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 smallest natural number with 6 as the last digit, such that if the final 6 is moved to the front of the number it is multiplied by 4."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"index": 10700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number with the following properties:\na) It ends in digit 6 in decimal notation;\nb) If you remove the last digit 6 and place this digit 6 at the beginning of the remaining number, the result is four times the original number."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"index": 10701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$ ."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 10702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$ , so that no two of the disks have a common interior point."
}
] | math | {
"style": "rule",
"ground_truth": "2 + 2\\sqrt{2}"
} | {
"index": 10703,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number $n \\geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.\n\n*Bulgaria*"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number \\( n > 1980 \\) such that the number\n\n$$\n\\frac{x_{1} + x_{2} + x_{3} + \\ldots + x_{n}}{5}\n$$\n\nis an integer for any given integer values \\( x_{1}, x_{2}, x_{3}, \\ldots, x_{n} \\), none of which is divisible by 5."
}
] | math | {
"style": "rule",
"ground_truth": "1985"
} | {
"index": 10705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number composed exclusively of ones that is divisible by 333...33 (where there are 100 threes in the number)."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 10706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number in which all digits are different and the sum of all digits equals 32."
}
] | math | {
"style": "rule",
"ground_truth": "26789"
} | {
"index": 10707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number that uses only the digits 2 and 3 in equal quantity and is divisible by both 2 and 3."
}
] | math | {
"style": "rule",
"ground_truth": "223332"
} | {
"index": 10708,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number which when successively divided by \\( 45,454,4545 \\) and 45454 leaves remainders of \\( 4, 45,454 \\) and 4545 respectively."
}
] | math | {
"style": "rule",
"ground_truth": "35641667749"
} | {
"index": 10709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number written using only the digits 0 and 1 that is divisible by the product of the six smallest natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "1111111110000"
} | {
"index": 10710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest number, written using only ones and zeros, that would be divisible by 225."
}
] | math | {
"style": "rule",
"ground_truth": "11111111100"
} | {
"index": 10711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest period of the function \\( y = \\cos^{10} x + \\sin^{10} x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 10712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 10713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive angle $\\theta,$ in degrees, for which\n\\[\\cos \\theta = \\sin 60^\\circ + \\cos 42^\\circ - \\sin 12^\\circ - \\cos 6^\\circ.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "66^\\circ"
} | {
"index": 10714,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive angle $x$ that satisfies $\\sin 2x \\sin 3x = \\cos 2x \\cos 3x,$ in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "18^\\circ"
} | {
"index": 10715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive four-digit number that is divisible by each of the four smallest prime numbers."
}
] | math | {
"style": "rule",
"ground_truth": "1050"
} | {
"index": 10716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\\{1,2,\\ldots,2012\\}$ with $|A|=k$ , there exist three elements $x,y,z$ in $A$ such that $x=a+b$ , $y=b+c$ , $z=c+a$ , where $a,b,c$ are in $S$ and are distinct integers.\n\n*Proposed... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"index": 10717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ for which $315^2-n^2$ evenly divides $315^3-n^3$ .\n\n*Proposed by Kyle Lee*"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 10718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 10719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \\ldots < d_k = n$ , $k \\geq 7$ , and for which the following equalities hold: $$ d_7 = 2d_5 + 1\\text{ and }d_7 = 3d_4 - 1 $$ *Proposed by Mykyta Kharin*"
}
] | math | {
"style": "rule",
"ground_truth": "2024"
} | {
"index": 10720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors."
}
] | math | {
"style": "rule",
"ground_truth": "162000"
} | {
"index": 10721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ with the property that in the set $\\{70, 71, 72,... 70 + n\\}$ you can choose two different numbers whose product is the square of an integer."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 10722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \\times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board."
}
] | math | {
"style": "rule",
"ground_truth": "\\boxed{1999}"
} | {
"index": 10723,
"solution": "We claim that $n = 1999$ is the smallest such number. For $n \\le 1998$ , we can simply color any of the $1998$ squares forming the top row and the left column, but excluding the top left corner square.\n[asy] for(int i = 0; i < 10; ++i){ for(int j = 0; j < 10; ++j){ if((i == 0 || ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n\\neq 2004$ for which there exists a polynomial $f\\in\\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions."
