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 remainder when the sum \\[75+76+77+78+79+80+81+82\\]is divided by 16."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 10600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the remainder when the value of $m$ is divided by 1000 in the number of increasing sequences of positive integers $a_1 \\le a_2 \\le a_3 \\le \\cdots \\le a_6 \\le 1500$ such that $a_i-i$ is odd for $1\\le i \\le 6$. The total number of sequences can be expressed as ${m \\... | math | {
"style": "rule",
"ground_truth": "752"
} | {
"index": 10601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the remainder when\\[\\binom{\\binom{3}{2}}{2} + \\binom{\\binom{4}{2}}{2} + \\dots + \\binom{\\binom{40}{2}}{2}\\]is divided by $1000$.\n ~ pi_is_3.14"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 10602,
"solution": "Doing simple algebra calculation will give the following equation: \\begin{align*} \\binom{\\binom{n}{2}}{2} = \\frac{\\frac{n(n-1)}{2} \\cdot (\\frac{n(n-1)}{2}-1)}{2} \\\\ = \\frac{n(n-1)(n^2-n-2)}{8} \\\\ = \\frac{(n+1)n(n-1)(n-2)}{8} \\\\ = \\frac{(n+1)!}{8\\cdot (n-3)!} = 3 \\cdo... |
DeepScaleR | [
{
"role": "user",
"content": "Find the repetend in the decimal representation of $\\frac{5}{17}$."
}
] | math | {
"style": "rule",
"ground_truth": "294117647058823529"
} | {
"index": 10603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the residue of $182\\cdot 12 - 15\\cdot 7 + 3\\pmod{14}$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the residue of the function\n\n$$\nw=z^{2} \\sin \\frac{1}{z+1}\n$$\n\nat its singular point."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 10605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the result of $(1011101_2 + 1101_2) \\times 101010_2 \\div 110_2$. Express your answer in base 2."
}
] | math | {
"style": "rule",
"ground_truth": "1110111100_2"
} | {
"index": 10606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the result of $46_8 - 63_8$ and express your answer in base 10."
}
] | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"index": 10607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the rightmost non-zero digit of the expansion of (20)(13!)."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 10608,
"solution": "We can rewrite this as $(10 \\times 2)(13 \\times 12 \\times 11 \\times 10 \\times 9 \\times 8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1)=\\left(10^{3}\\right)(2 \\times 13 \\times 12 \\times 11 \\times 9 \\times 8 \\times 7 \\times 6 \\times 4 \\times 3)$;... |
DeepScaleR | [
{
"role": "user",
"content": "Find the root $x$ of the equation $\\log x = 4 - x$ where $x \\in (k, k+1)$, and $k \\in \\mathbb{Z}$. What is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "k = 3"
} | {
"index": 10609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the root of the following equation to three significant digits:\n\n$$\n(\\sqrt{5}-\\sqrt{2})(1+x)=(\\sqrt{6}-\\sqrt{3})(1-x)\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "-0.068"
} | {
"index": 10610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the roots of\n\\[6x^4 - 35x^3 + 62x^2 - 35x + 6 = 0.\\]Enter the roots, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "2, 3, \\frac{1}{2}, \\frac{1}{3}"
} | {
"index": 10611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the roots of $6x^4+19x^3-51x^2+20x$. Enter your answer as a list of numbers separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "0, \\frac{1}{2}, \\frac{4}{3}, -5"
} | {
"index": 10612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the roots of $z^2 - z = 5 - 5i.$\n\nEnter the roots, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "3 - i, -2 + i"
} | {
"index": 10613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \\ne c$ (numbers $a, b, c, d$ are not given)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2015}{2}"
} | {
"index": 10614,
"solution": "\nTo find the roots of the equation \\((x-a)(x-b)=(x-c)(x-d)\\), given that \\(a + d = b + c = 2015\\) and \\(a \\neq c\\), we will simplify the equation and determine the solutions.\n\n### Step 1: Expand Both Sides\nExpanding both sides of the equation, we have:\n\\[\n(x-a)(x-b) = x^... |
DeepScaleR | [
{
"role": "user",
"content": "Find the set consisting of all real values of $x$ such that the three numbers $2^{x}, 2^{x^{2}}, 2^{x^{3}}$ form a non-constant arithmetic progression (in that order)."
