data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Find the largest five-digit number whose digits' product equals 120."
}
] | math | {
"style": "rule",
"ground_truth": "85311"
} | {
"index": 9800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11."
}
] | math | {
"style": "rule",
"ground_truth": "98765"
} | {
"index": 9801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest four-digit number in which all digits are different and which is divisible by 2, 5, 9, and 11."
}
] | math | {
"style": "rule",
"ground_truth": "8910"
} | {
"index": 9802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $n$ for which $2^n$ divides \\[ \\binom 21 \\binom 42 \\binom 63 \\dots \\binom {128}{64}. \\]*Proposed by Evan Chen*"
}
] | math | {
"style": "rule",
"ground_truth": "193"
} | {
"index": 9803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $k$ such that for all integers $x$ and $y$, if $xy + 1$ is divisible by $k$, then $x + y$ is also divisible by $k$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 9804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $n$ for which $12^n$ evenly divides $20!$."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 9805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $n$ satisfying the following conditions:\n (i) $n^2$ can be expressed as the difference of two consecutive cubes;\n (ii) $2n + 79$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "181"
} | {
"index": 9806,
"solution": "Let us generate numbers $1$ to $1000$ for the second condition, for squares. We know for $N$ to be integer, the squares must be odd. So we generate $N = 1, 21, 45, 73, 105, 141, 181, 381, 441, 721, 801$. $N$ cannot exceed $1000$ since it is AIME problem. Now take the first criterion, l... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $n$ satisfying the following conditions:\n(i) $n^2$ can be expressed as the difference of two consecutive cubes;\n(ii) $2n + 79$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "181"
} | {
"index": 9807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 9808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $n$ such that $3^{512}-1$ is divisible by $2^{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 9809,
"solution": "Write $$ \\begin{aligned} 3^{512}-1 & =\\left(3^{256}+1\\right)\\left(3^{256}-1\\right)=\\left(3^{256}+1\\right)\\left(3^{128}+1\\right)\\left(3^{128}-1\\right) \\\\ & =\\cdots=\\left(3^{256}+1\\right)\\left(3^{128}+1\\right) \\cdots(3+1)(3-1) \\end{aligned} $$ Now each factor $3^{2^{k... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\\sqrt[3]{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 9810,
"solution": "\nGiven the problem, we are tasked to find the largest integer \\( n \\) such that \\( n \\) is divisible by all positive integers less than \\( \\sqrt[3]{n} \\).\n\n### Step-by-Step Solution:\n\n1. **Understanding the condition:**\n\n We know \\( n \\) must be divisible by every int... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $n$ such that the following holds: there exists a set of $n$ points in the plane such that, for any choice of three of them, some two are unit distance apart."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 9811,
"solution": "We can obtain $n=7$ in the following way: Consider a rhombus $A B C D$ made up of two equilateral triangles of side length 1 , where $\\angle D A B=60^{\\circ}$. Rotate the rhombus clockwise about $A$ to obtain a new rhombus $A B^{\\prime} C^{\\prime} D^{\\prime}$ such that $D D^{\\pri... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer $x$ such that the number\n\n$$\n4^{27} + 4^{1000} + 4^{x}\n$$\n\nis a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "1972"
} | {
"index": 9812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer \\( a \\) such that the expression\n\n\\[ \na^2 - 15a - (\\tan x - 1)(\\tan x + 2)(\\tan x + 5)(\\tan x + 8) \n\\]\n\nis less than 35 for all values of \\( x \\in (-\\pi/2, \\pi/2) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 9813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer \\( k \\) such that for at least one natural number \\( n > 1000 \\), the number \\( n! = 1 \\cdot 2 \\cdot \\ldots \\cdot n \\) is divisible by \\( 2^{n+k+2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 9814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer less than 2012 all of whose divisors have at most two 1's in their binary representations."
