\ &=\\boxed{454}. \\end{align*} ~MRENTHUSIASM"
} |
DeepScaleR | [
{
"role": "user",
"content": "For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define \\[ S_n = \\sum | A \\cap B | , \\] where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\\left\\{ 1 , 2 , 3, \\cdots , n \\right\\}$ with $|A| = |B|$. For e... | math | {
"style": "rule",
"ground_truth": "245"
} | {
"index": 11520,
"solution": "We take cases based on the number of values in each of the subsets in the pair. Suppose we have $k$ elements in each of the subsets in a pair (for a total of n elements in the set). The expected number of elements in any random pair will be $n \\cdot \\frac{k}{n} \\cdot \\frac{k}{n}$ ... |
DeepScaleR | [
{
"role": "user",
"content": "For any four-digit number $m$, if the digits of $m$ are all non-zero and distinct, and the sum of the units digit and the thousands digit is equal to the sum of the tens digit and the hundreds digit, then this number is called a \"mirror number\". If we swap the units digit and... | math | {
"style": "rule",
"ground_truth": "\\frac{{11}}{8}"
} | {
"index": 11521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any integer $a$ , let $f(a) = |a^4 - 36a^2 + 96a - 64|$ . What is the sum of all values of $f(a)$ that are prime?\n\n*Proposed by Alexander Wang*"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 11522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any integer $k\\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2.$ Let $\\{x_n\\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x... | math | {
"style": "rule",
"ground_truth": "149"
} | {
"index": 11523,
"solution": "We go through the terms and look for a pattern. We find that\n$x_0 = 1$ $x_8 = 7$\n$x_1 = 2$ $x_9 = 14$\n$x_2 = 3$ $x_{10} = 21$\n$x_3 = 6$ $x_{11} = 42$\n$x_4 = 5$ $x_{12} = 35$\n$x_5 = 10$ $x_{13} = 70$\n$x_6 = 15$ $x_{14} = 105$\n$x_7 = 30$ $x_{15} = 210$\n Commit to the bash. Even... |
DeepScaleR | [
{
"role": "user",
"content": "For any integer $n \\ge2$, we define $ A_n$ to be the number of positive integers $ m$ with the following property: the distance from $n$ to the nearest multiple of $m$ is equal to the distance from $n^3$ to the nearest multiple of $ m$. Find all integers $n \\ge 2 $ for which ... | math | {
"style": "rule",
"ground_truth": "$\\boxed{n=(2k)^2}$"
} | {
"index": 11524,
"solution": "\nLet \\( n \\) be an integer such that \\( n \\ge 2 \\). We need to find the integers \\( n \\) for which the number of positive integers \\( m \\), denoted by \\( A_n \\), is odd. The integers \\( m \\) have the property that the distance from \\( n \\) to the nearest multiple of \\... |
DeepScaleR | [
{
"role": "user",
"content": "For any integer $n$, define $\\lfloor n\\rfloor$ as the greatest integer less than or equal to $n$. For any positive integer $n$, let $$f(n)=\\lfloor n\\rfloor+\\left\\lfloor\\frac{n}{2}\\right\\rfloor+\\left\\lfloor\\frac{n}{3}\\right\\rfloor+\\cdots+\\left\\lfloor\\frac{n}{n}... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 11525,
"solution": "55 Notice that, for fixed $a,\\lfloor n / a\\rfloor$ counts the number of integers $b \\in$ $\\{1,2, \\ldots, n\\}$ which are divisible by $a$; hence, $f(n)$ counts the number of pairs $(a, b), a, b \\in$ $\\{1,2, \\ldots, n\\}$ with $b$ divisible by $a$. For any fixed $b$, the number... |
DeepScaleR | [
{
"role": "user",
"content": "For any integer $n\\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \\ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied:\n[list][*] $a_i+b_i+c_i=n$ for all $i=1, \\ldots, N(n... | math | {
"style": "rule",
"ground_truth": "\\left\\lfloor \\frac{2n}{3} \\right\\rfloor + 1"
} | {
"index": 11526,
"solution": "\nTo determine \\( N(n) \\), the maximum number of triples \\((a_i, b_i, c_i)\\) where each \\( a_i, b_i, c_i \\) are nonnegative integers satisfying the conditions:\n\n1. \\( a_i + b_i + c_i = n \\) for all \\( i = 1, \\ldots, N(n) \\),\n2. If \\( i \\neq j \\) then \\( a_i \\neq a_j... |
DeepScaleR | [
{
"role": "user",
"content": "For any interval $\\mathcal{A}$ in the real number line not containing zero, define its *reciprocal* to be the set of numbers of the form $\\frac 1x$ where $x$ is an element in $\\mathcal{A}$ . Compute the number of ordered pairs of positive integers $(m,n)$ with $m<... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 11527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any natural number $n$ , expressed in base $10$ , let $S(n)$ denote the sum of all digits of $n$ . Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$ ."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 11528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any natural values \\( m \\) and \\( n \\) (with \\( n > 1 \\)), a function is defined as \\( f\\left(\\frac{m}{n}\\right)=\\frac{\\sqrt[n]{3^{m}}}{\\sqrt[n]{3^{m}}+3} \\). Compute the sum\n\n\\[ f\\left(\\frac{1}{2020}\\right)+f\\left(\\frac{2}{2020}\\right)+f\\left(\\frac... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"index": 11529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any number $x$, we are told that $x\\&=7-x$ and $\\&x = x -7$. What is the value of $\\&(12\\&)$?"