}
] | math | {
"style": "rule",
"ground_truth": "(1002!)^2 + 2004"
} | {
"index": 10724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $ K$ such that every $ K$-element subset of $ \\{1,2,...,50 \\}$ contains two distinct elements $ a,b$ such that $ a\\plus{}b$ divides $ ab$."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 10725,
"solution": "\nTo find the smallest positive integer \\( K \\) such that every \\( K \\)-element subset of \\( \\{1, 2, \\ldots, 50\\} \\) contains two distinct elements \\( a \\) and \\( b \\) such that \\( a + b \\) divides \\( ab \\), we need to analyze the properties of the set and the divisib... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $M$ such that both $M$ and $M^2$ end in the same sequence of three digits $xyz$ when written in base $10$, where $x$ is not zero."
}
] | math | {
"style": "rule",
"ground_truth": "376"
} | {
"index": 10726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $M$ such that the three numbers $M$, $M+1$, and $M+2$, one of them is divisible by $3^2$, one of them is divisible by $5^2$, and one is divisible by $7^2$."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 10727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $N$ such that among the four numbers $N$, $N+1$, $N+2$, and $N+3$, one is divisible by $3^2$, one by $5^2$, one by $7^2$, and one by $11^2$."
}
] | math | {
"style": "rule",
"ground_truth": "363"
} | {
"index": 10728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $N$ such that any \"hydra\" with 100 necks, where each neck connects two heads, can be defeated by cutting at most $N$ strikes. Here, one strike can sever all the necks connected to a particular head $A$, and immediately after, $A$ grows new n... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $N$ with the following property: of the three numbers $N$, $N+1$, and $N+2$, one of them is divisible by $2^2$, one of them is divisible by $3^2$, one is divisible by $5^2$, and one is divisible by $7^2$."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 10730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $Y$ such that a number $U$, made only of digits 0s and 1s, is divisible by 15, and $U = 15Y$."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 10731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $b$ for which $x^2 + bx + 1760$ factors into a product of two polynomials, each having integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 10733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $b$ for which $x^2 + bx + 1764$ factors into a product of two polynomials, each having integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 10734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $b$ for which $x^2 + bx + 2008$ factors into a product of two polynomials, each having integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "259"
} | {
"index": 10735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "136"
} | {
"index": 10736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $b$ for which $x^2+bx+2008$ factors into a product of two binomials, each having integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "259"
} | {
"index": 10737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $b$ such that $1111_{b}$ ( 1111 in base $b$) is a perfect square. If no such $b$ exists, write \"No solution\"."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 10738,
"solution": "We have $1111_{b}=b^{3}+b^{2}+b+1=\\left(b^{2}+1\\right)(b+1)$. Note that $\\operatorname{gcd}\\left(b^{2}+1, b+1\\right)=\\operatorname{gcd}\\left(b^{2}+1-(b+1)(b-1), b+1\\right)=\\operatorname{gcd}(2, b+1)$, which is either 1 or 2 . If the gcd is 1 , then there is no solution as thi... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $\nz^{10} + z^9 + z^6+z^5+z^4+z+1\n$ divides $z^k-1$."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 10739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \\ldots + k^2$ is a multiple of $150$."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 10740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \\ldots + k^2$ is a multiple of $360$."
}
] | math | {
"style": "rule",
"ground_truth": "175"
} | {
"index": 10741,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\\ldots+k^2$ is a multiple of $360$."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 10742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $z^{10}+z^{9}+z^{6}+z^{5}+z^{4}+z+1$ divides $z^{k}-1$."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 10743,
"solution": "Let $Q(z)$ denote the polynomial divisor. We need that the roots of $Q$ are $k$-th roots of unity. With this in mind, we might observe that solutions to $z^{7}=1$ and $z \\neq 1$ are roots of $Q$, which leads to its factorization. Alternatively, we note that $$(z-1) Q(z)=z^{11}-z^{9}+... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $m$ such that for all positive integers $n \\geq m$, there exists a positive integer $l$ satisfying \n$$\nn < l^2 < \\left(1+\\frac{1}{2009}\\right)n.\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "16144325"