}
] | math | {
"style": "rule",
"ground_truth": "1536"
} | {
"index": 9815,
"solution": "Call a number good if all of its positive divisors have at most two 1's in their binary representations. Then, if $p$ is an odd prime divisor of a good number, $p$ must be of the form $2^{k}+1$. The only such primes less than 2012 are $3,5,17$, and 257 , so the only possible prime divi... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer less than 74 that leaves a remainder of 3 when divided by 7."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 9816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer less than 80 that leaves a remainder of 3 when divided by 5."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 9817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer that divides $m^{5}-5 m^{3}+4 m$ for all $m \\geq 5$."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 9818,
"solution": "5!=120."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integer value of $n$ such that $n^2-9n+18$ is negative."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 9819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest integral value of $x$ which solves: $\\frac{1}{3}<{\\frac{x}{5}}<{\\frac{5}{8}}$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 9820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number \\( n \\) for which the product of the numbers \\( n, n+1, n+2, \\ldots, n+20 \\) is divisible by the square of one of these numbers."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 9821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number \\( n \\) with the following property: for any odd prime number \\( p \\) less than \\( n \\), the difference \\( n - p \\) is also a prime number."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 9822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number consisting of distinct digits such that the product of its digits equals 2016."
}
] | math | {
"style": "rule",
"ground_truth": "876321"
} | {
"index": 9823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number from which it is impossible to obtain a number divisible by 11 by deleting some of its digits."
}
] | math | {
"style": "rule",
"ground_truth": "987654321"
} | {
"index": 9824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7."
}
] | math | {
"style": "rule",
"ground_truth": "60718293"
} | {
"index": 9825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number."
}
] | math | {
"style": "rule",
"ground_truth": "520"
} | {
"index": 9826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number in which all the digits are different and each pair of adjacent digits differs by 6 or 7."
}
] | math | {
"style": "rule",
"ground_truth": "60718293"
} | {
"index": 9827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number in which each digit, starting from the third, is equal to the sum of all previous digits of the number."
}
] | math | {
"style": "rule",
"ground_truth": "101248"
} | {
"index": 9828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest natural number whose all digits in its decimal representation are different and which decreases 5 times if you cross out the first digit."
}
] | math | {
"style": "rule",
"ground_truth": "3750"
} | {
"index": 9829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest negative integer $x$ which satisfies the congruence $34x+6\\equiv 2\\pmod {20}$."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 9830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest negative root \\( x_{0} \\) of the equation \\( \\frac{\\sin x}{1+\\cos x} = 2 - \\operatorname{ctg} x \\). Write the answer as \\( x_{0} \\cdot \\frac{3}{\\pi} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-3.5"
} | {
"index": 9831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest negative root of the equation\n\n$$\n4 \\sin (3 x) + 13 \\cos (3 x) = 8 \\sin (x) + 11 \\cos (x)\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "-0.1651"
} | {
"index": 9832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 9833,
"solution": "For positive integers $a, b$, we have $$a!\\mid b!\\quad \\Leftrightarrow a!\\leq b!\\quad \\Leftrightarrow \\quad a \\leq b$$ Thus, $$((n!)!)!\\mid(2004!)!\\Leftrightarrow(n!)!\\leq 2004!\\Leftrightarrow n!\\leq 2004 \\Leftrightarrow n \\leq 6$$"
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest odd natural number that cannot be expressed as the sum of three distinct composite numbers."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 9834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest perfect square such that when the last two digits are removed, it is still a perfect square. (It is assumed that one of the digits removed is not zero.)"
}
] | math | {
"style": "rule",
"ground_truth": "1681"
} | {
"index": 9835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $N $ for which one can choose $N $ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$ .\n\nProposed by Mikhail Evdokimov"
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 9836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $N$ so that the number of integers in the set $\\{1,2,\\dots,N\\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both)."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 9837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $k$ such that $\\phi ( \\sigma ( 2^k)) = 2^k$ . ( $\\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$ , and $\\sigma(n)$ denotes the sum of divisors of $n$ ). As a hint, you are giv... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 9838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.)"
}
] | math | {
"style": "rule",
"ground_truth": "9999"
} | {
"index": 9839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $n$ such that $\\sigma(n) = 28$ , where $\\sigma(n)$ is the sum of the divisors of $n$ , including $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 9840,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 9841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .\n\n*Proposed by Andy Xu*"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 9842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\\dfrac n2$ is an odd number."