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 11530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any number $y$, define the operations $\\&y = 2(7-y)$ and $\\&y = 2(y-7)$. What is the value of $\\&(-13\\&)$?"
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 11531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any permutation $p$ of set $\\{1, 2, \\ldots, n\\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \\ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \\leqq < j \\leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, ... | math | {
"style": "rule",
"ground_truth": "$p=(1 \\; n)$."
} | {
"index": 11532,
"solution": "\nTo solve this problem, we need to understand the relationship between \\(d(p)\\) and \\(i(p)\\) for any permutation \\(p\\) of the set \\(\\{1, 2, \\ldots, n\\}\\).\n\n### Definitions:\n- A permutation \\(p\\) of a set \\(\\{1, 2, \\ldots, n\\}\\) is a bijection from the set to itse... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $a$ , define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$ . Find all integer(s) $a$ with $1\\le a\\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 11533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$ , let $a_n=\\sum_{k=1}^{\\infty}[\\frac{n+2^{k-1}}{2^k}]$ , where $[x]$ is the largest integer that is equal or less than $x$ . Determine the value of $a_{2015}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"index": 11534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $x$ , let $f(x)=x^x$ . Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \\neq 1$ such that $f(f(f(m)))=m^{m^{n+2020}}$ . Compute the smallest possible value of $n$ .\n\n*Proposed by Luke Robitaille*... | math | {
"style": "rule",
"ground_truth": "13611"
} | {
"index": 11535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98! + 99! + 100!$?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 11536,
"solution": "1. **Factor out $98!$ from the sum**: \n \\[\n 98! + 99! + 100! = 98!(1 + 99 + 100 \\times 99)\n \\]\n Simplifying the expression inside the parentheses:\n \\[\n 1 + 99 + 9900 = 10000\n \\]\n Therefore, the expression becomes:\n \\[\n 98! \\times 10000\n \\]\n\n2... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $a, \\sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization ... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11537,
"solution": "Since the problem works for all positive integers $a$, let's plug in $a=2$ and see what we get. Since $\\sigma({2^n}) = 2^{n+1}-1,$ we have $2^{n+1} \\equiv 2 \\pmod{2021}.$ Simplifying using CRT and Fermat's Little Theorem, we get that $2^n \\equiv 0 \\pmod{42}$ and $2^n \\equiv 0 \\... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \\ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 11538,
"solution": "We see that $f_{1}(11)=4$\n$f_2(11) = f_1(4)=16$\n$f_3(11) = f_1(16)=49$\n$f_4(11) = f_1(49)=169$\n$f_5(11) = f_1(169)=256$\n$f_6(11) = f_1(256)=169$\n Note that this revolves between the two numbers. Since $1988$ is even, we thus have $f_{1988}(11) = f_{4}(11) = \\boxed{169}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, define $\\boxed{n}$ to be the sum of the positive factors of $n$.\nFor example, $\\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\\boxed{\\boxed{11}}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 11539,
"solution": "To solve for $\\boxed{\\boxed{11}}$, we first need to determine $\\boxed{11}$, which is the sum of the positive factors of $11$.\n\n1. **Finding $\\boxed{11}$**:\n Since $11$ is a prime number, its only positive factors are $1$ and $11$ itself. Therefore, the sum of the factors of $... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, let \n$f(n) =\\begin{cases}\\log_{8}{n}, &\\text{if }\\log_{8}{n}\\text{ is rational,}\\\\ 0, &\\text{otherwise.}\\end{cases}$\nWhat is $\\sum_{n = 1}^{1997}{f(n)}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{55}{3}"
} | {
"index": 11540,
"solution": "1. **Identify when $\\log_8 n$ is rational**: \n For $\\log_8 n$ to be rational, $n$ must be a power of $8$'s base, which is $2$. Thus, $n = 2^k$ for some integer $k$. This is because $\\log_8 2^k = k \\log_8 2 = \\frac{k}{3}$, which is rational.\n\n2. **Determine the range of $k$**... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, let \n\\[f(n) =\\left\\{\\begin{matrix}\\log_{4}{n}, &\\text{if }\\log_{4}{n}\\text{ is rational,}\\\\ 0, &\\text{otherwise.}\\end{matrix}\\right.\\]\nWhat is $\\sum_{n = 1}^{1023}{f(n)}$?\nA) $\\frac{40}{2}$\nB) $\\frac{45}{2}$\nC) $\\frac{21}{2}$... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{2}"