} | {
"index": 10744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ for which $$1!2!\\cdots(n-1)!>n!^{2}$$"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10745,
"solution": "Dividing both sides by $n!^{2}$, we obtain $$\\begin{aligned} \\frac{1!2!\\ldots(n-3)!(n-2)!(n-1)!}{[n(n-1)!][n(n-1)(n-2)!]} & >1 \\\\ \\frac{1!2!\\ldots(n-3)!}{n^{2}(n-1)} & >1 \\\\ 1!2!\\ldots(n-3)! & >n^{2}(n-1) \\end{aligned}$$ Factorials are small at first, so we can rule out som... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 10746,
"solution": "Using Simon's Favorite Factoring Trick, we rewrite as $[(x+7)(y-3)]^n = (x+7)^n(y-3)^n$. Both binomial expansions will contain $n+1$ non-like terms; their product will contain $(n+1)^2$ terms, as each term will have an unique power of $x$ or $y$ and so none of the terms will need to b... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that\n\\[\\begin{pmatrix} \\cos 170^\\circ & -\\sin 170^\\circ \\\\ \\sin 170^\\circ & \\cos 170^\\circ \\end{pmatrix}^n = \\mathbf{I}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 10747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that\n\\[\\begin{pmatrix} \\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\ -\\frac{\\sqrt{3}}{2} & \\frac{1}{2} \\end{pmatrix}^n = \\mathbf{I}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 10748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $$\\underbrace{2^{2^{2^{2}}}}_{n 2^{\\prime} s}>\\underbrace{((\\cdots((100!)!)!\\cdots)!)!}_{100 \\text { factorials }}$$"
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"index": 10749,
"solution": "Note that $2^{2^{2^{2}}}>100^{2}$. We claim that $a>b^{2} \\Longrightarrow 2^{a}>(b!)^{2}$, for $b>2$. This is because $$2^{a}>b^{2 b} \\Longleftrightarrow a>2 b \\log _{2}(b)$$ and $\\log _{2}(b)<b^{2} / 2$ for $b>2$. Then since $b^{b}>b$ ! this bound works. Then $$\\underbrace{\\lef... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $1^{2}+2^{2}+3^{2}+4^{2}+\\cdots+n^{2}$ is divisible by 100."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 10750,
"solution": "The sum of the first $n$ squares equals $n(n+1)(2 n+1) / 6$, so we require $n(n+1)(2 n+1)$ to be divisible by $600=24 \\cdot 25$. The three factors are pairwise relatively prime, so one of them must be divisible by 25 . The smallest $n$ for which this happens is $n=12$ $(2 n+1=25)$, b... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $\\frac{5^{n+1}+2^{n+1}}{5^{n}+2^{n}}>4.99$."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 10751,
"solution": "Writing $5^{n+1}=5 \\cdot 5^{n}$ and $2^{n+1}=2 \\cdot 2^{n}$ and cross-multiplying yields $0.01 \\cdot 5^{n}>2.99 \\cdot 2^{n}$, and re-arranging yields $(2.5)^{n}>299$. A straightforward calculation shows that the smallest $n$ for which this is true is $n=7$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $\\underbrace{2^{2 \\cdot 2}}_{n}>3^{3^{3^{3}}}$. (The notation $\\underbrace{2^{2^{2}}}_{n}$, is used to denote a power tower with $n 2$ 's. For example, $\\underbrace{2^{22^{2}}}_{n}$ with $n=4$ would equal $2^{2^{2^{2}}}$.)"
... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 10752,
"solution": "Clearly, $n \\geq 5$. When we take $n=5$, we have $$2^{2^{2^{2^{2}}}}=2^{2^{16}}<3^{3^{27}}=3^{3^{3^{3}}}.$$ On the other hand, when $n=6$, we have $$2^{2^{2^{2^{2^{2}}}}}=2^{2^{65536}}=4^{2^{65535}}>4^{4^{27}}>3^{3^{27}}=3^{3^{3^{3}}}.$$ Our answer is thus $n=6$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that for any $n$ mutually coprime integers greater than 1 and not exceeding 2009, there is at least one prime number among them."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 10753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that the $73$ fractions $\\frac{19}{n+21}, \\frac{20}{n+22},\\frac{21}{n+23},...,\\frac{91}{n+93}$ are all irreducible."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"index": 10754,
"solution": "\nTo solve this problem, we need to find the smallest positive integer \\( n \\) such that all 73 fractions of the form \\(\\frac{k}{n+k+20}\\) for \\( k = 19, 20, \\ldots, 91 \\) are irreducible. A fraction \\(\\frac{a}{b}\\) is irreducible if and only if \\(\\gcd(a, b) = 1\\).\n\nFo... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that the divisors of $n$ can be partitioned into three sets with equal sums."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 10755,