}
] | math | {
"style": "rule",
"ground_truth": "505"
} | {
"index": 9843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\\mathcal{P}$ with 2022 edges, which satisfies the following properties:\n[list]\n[*]The degrees of the vertices of $\\mathcal{P}$ don’t differ by more than one, and\n[*]It is possible to colour... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 9844,
"solution": "\nWe are tasked with finding the largest positive integer \\( k \\) such that there exists a convex polyhedron \\(\\mathcal{P}\\) with 2022 edges, which satisfies the following conditions:\n1. The degrees of the vertices of \\(\\mathcal{P}\\) do not differ by more than one.\n2. It is p... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $m$ such that an $m \\times m$ square can be exactly divided into 7 rectangles with pairwise disjoint interiors, and the lengths of the 14 sides of these 7 rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14$."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 9845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $m$ which makes it possible to color several cells of a $70\\times 70$ table red such that [list] [*] There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they a... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 9846,
"solution": "\nTo find the largest positive integer \\( m \\) that allows coloring several cells of a \\( 70 \\times 70 \\) table red such that:\n1. There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $n$ such that\n\\[\\sin^n x + \\cos^n x \\ge \\frac{1}{n}\\]for all real numbers $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 9847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer $n$ such that there exist $n$ distinct positive integers $x_{1}, x_{2}, \\cdots, x_{n}$ satisfying\n$$\nx_{1}^{2}+x_{2}^{2}+\\cdots+x_{n}^{2}=2017.\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 9848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer \\( n \\) for which we can find a set of distinct positive integers such that each integer is at most 2002 and if \\( a \\) and \\( b \\) are in the set, then \\( a^2 \\) and \\( ab \\) are not in the set."
}
] | math | {
"style": "rule",
"ground_truth": "1958"
} | {
"index": 9849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer \\( n \\) such that \\( n^{3} + 4n^{2} - 15n - 18 \\) is the cube of an integer."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 9850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer \\(n\\) for which there exist \\(n\\) finite sets \\(X_{1}, X_{2}, \\ldots, X_{n}\\) with the property that for every \\(1 \\leq a<b<c \\leq n\\), the equation \\(\\left|X_{a} \\cup X_{b} \\cup X_{c}\\right|=\\lceil\\sqrt{a b c}\\rceil\\) holds... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 9851,
"solution": "First, we construct an example for \\(N=4\\). Let \\(X_{1}, X_{2}, X_{3}, X_{4}\\) be pairwise disjoint sets such that \\(X_{1}=\\varnothing,\\left|X_{2}\\right|=1,\\left|X_{3}\\right|=2\\), and \\(\\left|X_{4}\\right|=2\\). It is straightforward to verify the condition. We claim that ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive integer solution of the equation $\\left\\lfloor\\frac{N}{3}\\right\\rfloor=\\left\\lfloor\\frac{N}{5}\\right\\rfloor+\\left\\lfloor\\frac{N}{7}\\right\\rfloor-\\left\\lfloor\\frac{N}{35}\\right\\rfloor$."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 9852,
"solution": "For $N$ to be a solution, it is necessary that $\\frac{N-2}{3}+\\frac{N-34}{35} \\leq \\frac{N}{5}+\\frac{N}{7}$, which simplifies to $N \\leq 86$. However, if $N \\geq 70$, then $N \\leq 59$, contradicting $N \\geq 70$. It follows that $N$ must be at most 69. Checking for $N \\leq 69$... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest positive number \\( c \\) such that for every natural number \\( n \\), the inequality \\( \\{n \\sqrt{2}\\} \\geqslant \\frac{c}{n} \\) holds, where \\( \\{n \\sqrt{2}\\} = n \\sqrt{2} - \\lfloor n \\sqrt{2} \\rfloor \\) and \\( \\lfloor x \\rfloor \\) denotes... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2\\sqrt{2}}"
} | {
"index": 9853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest possible number in decimal notation where all the digits are different, and the sum of its digits is 37."
}
] | math | {
"style": "rule",
"ground_truth": "976543210"
} | {
"index": 9854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest possible subset of {1, 2, ... , 15} such that the product of any three distinct elements of the subset is not a square."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 9855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "486"
} | {
"index": 9856,
"solution": "Let us write down one such sum, with $m$ terms and first term $n + 1$:\n$3^{11} = (n + 1) + (n + 2) + \\ldots + (n + m) = \\frac{1}{2} m(2n + m + 1)$.\n Thus $m(2n + m + 1) = 2 \\cdot 3^{11}$ so $m$ is a divisor of $2\\cdot 3^{11}$. However, because $n \\geq 0$ we have $m^2 < m(m + 1) ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest possible value of $k$ for which $3^{13}$ is expressible as the sum of $k$ consecutive positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "1458"
} | {
"index": 9857,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest possible value of $x$ in the simplified form $x=\\frac{a+b\\sqrt{c}}{d}$ if $\\frac{5x}{6}+1=\\frac{3}{x}$, where $a,b,c,$ and $d$ are integers. What is $\\frac{acd}{b}$?"