} | {
"index": 11541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, let $\\tau(n)$ denote the number of positive divisors of $n$. If $n$ is a positive integer such that $\\frac{\\tau\\left(n^{2}\\right)}{\\tau(n)}=3$, compute $\\frac{\\tau\\left(n^{7}\\right)}{\\tau(n)}$."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 11542,
"solution": "Let the prime factorization of $n$ be $n=p_{1}^{e_{1}} p_{2}^{e_{2}} \\cdots p_{k}^{e_{k}}$. Then, the problem condition is equivalent to $$\\prod_{i=1}^{k} \\frac{2 e_{i}+1}{e_{i}+1}=3$$ Note that since $\\frac{2 x+1}{x+1} \\geq 1.5$ for $x \\geq 1$, and $1.5^{3}>3$, we have $k \\leq... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, let $a_n$ be the $y$-coordinate of the intersection point between the tangent line of the curve $y=x^n(1-x)$ at $x=2$ and the $y$-axis in the Cartesian coordinate system. Calculate the sum of the first 10 terms of the sequence $\\{\\log_2 \\frac{a_... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 11543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, let $f(n)$ denote the number of 1's in the base-2 representation of $n$. For how many values of $n$ with $1 \\leq n \\leq 2002$ do we have $f(n)=f(n+1)$?"
}
] | math | {
"style": "rule",
"ground_truth": "501"
} | {
"index": 11544,
"solution": "If $n$ is even, then $n+1$ is obtained from $n$ in binary by changing the final 0 to a 1; thus $f(n+1)=f(n)+1$. If $n$ is odd, then $n+1$ is obtained by changing the last 0 to a 1, the ensuing string of 1's to 0's, and then changing the next rightmost 0 to a 1. This produces no net ch... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, let \\langle n\\rangle denote the closest integer to \\sqrt{n}. Evaluate\n\\[\\sum_{n=1}^\\infty \\frac{2^{\\langle n\\rangle}+2^{-\\langle n\\rangle}}{2^n}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11545,
"solution": "Since $(k-1/2)^2 = k^2-k+1/4$ and $(k+1/2)^2 = k^2+k+1/4$, we have that $\\langle n \\rangle = k$ if and only if $k^2-k+1 \\leq n \\leq k^2+k$. Hence\n\\begin{align*}\n\\sum_{n=1}^\\infty \\frac{2^{\\langle n \\rangle} + 2^{-\\langle n \\rangle}}{2^n}\n&= \\sum_{k=1}^\\infty \\sum_{n... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. For example, $4! = 4\\cdot 3\\cdot 2\\cdot 1 =24$. What is the greatest common divisor of $5!$ and $7!$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 11546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. Calculate the greatest common divisor of $8!$ and $10!$."
}
] | math | {
"style": "rule",
"ground_truth": "40320"
} | {
"index": 11547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n$, we define the integer $P(n)$ by :\n\n$P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$.\n\nFind the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$."
}
] | math | {
"style": "rule",
"ground_truth": "510510"
} | {
"index": 11548,
"solution": "\nTo find the greatest common divisor (GCD) of the integers \\( P(1), P(2), P(3), \\ldots, P(2016) \\), where \\( P(n) = n(n+1)(2n+1)(3n+1)\\cdots(16n+1) \\), we will first consider each part of the product and determine if there is a consistent factor across all \\( P(n) \\).\n\n### ... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $n,$ let $\\langle n \\rangle$ denote the closest integer to $\\sqrt{n}.$ Evaluate\n\\[\\sum_{n = 1}^\\infty \\frac{2^{\\langle n \\rangle} + 2^{-\\langle n \\rangle}}{2^n}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $x$, define $\\operatorname{Accident}(x)$ to be the set of ordered pairs $(s, t)$ with $s \\in \\{0,2,4,5,7,9,11\\}$ and $t \\in\\{1,3,6,8,10\\}$ such that $x+s-t$ is divisible by 12. For any nonnegative integer $i$, let $a_{i}$ denote the number of $x ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 11550,
"solution": "Modulo twelve, the first set turns out to be $\\{-1 \\cdot 7,0 \\cdot 7, \\ldots, 5 \\cdot 7\\}$ and the second set turns out to be be $\\{6 \\cdot 7, \\ldots, 10 \\cdot 7\\}$. We can eliminate the factor of 7 and shift to reduce the problem to $s \\in\\{0,1, \\ldots, 6\\}$ and $t \\i... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer $x_{}$, let $S(x)$ be the sum of the digits of $x_{}$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values of $T(x)$ do not exceed 1999?"