"solution": "I claim the answer is 120. First, note that $120=2^{3} \\cdot 3 \\cdot 5$, so the sum of divisors is $(1+2+4+8)(1+3)(1+5)=15 \\cdot 4 \\cdot 6=360$. Thus, we need to split the divisors into groups summing to 120 . But then we can just take $\\{120\\},\\{20,40,60\\},\\{1,2,3,4,5,6,8,10... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ is \"divisible by $x^{2}+1$ modulo 3\", or more precisely, either of the following equivalent conditions holds: there exist polynomials $P, Q$ with integer coefficients such that $(x+1)^{n}-1=\\left(x... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10756,
"solution": "We have $(x+1)^{2}=x^{2}+2 x+1 \\equiv 2 x,(x+1)^{4} \\equiv(2 x)^{2} \\equiv-4 \\equiv-1$, and $(x+1)^{8} \\equiv(-1)^{2}=1$. So the order $n$ divides 8, as $x+1$ and $x^{2}+1$ are relatively prime polynomials modulo 3 (or more conceptually, in $\\mathbb{F}_{3}[x]$ ), but cannot be s... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that there exists a complex number $z$, with positive real and imaginary part, satisfying $z^{n}=(\\bar{z})^{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 10757,
"solution": "Since $|z|=|\\bar{z}|$ we may divide by $|z|$ and assume that $|z|=1$. Then $\\bar{z}=\\frac{1}{z}$, so we are looking for the smallest positive integer $n$ such that there is a $2 n^{\\text {th }}$ root of unity in the first quadrant. Clearly there is a sixth root of unity in the fir... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that there exists a sequence of $n+1$ terms $a_{0}, a_{1}, \\cdots, a_{n}$ satisfying $a_{0}=0, a_{n}=2008$, and $\\left|a_{i}-a_{i-1}\\right|=i^{2}$ for $i=1,2, \\cdots, n$."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 10758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that, if there are initially $2n$ townspeople and 1 goon, then the probability the townspeople win is greater than $50\\%$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 10759,
"solution": "We instead consider the probability the goon wins. The game clearly must last $n$ days. The probability the goon is not sent to jail on any of these $n$ days is then $$\\frac{2n}{2n+1} \\cdot \\frac{2n-2}{2n-1} \\cdots \\cdot \\frac{2}{3}$$ If $n=2$ then the probability the goon wins ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that, if there are initially $n+1$ townspeople and $n$ goons, then the probability the townspeople win is less than $1\\%$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 10760,
"solution": "By a similar inductive argument, the probability for a given $n$ is $$p_{n}=\\frac{n!}{(2n+1)!!}$$ Clearly this is decreasing in $n$. It is easy to see that $$p_{5}=\\frac{1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5}{3 \\cdot 5 \\cdot 7 \\cdot 9 \\cdot 11}=\\frac{8}{693}>0.01$$ and $$p_{6}=... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ that satisfies the following conditions: For $n$, there exists a positive integer $k$ such that $\\frac{8}{15} < \\frac{n}{n+k} < \\frac{7}{13}$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 10761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ that satisfies the following two properties:\n\n1. $n$ has exactly 144 distinct positive divisors.\n2. Among the positive divisors of $n$, there are ten consecutive integers."
}
] | math | {
"style": "rule",
"ground_truth": "110880"
} | {
"index": 10762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ with the property that the polynomial $x^4 - nx + 63$ can be written as a product of two nonconstant polynomials with integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10763,
"solution": "You can factor the polynomial into two quadratic factors or a linear and a cubic factor.\n For two quadratic factors, let $x^2+ax+b$ and $x^2+cx+d$ be the two quadratics, so that\n\\[(x^2 + ax + b )(x^2 + cx + d) = x^4 + (a + c)x^3 + (b + d + ac)x^2 + (ad + bc)x + bd.\\]\n Therefore, ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$)"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 10764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( k \\) such that \\( (k-10)^{5026} \\geq 2013^{2013} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 10765,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( m \\) such that the equation regarding \\( x, y, \\) and \\( z \\):\n\\[ 2^x + 3^y - 5^z = 2m \\]\nhas no positive integer solutions."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10766,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n > 1 \\) such that the arithmetic mean of \\( 1^2, 2^2, 3^2, \\cdots, n^2 \\) is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 10767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n > 1 \\) such that the arithmetic mean of the squares of the integers \\( 1^2, 2^2, 3^2, \\ldots, n^2 \\) is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 10768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) for which there are exactly 2323 positive integers less than or equal to \\( n \\) that are divisible by 2 or 23, but not both."