}
] | math | {
"style": "rule",
"ground_truth": "-55"
} | {
"index": 9858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest possible value of \\( n \\) such that there exist \\( n \\) consecutive positive integers whose sum is equal to 2010."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 9859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest possible value of the expression $$\\frac{(2a+3b)^2 + (b-c)^2 + (2c-a)^2}{b^2}$$ where \\( b > a > c \\) are real numbers, and \\( b \\neq 0 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 9860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime $p$ such that $p$ divides $2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 9861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime divisor of $25^2+72^2$."
}
] | math | {
"style": "rule",
"ground_truth": "157"
} | {
"index": 9862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime divisor of $36^2 + 49^2$."
}
] | math | {
"style": "rule",
"ground_truth": "3697"
} | {
"index": 9863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime divisor of 11! + 12!"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 9864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime divisor of \\( 16^2 + 81^2 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 9865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime factor of $-x^{10}-x^{8}-x^{6}-x^{4}-x^{2}-1$, where $x=2 i$, $i=\\sqrt{-1}$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 9866,
"solution": "13."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime factor of $11236$."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 9867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime factor of $15^3+10^4-5^5$."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 9868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime factor of $9879$."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 9869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying\nthe real and imaginary part of $z$ are both integers;\n$|z|=\\sqrt{p},$ and\nthere exists a triangle whose three side lengths are $p,$ the real part of $z^{3},$ and the imaginary part... | math | {
"style": "rule",
"ground_truth": "349"
} | {
"index": 9870,
"solution": "Assume that $z=a+bi$. Then, \\[z^3=(a^3-3ab^2)+(3a^2b-b^3)i\\]Note that by the Triangle Inequality, \\[|(a^3-3ab^2)-(3a^2b-b^3)|<p\\implies |a^3+b^3-3ab^2-3a^2b|<a^2+b^2\\]Thus, we know \\[|a+b||a^2+b^2-4ab|<a^2+b^2\\]Without loss of generality, assume $a>b$ (as otherwise, consider $i^... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ such that $ a\\plus{}b\\equal{}c\\plus{}d\\plus{}e$ : \\[ \\sqrt{a^{2}\\plus{}b^{2}\\plus{}c^{2}\\plus{}d^{2}\\plus{}e^{2}}\\geq T(\\sqrt a\\plus{}\\sqrt b\\plus{}\\sqrt c\\plus{}\\sqrt d\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{30}}{30 + 12\\sqrt{6}}"
} | {
"index": 9871,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real $C$ such that for all pairwise distinct positive real $a_{1}, a_{2}, \\ldots, a_{2019}$ the following inequality holds $$\\frac{a_{1}}{\\left|a_{2}-a_{3}\\right|}+\\frac{a_{2}}{\\left|a_{3}-a_{4}\\right|}+\\ldots+\\frac{a_{2018}}{\\left|a_{2019}-a_{1}\\rig... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 9872,
"solution": "Without loss of generality we assume that $\\min \\left(a_{1}, a_{2}, \\ldots, a_{2019}\\right)=a_{1}$. Note that if $a, b, c$ $(b \\neq c)$ are positive, then $\\frac{a}{|b-c|}>\\min \\left(\\frac{a}{b}, \\frac{a}{c}\\right)$. Hence $$S=\\frac{a_{1}}{\\left|a_{2}-a_{3}\\right|}+\\cdot... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real \\( k \\) such that if \\( a, b, c, d \\) are positive integers such that \\( a + b = c + d \\), \\( 2ab = cd \\) and \\( a \\geq b \\), then \\(\\frac{a}{b} \\geq k\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"index": 9873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real constant $a$ such that for all $n \\geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \\cdots < x_n$ we have\n\\[\\frac{1}{x_1-x_0} + \\frac{1}{x_2-x_1} + \\dots + \\frac{1}{x_n-x_{n-1}} \\geq a \\left( \\frac{2}{x_1} ... | math | {