}
] | math | {
"style": "rule",
"ground_truth": "223"
} | {
"index": 11551,
"solution": "For most values of $x$, $T(x)$ will equal $2$. For those that don't, the difference must be bumping the number up a ten, a hundred, etc. If we take $T(a999)$ as an example, \\[|(a + 1) + 0 + 0 + 1 - (a + 9 + 9 + 9)| = |2 - 9(3)|\\] And in general, the values of $T(x)$ will then be in ... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer \\( k \\), let \\( f_{1}(k) \\) be the square of the sum of the digits of \\( k \\) when written in decimal notation. For \\( n > 1 \\), let \\( f_{n}(k) = f_{1}\\left(f_{n-1}(k)\\right) \\). What is \\( f_{1992}\\left(2^{1991}\\right) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 11552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer \\( n \\), define \n\\( g(n) =\\left\\{\\begin{matrix}\\log_{4}{n}, &\\text{if }\\log_{4}{n}\\text{ is rational,}\\\\ 0, &\\text{otherwise.}\\end{matrix}\\right. \\)\nWhat is \\( \\sum_{n = 1}^{1023}{g(n)} \\)?\n**A** \\( \\frac{40}{2} \\) \n**B** \\( ... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{2}"
} | {
"index": 11553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer \\( n \\), let \\( f(n) = 70 + n^2 \\) and let \\( g(n) \\) be the greatest common divisor (GCD) of \\( f(n) \\) and \\( f(n+1) \\). Find the greatest possible value of \\( g(n) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "281"
} | {
"index": 11554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer \\( n \\), let \\( f(n) \\) represent the last digit of \\( 1 + 2 + 3 + \\cdots + n \\). For example, \\( f(1) = 1 \\), \\( f(2) = 3 \\), \\( f(5) = 5 \\), and so on. Find the value of \\( f(2) + f(4) + f(6) + \\cdots + f(2012) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "3523"
} | {
"index": 11555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integer \\( n \\), the value of \\( n! \\) is the product of the first \\( n \\) positive integers. Calculate the greatest common divisor of \\( 8! \\) and \\( 10! \\)."
}
] | math | {
"style": "rule",
"ground_truth": "40320"
} | {
"index": 11556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integers $a$ and $b$ with $b>1$, let $s_{b}(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\\sum_{i=1}^{\\left\\lfloor\\log _{23} n\\right\\rfloor} s_{20}\\left(\\left\\lfloor\\frac{n}{23^{i}}\\... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 11557,
"solution": "First we will prove that $$s_{a}(n)=n-(a-1)\\left(\\sum_{i=1}^{\\infty}\\left\\lfloor\\frac{n}{a^{i}}\\right\\rfloor\\right)$$ If $n=\\left(n_{k} n_{k-1} \\cdots n_{1} n_{0}\\right)_{a}$, then the digit $n_{i}$ contributes $n_{i}$ to the left side of the sum, while it contributes $$n_... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integers \\( m \\) and \\( n \\), define \\( r(m, n) \\) as the remainder of \\( m \\div n \\) (for example, \\( r(8,3) \\) represents the remainder of \\( 8 \\div 3 \\), so \\( r(8,3)=2 \\)). What is the smallest positive integer solution satisfying the equati... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 11558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive integers \\( n \\) and \\( k \\) (where \\( k \\leqslant n \\)), let \\( f(n, k) \\) denote the number of positive integers that do not exceed \\( \\left\\lfloor \\frac{n}{k} \\right\\rfloor \\) and are coprime with \\( n \\). Determine the value of \\( f(100, ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 11559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any positive real number $x, \\lfloor x \\rfloor$ denotes the largest integer less than or equal to $x$. If $\\lfloor x \\rfloor \\cdot x = 36$ and $\\lfloor y \\rfloor \\cdot y = 71$ where $x, y > 0$, what is $x + y$ equal to?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{119}{8}"
} | {
"index": 11560,