}
] | math | {
"style": "rule",
"ground_truth": "4644"
} | {
"index": 10769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that\n\\[\n\\sqrt{5 n}-\\sqrt{5 n-4}<0.01\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "8001"
} | {
"index": 10770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that \\( n(n+1)(n+2) \\) is divisible by 247."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 10771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that a cube with side length \\( n \\) can be divided into 1996 smaller cubes, each with side length a positive integer."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 10772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that every \\( n \\)-element subset of \\( S = \\{1, 2, \\ldots, 150\\} \\) contains 4 numbers that are pairwise coprime (it is known that there are 35 prime numbers in \\( S \\))."
}
] | math | {
"style": "rule",
"ground_truth": "111"
} | {
"index": 10773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that for any given \\( n \\) rectangles with side lengths not exceeding 100, there always exist 3 rectangles \\( R_{1}, R_{2}, R_{3} \\) such that \\( R_{1} \\) can be nested inside \\( R_{2} \\) and \\( R_{2} \\) can be nested ... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 10774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that for any set of \\( n \\) distinct integers \\( a_{1}, a_{2}, \\ldots, a_{n} \\), the product of all differences \\( a_{i} - a_{j} \\) for \\( i < j \\) is divisible by 1991."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 10775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that the mean of the squares of the first \\( n \\) natural numbers (\\( n > 1 \\)) is an integer.\n\n(Note: The mean of the squares of \\( n \\) numbers \\( a_1, a_2, \\cdots, a_n \\) is given by \\( \\sqrt{\\frac{a_{1}^2 + a_{... | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 10776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that:\n1. \\( n \\) has exactly 144 distinct positive divisors.\n2. There are ten consecutive integers among the positive divisors of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "110880"
} | {
"index": 10777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) that is not less than 9, such that for any \\( n \\) integers (which can be the same) \\( a_{1}, a_{2}, \\cdots, a_{n} \\), there always exist 9 numbers \\( a_{i_{1}}, a_{i_{2}}, \\cdots, a_{i_{9}} \\) (where \\(1 \\leq i_{1} < i_{2}... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 10778,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) that satisfies the following condition: If the set \\( A = \\{1, 2, 3, \\ldots, n\\} \\) is arbitrarily divided into 63 pairwise disjoint subsets \\( A_1, A_2, A_3, \\ldots, A_{63} \\) (which are non-empty and their union is the set ... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 10779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer divisible by $10$, $11$, and $12$."
}
] | math | {
"style": "rule",
"ground_truth": "660"
} | {
"index": 10780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer n such that n has exactly 144 positive divisors including 10 consecutive integers."
}
] | math | {
"style": "rule",
"ground_truth": "110880"
} | {
"index": 10781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer solution to $\\tan{19x^{\\circ}}=\\dfrac{\\cos{96^{\\circ}}+\\sin{96^{\\circ}}}{\\cos{96^{\\circ}}-\\sin{96^{\\circ}}}$."
}
] | math | {
"style": "rule",
"ground_truth": "159"
} | {
"index": 10782,
"solution": "$\\dfrac{\\cos{96^{\\circ}}+\\sin{96^{\\circ}}}{\\cos{96^{\\circ}}-\\sin{96^{\\circ}}} = \\dfrac{1 + \\tan{96^{\\circ}}}{1-\\tan{96^{\\circ}}}$ which is the same as $\\dfrac{\\tan{45^{\\circ}} + \\tan{96^{\\circ}}}{1-\\tan{45^{\\circ}}\\tan{96^{\\circ}}} = \\tan{141{^\\circ}}$.\n So $... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer that cannot be expressed in the form $\\frac{2^a - 2^b}{2^c - 2^d}$ , where $a$ , $ b$ , $c$ , $d$ are non-negative integers."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer that cannot be expressed in the form $\\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer that is both an integer power of 11 and is not a palindrome."
}
] | math | {
"style": "rule",
"ground_truth": "161051"
} | {
"index": 10785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer that is both an integer power of 13 and is not a palindrome."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 10786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer that is both an integer power of 7 and is not a palindrome."
}
] | math | {
"style": "rule",
"ground_truth": "2401"
} | {
"index": 10787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer which cannot be expressed in the form \\(\\frac{2^{a}-2^{b}}{2^{c}-2^{d}}\\) where \\(a, b, c, d\\) are non-negative integers."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive integer whose cube ends in $888$."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 10789,
"solution": "Let $x^3 = 1000a + 888$. We factor an $8$ out of the right hand side, and we note that $x$ must be of the form $x = 2y$, where $y$ is a positive integer. Then, this becomes $y^3 = 125a + 111$. Taking mod $5$, $25$, and $125$, we get $y^3 \\equiv 1\\pmod 5$, $y^3 \\equiv 11\\pmod{25}$,... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive multiple of 9 that can be written using only the digits: (a) 0 and 1; (b) 1 and 2."