"style": "rule",
"ground_truth": "a = 4/9"
} | {
"index": 9874,
"solution": "\nLet's consider \\( n \\geq 1 \\) and real numbers \\( x_0, x_1, \\ldots, x_n \\) such that \\( 0 = x_0 < x_1 < x_2 < \\cdots < x_n \\). We need to find the largest real constant \\( a \\) such that the inequality holds:\n\n\\[\n\\frac{1}{x_1 - x_0} + \\frac{1}{x_2 - x_1} + \\cdots + ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $\\lambda$ such that\n\n\\[a_1^2 + \\cdots + a_{2019}^2 \\ge a_1a_2 + a_2a_3 + \\cdots + a_{1008}a_{1009} + \\lambda a_{1009}a_{1010} + \\lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \\cdots + a_{2018}a_{2019}\\]\nfor all real numbers $a_1, \\ldo... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"index": 9875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $k$ , such that for any positive real numbers $a,b$ , $$ (a+b)(ab+1)(b+1)\\geq kab^2 $$"
}
] | math | {
"style": "rule",
"ground_truth": "27/4"
} | {
"index": 9876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $\\lambda$ such that\n\\[a^2 + b^2 + c^2 + d^2 \\ge ab + \\lambda bc + cd\\]for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 9877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $\\lambda$ such that $a^{2}+b^{2}+c^{2}+d^{2} \\geq a b+\\lambda b c+c d$ for all real numbers $a, b, c, d$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 9878,
"solution": "Let $f(a, b, c, d)=\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)-(a b+\\lambda b c+c d)$. For fixed $(b, c, d), f$ is minimized at $a=\\frac{b}{2}$, and for fixed $(a, b, c), f$ is minimized at $d=\\frac{c}{2}$, so simply we want the largest $\\lambda$ such that $f\\left(\\frac{b}{2}, b, c, \... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $\\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\\in \\mathbb{R})$ such that $$ |b|\\ge \\lambda |a| \\quad \\text{and} \\quad (pz^3+2qz^2+2rz+s) \\cdot (qz^3+2pz^2+2sz+r) =... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 9879,
"solution": "\nTo find the largest real number \\(\\lambda\\) such that for any positive real numbers \\(p, q, r, s\\), there exists a complex number \\(z = a + bi\\) (\\(a, b \\in \\mathbb{R}\\)) satisfying\n\\[\n|b| \\ge \\lambda |a|\n\\]\nand\n\\[\n(pz^3 + 2qz^2 + 2rz + s) \\cdot (qz^3 + 2pz^2 +... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $c$ such that $$\\sum_{i=1}^{101} x_{i}^{2} \\geq c M^{2}$$ whenever $x_{1}, \\ldots, x_{101}$ are real numbers such that $x_{1}+\\cdots+x_{101}=0$ and $M$ is the median of $x_{1}, \\ldots, x_{101}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5151}{50}"
} | {
"index": 9880,
"solution": "Suppose without loss of generality that $x_{1} \\leq \\cdots \\leq x_{101}$ and $M=x_{51} \\geq 0$. Note that $f(t)=t^{2}$ is a convex function over the reals, so we may \"smooth\" to the case $x_{1}=\\cdots=x_{50}=-51 r$ and $x_{51}=\\cdots=x_{101}=50 r$ for some $r \\geq 0$, and by h... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $c$ such that \\[x_1^2 + x_2^2 + \\dots + x_{101}^2 \\geq cM^2\\]whenever $x_1,x_2,\\ldots,x_{101}$ are real numbers such that $x_1+x_2+\\cdots+x_{101}=0$ and $M$ is the median of $x_1,x_2,\\ldots,x_{101}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{5151}{50}"
} | {
"index": 9881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $k$ such that \\[x_1^2 + x_2^2 + \\dots + x_{11}^2 \\geq kx_6^2\\] whenever $x_1, x_2, \\ldots, x_{11}$ are real numbers such that $x_1 + x_2 + \\cdots + x_{11} = 0$ and $x_6$ is the median of $x_1, x_2, \\ldots, x_{11}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{66}{5}"
} | {
"index": 9882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $k$ such that there exists a sequence of positive reals $\\left\\{a_{i}\\right\\}$ for which $\\sum_{n=1}^{\\infty} a_{n}$ converges but $\\sum_{n=1}^{\\infty} \\frac{\\sqrt{a_{n}}}{n^{k}}$ does not."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 9883,