"solution": "For any positive real number $x, \\lfloor x \\rfloor$ equals the largest integer less than or equal to $x$ and so $\\lfloor x \\rfloor \\leq x$. In particular, $\\lfloor x \\rfloor \\cdot x \\leq x \\cdot x = x^{2}$. Thus, if $\\lfloor x \\rfloor \\cdot x = 36$, then $36 \\leq x^{2}$.... |
DeepScaleR | [
{
"role": "user",
"content": "For any positive real numbers \\(a\\) and \\(b\\), define \\(a \\circ b=a+b+2 \\sqrt{a b}\\). Find all positive real numbers \\(x\\) such that \\(x^{2} \\circ 9x=121\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31-3\\sqrt{53}}{2}"
} | {
"index": 11561,
"solution": "Since \\(a \\circ b=(\\sqrt{a}+\\sqrt{b})^{2}\\), we have \\(x^{2} \\circ 9x=(x+3\\sqrt{x})^{2}\\). Moreover, since \\(x\\) is positive, we have \\(x+3\\sqrt{x}=11\\), and the only possible solution is that \\(\\sqrt{x}=\\frac{-3+\\sqrt{53}}{2}\\), so \\(x=\\frac{31-3\\sqrt{53}}{2}\\)... |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $t$ , let $\\lfloor t \\rfloor$ denote the largest integer $\\le t$ . Suppose that $N$ is the greatest integer such that $$ \\left \\lfloor \\sqrt{\\left \\lfloor \\sqrt{\\left \\lfloor \\sqrt{N} \\right \\rfloor}\\right \\rfloor}\\right \\rfloor = 4... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 11562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $x$ , we let $\\lfloor x \\rfloor$ be the unique integer $n$ such that $n \\leq x < n+1$ . For example. $\\lfloor 31.415 \\rfloor = 31$ . Compute \\[2020^{2021} - \\left\\lfloor\\frac{2020^{2021}}{2021} \\right \\rfloor (2021).\\]\n\n\n*2021 CCA Math... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"index": 11563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $\\alpha$, define $$\\operatorname{sign}(\\alpha)= \\begin{cases}+1 & \\text { if } \\alpha>0 \\\\ 0 & \\text { if } \\alpha=0 \\\\ -1 & \\text { if } \\alpha<0\\end{cases}$$ How many triples $(x, y, z) \\in \\mathbb{R}^{3}$ satisfy the following system of e... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11564,
"solution": "Since $\\operatorname{sign}(x+y)$ can take one of 3 values, $z$ can be one of 3 values: 4037,2018, or -1. The same is true of $x$ and $y$. However, this shows that $x+y$ cannot be 0, so $z$ can only be 4037 or -1. The same is true of $x$ and $y$. Now note that, if any two of $x, y, z$... |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $a$, let $\\left[a\\right]$ denote the largest integer not exceeding $a$. For example, $\\left[4\\right]=4$, $[\\sqrt{3}]=1$. Now, for the number $72$, the following operations are performed: $72\\stackrel{1st}{→}[\\sqrt{72}]=8\\stackrel{2nd}{→}[\\sqrt{8}]=2... | math | {
"style": "rule",
"ground_truth": "6560"
} | {
"index": 11565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $k,$ the graph of\n\\[y = 7x^2 + kx - 4k\\]passes through a fixed point $(a,b).$ Find $(a,b).$"
}
] | math | {
"style": "rule",
"ground_truth": "(4,112)"
} | {
"index": 11566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $x$, $\\lfloor x \\rfloor$ represents the largest integer not exceeding $x$, for example: $\\lfloor 2 \\rfloor = 2$, $\\lfloor 3.2 \\rfloor = 3$. Then $\\lfloor \\log_{2}1 \\rfloor + \\lfloor \\log_{2}2 \\rfloor + \\lfloor \\log_{2}3 \\rfloor + \\ldots + \\l... | math | {
"style": "rule",
"ground_truth": "264"
} | {
"index": 11567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $x$, the symbol $[x]$ represents the integer part of $x$, i.e., $[x]$ is the largest integer not exceeding $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. This function $[x]$ is called the \"floor function\", which has wide applications in mathematics and... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 11568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $x$, the symbol $[x]$ represents the largest integer not greater than $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. The function $y=[x]$ is called the \"floor function\", which has wide applications in mathematics and practical production. Then, the val... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $x$, the symbol $\\lfloor x \\rfloor$ represents the integer part of $x$, that is, $\\lfloor x \\rfloor$ is the largest integer not exceeding $x$. Calculate the value of $\\lfloor \\log_{2}1 \\rfloor + \\lfloor \\log_{2}2 \\rfloor + \\lfloor \\log_{2}3 \\rfl... | math | {