}
] | math | {
"style": "rule",
"ground_truth": "12222"
} | {
"index": 10790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive number $\\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\\in\\{z\\in C\\big| |z|<1\\}$ ,if $z_1+z_2+z_3=0$, then $$\\left|z_1z_2 +z_2z_3+z_3z_1\\right|^2+\\left|z_1z_2z_3\\right|^2 <\\lambda .$$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10791,
"solution": "\nWe aim to find the smallest positive number \\(\\lambda\\) such that for any complex numbers \\(z_1, z_2, z_3 \\in \\{z \\in \\mathbb{C} \\mid |z| < 1\\}\\) with \\(z_1 + z_2 + z_3 = 0\\), the following inequality holds:\n\\[\n\\left|z_1z_2 + z_2z_3 + z_3z_1\\right|^2 + \\left|z_1z_... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive number $\\lambda$ such that for any triangle with side lengths $a, b, c$, given $a \\geqslant \\frac{b+c}{3}$, it holds that\n$$\na c + b c - c^{2} \\leqslant \\lambda\\left(a^{2} + b^{2} + 3 c^{2} + 2 a b - 4 b c\\right).\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} + 1}{7}"
} | {
"index": 10792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive number $\\lambda$ such that for any triangle with side lengths $a, b, c$, if $a \\geqslant \\frac{b+c}{3}$, then the following inequality holds:\n$$\nac + bc - c^2 \\leqslant \\lambda \\left( a^2 + b^2 + 3c^2 + 2ab - 4bc \\right).\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} + 1}{7}"
} | {
"index": 10793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive number $\\lambda$, such that for any $12$ points on the plane $P_1,P_2,\\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\\sum_{1\\le i<j\\le 12} |P_iP_j|^2\\le \\lambda$."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 10794,
"solution": "\nWe are tasked with finding the smallest positive number \\(\\lambda\\) such that for any 12 points on the plane \\(P_1, P_2, \\ldots, P_{12}\\) (which can overlap), if the distance between any two of them does not exceed 1, then \\(\\sum_{1 \\le i < j \\le 12} |P_iP_j|^2 \\le \\lamb... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive number \\( c \\) with the following property: For any integer \\( n \\geqslant 4 \\) and any set \\( A \\subseteq \\{1, 2, \\ldots, n\\} \\), if \\( |A| > c n \\), then there exists a function \\( f: A \\rightarrow \\{1, -1\\} \\) such that \\( \\left... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 10795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive prime that divides \\( n^2 + 5n + 23 \\) for some integer \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 10796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive real $\\alpha$, such that $$\\frac{x+y} {2}\\geq \\alpha\\sqrt{xy}+(1 - \\alpha)\\sqrt{\\frac{x^2+y^2}{2}}$$ for all positive reals $x, y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 10797,
"solution": "\nTo solve the problem of finding the smallest positive real \\(\\alpha\\) such that \n\n\\[\n\\frac{x+y}{2} \\geq \\alpha\\sqrt{xy} + (1 - \\alpha)\\sqrt{\\frac{x^2 + y^2}{2}}\n\\]\n\nfor all positive reals \\(x\\) and \\(y\\), we proceed as follows:\n\n### Step 1: Analyze Special Ca... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive real number $\\lambda$ such that for every numbers $a_1,a_2,a_3 \\in \\left[0, \\frac{1}{2} \\right]$ and $b_1,b_2,b_3 \\in (0, \\infty)$ with $\\sum\\limits_{i=1}^3a_i=\\sum\\limits_{i=1}^3b_i=1,$ we have $$ b_1b_2b_3 \\le \\lambda (a_1b_1+... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"index": 10798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest positive real number $r$ with the following property: For every choice of $2023$ unit vectors $v_1,v_2, \\dots ,v_{2023} \\in \\mathbb{R}^2$ , a point $p$ can be found in the plane such that for each subset $S$ of $\\{1,2, \\dots , 2023\\}$ , the su... | math | {
"style": "rule",
"ground_truth": "\\frac{2023}{2}"
} | {
"index": 10799,
"solution": ""
} |
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