"solution": "For $k>\\frac{1}{2}$, I claim that the second sequence must converge. The proof is as follows: by the CauchySchwarz inequality, $$\\left(\\sum_{n \\geq 1} \\frac{\\sqrt{a_{n}}}{n^{k}}\\right)^{2} \\leq\\left(\\sum_{n \\geq 1} a_{n}\\right)\\left(\\sum_{n \\geq 1} \\frac{1}{n^{2 k}}\\ri... |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number $x$ such that\n\\[\\frac{\\lfloor x \\rfloor}{x} = \\frac{9}{10}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{80}{9}"
} | {
"index": 9884,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number \\( C \\) such that for any \\( n \\in \\mathbf{Z}_{+} \\) and any sequence \\(\\{x_{k}\\}\\) satisfying \\(0 = x_{0} < x_{1} < x_{2} < \\cdots < x_{n} = 1\\), the following inequality holds:\n\\[ \n\\sum_{k=1}^{n} x_{k}^{2} \\left(x_{k} - x_{k-1}\\... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 9885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number \\( p \\) such that all three roots of the equation below are positive integers:\n\\[ \n5x^{3} - 5(p+1)x^{2} + (71p-1)x + 1 = 66p .\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 9886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number \\(\\lambda\\) such that for the real coefficient polynomial \\(f(x) = x^3 + ax^2 + bx + c\\) with all non-negative real roots, it holds that \\(f(x) \\geqslant \\lambda(x - a)^3\\) for all \\(x \\geqslant 0\\). Additionally, determine when the equa... | math | {
"style": "rule",
"ground_truth": "-1/27"
} | {
"index": 9887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number \\(\\lambda\\) such that for the real-coefficient polynomial \\(f(x) = x^3 + ax^2 + bx + c\\) with all roots non-negative real numbers, we have \\(f(x) \\geqslant \\lambda(x - a)^3\\) for all \\(x \\geqslant 0\\). Also, determine when the equality h... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{27}"
} | {
"index": 9888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest real number \\(x\\) such that\n\\[ \n\\frac{x^{2} + x - 1 + \\left|x^{2} - (x - 1)\\right|}{2} = 35x - 250. \n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 9889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest root of the equation $|\\sin (2 \\pi x) - \\cos (\\pi x)| = ||\\sin (2 \\pi x)| - |\\cos (\\pi x)|$, which belongs to the interval $\\left(\\frac{1}{4}, 2\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 9890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits."
}
] | math | {
"style": "rule",
"ground_truth": "972538"
} | {
"index": 9891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits."
}
] | math | {
"style": "rule",
"ground_truth": "972538"
} | {
"index": 9892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest six-digit number in which each digit, starting from the third, is the sum of the two preceding digits."
}
] | math | {
"style": "rule",
"ground_truth": "303369"
} | {
"index": 9893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest solution to \\[\n\\lfloor x \\rfloor = 7 + 150 \\{ x \\},\n\\] where $\\{x\\} = x - \\lfloor x \\rfloor$."
}
] | math | {
"style": "rule",
"ground_truth": "156.9933"
} | {
"index": 9894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest solution to \\[\\lfloor x \\rfloor = 5 + 100 \\{ x \\},\\]where $\\{x\\} = x - \\lfloor x \\rfloor.$"
}
] | math | {
"style": "rule",
"ground_truth": "104.99"
} | {
"index": 9895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest solution to the equation \\[\\lfloor x \\rfloor = 8 + 50 \\{ x \\},\\] where $\\{x\\} = x - \\lfloor x \\rfloor.$"
}
] | math | {
"style": "rule",
"ground_truth": "57.98"
} | {
"index": 9896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest three-digit integer starting with 8 that is divisible by each of its distinct, non-zero digits except for 7."
}
] | math | {
"style": "rule",
"ground_truth": "864"
} | {
"index": 9897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest three-digit integer that is divisible by each of its digits and the sum of the digits is divisible by 6."
}
] | math | {
"style": "rule",
"ground_truth": "936"
} | {
"index": 9898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits."
}
] | math | {
"style": "rule",
"ground_truth": "936"
} | {
"index": 9899,
"solution": ""
} |
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