"style": "rule",
"ground_truth": "8204"
} | {
"index": 11570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $x$, the symbol $\\lfloor x \\rfloor$ represents the integer part of $x$, which is the greatest integer not exceeding $x$. This function, $\\lfloor x \\rfloor$, is called the \"floor function\". Calculate the sum $\\lfloor \\log_3 1 \\rfloor + \\lfloor \\log... | math | {
"style": "rule",
"ground_truth": "857"
} | {
"index": 11571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number $x$, the symbol $\\lfloor x \\rfloor$ represents the largest integer not exceeding $x$. Evaluate the expression $\\lfloor \\log_{2}1 \\rfloor + \\lfloor \\log_{2}2 \\rfloor + \\lfloor \\log_{2}3 \\rfloor + \\ldots + \\lfloor \\log_{2}1023 \\rfloor + \\lfloor... | math | {
"style": "rule",
"ground_truth": "8204"
} | {
"index": 11572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number \\( x \\), let \\( \\lfloor x \\rfloor \\) denote the greatest integer less than or equal to \\( x \\), and let \\( \\{ x \\} \\) denote the fractional part of \\( x \\). Then\n$$\n\\left\\{\\frac{2014}{2015}\\right\\}+\\left\\{\\frac{2014^{2}}{2015}\\right\... | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"index": 11573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number \\( x \\), let \\( f(x) \\) be the minimum of the values \\( 4x + 1 \\), \\( x + 2 \\), and \\( -2x + 4 \\). What is the maximum value of \\( f(x) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 11574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number \\( x \\), let \\([x]\\) denote the greatest integer less than or equal to \\( x \\). When \\( 0 \\leqslant x \\leqslant 100 \\), how many different integers are in the range of the function \\( f(x) = [2x] + [3x] + [4x] + [5x] \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 11575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number \\(x\\), \\(\\lfloor x \\rfloor\\) denotes the largest integer less than or equal to \\(x\\). For example, \\(\\lfloor 4.2 \\rfloor = 4\\) and \\(\\lfloor 0.9 \\rfloor = 0\\).\n\nIf \\(S\\) is the sum of all integers \\(k\\) with \\(1 \\leq k \\leq 999999\\)... | math | {
"style": "rule",
"ground_truth": "999999000"
} | {
"index": 11576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real number a and positive integer k, define\n$\\binom{a}{k} = \\frac{a(a-1)(a-2)\\cdots(a-(k-1))}{k(k-1)(k-2)\\cdots(2)(1)}$\nWhat is\n$\\binom{-\\frac{1}{2}}{100} \\div \\binom{\\frac{1}{2}}{100}$?"
}
] | math | {
"style": "rule",
"ground_truth": "-199"
} | {
"index": 11577,
"solution": "1. **Define the binomial coefficient for non-integer upper index**:\n \\[\n {a \\choose k} = \\frac{a(a-1)(a-2)\\cdots(a-(k-1))}{k(k-1)(k-2)\\cdots(2)(1)}\n \\]\n This definition extends the binomial coefficient to cases where $a$ is not necessarily a non-negative integer.\n\n... |
DeepScaleR | [
{
"role": "user",
"content": "For any real numbers $x,y$ that satisfies the equation $$ x+y-xy=155 $$ and $$ x^2+y^2=325 $$ , Find $|x^3-y^3|$"
}
] | math | {
"style": "rule",
"ground_truth": "4375"
} | {
"index": 11578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real numbers \\( a \\) and \\( b \\), the inequality \\( \\max \\{|a+b|,|a-b|,|2006-b|\\} \\geq C \\) always holds. Find the maximum value of the constant \\( C \\). (Note: \\( \\max \\{x, y, z\\} \\) denotes the largest among \\( x, y, \\) and \\( z \\).)"
}
] | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"index": 11579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real numbers \\( x \\) and \\( y \\), the operation is defined as\n\\[ x \\oplus y = x + 2y + 3. \\]\n\nGiven that real numbers \\( a \\) and \\( b \\) satisfy\n\\[ \\left(a^3 \\oplus a^2\\right) \\oplus a = a^3 \\oplus \\left(a^2 \\oplus a\\right) = b, \\]\n\nfind the ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{8}"
} | {
"index": 11580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any real value of $x$ the maximum value of $8x - 3x^2$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"index": 11581,
"solution": "1. **Identify the function and its type**: \n Let $f(x) = 8x - 3x^2$. This is a quadratic function.\n\n2. **Determine the nature of the quadratic**:\n Since the coefficient of $x^2$ is negative ($-3$), the parabola opens downwards. This means the vertex of the parabola will give t... |
DeepScaleR | [
{
"role": "user",
"content": "For any sequence of real numbers $A=(a_1,a_2,a_3,\\ldots)$, define $\\Delta A^{}_{}$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\\ldots)$, whose $n^{\\mbox{th}}_{}$ term is $a_{n+1}-a_n^{}$. Suppose that all of the terms of the sequence $\\Delta(\\Delta A^{}_{})$ are $1^{}_{}... | math | {
"style": "rule",
"ground_truth": "819"
} | {
"index": 11582,
"solution": "Note that the $\\Delta$s are reminiscent of differentiation; from the condition $\\Delta(\\Delta{A}) = 1$, we are led to consider the differential equation \\[\\frac{d^2 A}{dn^2} = 1\\] This inspires us to guess a quadratic with leading coefficient 1/2 as the solution; \\[a_{n} = \\fr... |
DeepScaleR | [
{
"role": "user",
"content": "For any sequence of real numbers $A=\\{a_1, a_2, a_3, \\ldots\\}$, define $\\triangle A$ as the sequence $\\{a_2 - a_1, a_3 - a_2, a_4 - a_3, \\ldots\\}$, where the $n$-th term is $a_{n+1} - a_n$. Assume that all terms of the sequence $\\triangle (\\triangle A)$ are $1$ and $a_... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 11583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S)/m(S)$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "\\phi=\\frac{1+\\sqrt{5}}{2}"
} | {
"index": 11584,
"solution": "\nLet \\( S \\) be a set of five points in the plane, no three of which are collinear. We need to determine the minimum possible value of \\( \\frac{M(S)}{m(S)} \\) where \\( M(S) \\) and \\( m(S) \\) are the maximum and minimum areas of triangles that can be formed by any three point... |
DeepScaleR | [
{
"role": "user",
"content": "For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\\cup B\\cup C)$ and $|A|=|B|=100$, then what is the minim... | math | {
"style": "rule",
"ground_truth": "97"
} | {
"index": 11585,
"solution": "1. **Given Information and Basic Calculations**:\n - $|A| = |B| = 100$, so $n(A) = n(B) = 2^{100}$.\n - The formula for the number of subsets of a set $S$ is $n(S) = 2^{|S|}$.\n - Given $n(A) + n(B) + n(C) = n(A \\cup B \\cup C)$, substituting the values, we get:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "For any set \\( S \\), let \\( |S| \\) represent the number of elements in set \\( S \\) and let \\( n(S) \\) represent the number of subsets of set \\( S \\). If \\( A \\), \\( B \\), and \\( C \\) are three finite sets such that:\n\n(1) \\( |A|=|B|=2016 \\);\n(2) \\( n(A) + n... | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"index": 11586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any subset \\( S \\subseteq \\{1, 2, \\ldots, 15\\} \\), a number \\( n \\) is called an \"anchor\" for \\( S \\) if \\( n \\) and \\( n+|S| \\) are both members of \\( S \\), where \\( |S| \\) denotes the number of members of \\( S \\). Find the average number of anchors o... | math | {
"style": "rule",
"ground_truth": "13/8"
} | {
"index": 11587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any three real numbers $a$, $b$, and $c$, with $b\\neq c$, the operation $\\otimes$ is defined by:\n\\[\\otimes(a,b,c)=\\frac{a}{b-c}\\]\nWhat is $\\otimes(\\otimes(1,2,3),\\otimes(2,3,1),\\otimes(3,1,2))$?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"index": 11588,
"solution": "1. **Calculate each $\\otimes$ operation inside the main operation**:\n - $\\otimes(1,2,3) = \\frac{1}{2-3} = \\frac{1}{-1} = -1$\n - $\\otimes(2,3,1) = \\frac{2}{3-1} = \\frac{2}{2} = 1$\n - $\\otimes(3,1,2) = \\frac{3}{1-2} = \\frac{3}{-1} = -3$\n\n2. **Substitute these values... |
DeepScaleR | [
{
"role": "user",
"content": "For any two non-zero plane vectors $\\overrightarrow \\alpha$ and $\\overrightarrow \\beta$, a new operation $\\odot$ is defined as $\\overrightarrow \\alpha ⊙ \\overrightarrow \\beta = \\frac{\\overrightarrow \\alpha • \\overrightarrow \\beta}{\\overrightarrow \\beta • \\overr... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 11589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any two non-zero plane vectors $\\overrightarrow{\\alpha}$ and $\\overrightarrow{\\beta}$, define $\\overrightarrow{\\alpha}○\\overrightarrow{\\beta}=\\dfrac{\\overrightarrow{\\alpha}⋅\\overrightarrow{\\beta}}{\\overrightarrow{\\beta}⋅\\overrightarrow{\\beta}}$. If plane ve... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{2}"
} | {
"index": 11590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any two positive integers m and n, define an operation \"※\" as follows: when m and n are both positive even numbers or positive odd numbers, m※n=m+n; when one of m and n is a positive even number and the other is a positive odd number, m※n=mn. Determine the number of eleme... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 11591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For any two positive integers, define the operation (represented by the operator ⊕): when both $m$ and $n$ are positive even numbers or both are positive odd numbers, $m⊕n=m+n$; when one of $m$ and $n$ is a positive even number and the other is a positive odd number, $m⊕n=m×n$.... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 11592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For arbitrary real numbers \\(a\\) and \\(b\\) (\\(a \\neq 0\\)), find the minimum value of the expression \\(\\frac{1}{a^{2}} + 2a^{2} + 3b^{2} + 4ab\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{8}{3}}"
} | {
"index": 11593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively. If $\\frac{{S}_{n}}{{T}_{n}}=\\frac{2n+1}{3n+2}$, then $\\frac{{a}_{2}+{a}_{5}+{a}_{17}+{a}_{20}}{{b}_{8}+{b}_{10}+{b}_{12}+{b}_{14}}=\\_\\_\\_\\_\\_... | math | {
"style": "rule",
"ground_truth": "\\frac{43}{65}"
} | {
"index": 11594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For breakfast, Milan is eating a piece of toast shaped like an equilateral triangle. On the piece of toast rests a single sesame seed that is one inch away from one side, two inches away from another side, and four inches away from the third side. He places a circular piece of ... | math | {
"style": "rule",
"ground_truth": "\\frac{49 \\pi}{9}"
} | {
"index": 11595,
"solution": "Suppose the toast has side length $s$. If we draw the three line segments from the sesame seed to the three vertices of the triangle, we partition the triangle into three smaller triangles, with areas $\\frac{s}{2}, s$, and $2 s$, so the entire piece of toast has area $\\frac{7 s}{2}$... |
DeepScaleR | [
{
"role": "user",
"content": "For certain ordered pairs $(a,b)\\,$ of real numbers, the system of equations\n$ax+by=1\\,$\n$x^2+y^2=50\\,$\n has at least one solution, and each solution is an ordered pair $(x,y)\\,$ of integers. How many such ordered pairs $(a,b)\\,$ are there?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 11596,
"solution": "$x^2+y^2=50$ is the equation of a circle of radius $\\sqrt{50}$, centered at the origin. The lattice points on this circle are $(\\pm1,\\pm7)$, $(\\pm5,\\pm5)$, and $(\\pm7,\\pm1)$.\n$ax+by=1$ is the equation of a line that does not pass through the origin. (Since $(x,y)=(0,0)$ yields... |
DeepScaleR | [
{
"role": "user",
"content": "For certain ordered pairs $(a,b)\\,$ of real numbers, the system of equations\n\\[\\begin{aligned} ax+by&=1 \\\\ x^2 + y^2 &= 50 \\end{aligned}\\]has at least one solution, and each solution is an ordered pair $(x,y)\\,$ of integers. How many such ordered pairs $(a,b)\\,$ are ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 11597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For certain pairs $(m,n)$ of positive integers with $m\\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\\log m - \\log k| < \\log n$. Find the sum of all possible values of the product $mn$."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11598,
"solution": "We have $\\log m - \\log k = \\log \\left( \\frac mk \\right)$, hence we can rewrite the inequality as follows: \\[- \\log n < \\log \\left( \\frac mk \\right) < \\log n\\] We can now get rid of the logarithms, obtaining: \\[\\frac 1n < \\frac mk < n\\] And this can be rewritten in te... |
DeepScaleR | [
{
"role": "user",
"content": "For certain real numbers $a$, $b$, and $c$, the polynomial \\[g(x) = x^3 + ax^2 + 2x + 15\\] has three distinct roots, which are also roots of the polynomial \\[f(x) = x^4 + x^3 + bx^2 + 75x + c.\\] Determine the value of $f(-1)$."
}
] | math | {
"style": "rule",
"ground_truth": "-2773"
} | {
"index": 11599,
"solution": ""
} |
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