}
] | math | {
"style": "rule",
"ground_truth": "\\varnothing"
} | {
"index": 10615,
"solution": "The empty set, $\\varnothing$. Trivially, $x=0,1$ yield constant arithmetic progressions; we show that there are no other possibilities. If these numbers do form a progression, then, by the AM-GM (arithmetic mean-geometric mean) inequality, $$2 \\cdot 2^{x^{2}}=2^{x}+2^{x^{3}} \\geq 2... |
DeepScaleR | [
{
"role": "user",
"content": "Find the set of $x$-values satisfying the inequality $|\\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive, \n$-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $1<a<2$ means that a can have any value between $1$ and $2$, excluding $1$ and $2$. ]"
}
] | math | {
"style": "rule",
"ground_truth": "-1 < x < 11"
} | {
"index": 10616,
"solution": "\nTo solve the inequality $|\\frac{5-x}{3}| < 2$, we start by understanding the absolute value inequality. The inequality $|A| < B$ (where $B > 0$) implies that $-B < A < B$. Applying this to our given inequality:\n\n$$ -2 < \\frac{5-x}{3} < 2 $$\n\nWe will solve this compound inequal... |
DeepScaleR | [
{
"role": "user",
"content": "Find the set of all attainable values of $\\frac{ab+b^{2}}{a^{2}+b^{2}}$ for positive real $a, b$."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(0, \\frac{1+\\sqrt{2}}{2}\\right]"
} | {
"index": 10617,
"solution": "Suppose that $k=\\frac{ab+b^{2}}{a^{2}+b^{2}}$ for some positive real $a, b$. We claim that $k$ lies in $\\left(0, \\frac{1+\\sqrt{2}}{2}\\right]$. Let $x=\\frac{a}{b}$. We have that $\\frac{ab+b^{2}}{a^{2}+b^{2}}=\\frac{\\frac{a}{b}+1}{\\left(\\frac{a}{b}\\right)^{2}+1}=\\frac{x+1}{x... |
DeepScaleR | [
{
"role": "user",
"content": "Find the set of solutions for $x$ in the inequality $\\frac{x+1}{x+2} > \\frac{3x+4}{2x+9}$ when $x \\neq -2, x \\neq \\frac{9}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-9}{2} \\leq x \\leq -2 \\cup \\frac{1-\\sqrt{5}}{2} < x < \\frac{1+\\sqrt{5}}{2}"
} | {
"index": 10618,
"solution": "There are 3 possible cases of $x$: 1) $-\\frac{9}{2} < x$, 2) $\\frac{9}{2} \\leq x \\leq -2$, 3) $-2 < x$. For the cases (1) and (3), $x+2$ and $2x+9$ are both positive or negative, so the following operation can be carried out without changing the inequality sign: $$\\begin{aligned}... |
DeepScaleR | [
{
"role": "user",
"content": "Find the set of values for parameter \\(a\\) for which the sum of the cubes of the roots of the equation \\(x^{2} + ax + a + 1 = 0\\) is equal to 1."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 10619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the set of values of the parameter \\(a\\) for which the sum of the cubes of the roots of the equation \\(x^{2}-a x+a+2=0\\) is equal to -8."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 10620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the shortest distance between the lines $\\frac{x+2}{2}=\\frac{y-1}{3}=\\frac{z}{1}$ and $\\frac{x-3}{-1}=\\frac{y}{1}=\\frac{z+1}{2}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{3}}{3}"
} | {
"index": 10621,
"solution": "First we find the direction of a line perpendicular to both of these lines. By taking the cross product $(2,3,1) \\times(-1,1,2)=(5,-5,5)$ we find that the plane $x-y+z+3=0$ contains the first line and is parallel to the second. Now we take a point on the second line, say the point $(... |
DeepScaleR | [
{
"role": "user",
"content": "Find the shortest distance between the point $(5,10)$ and the parabola given by the equation $x = \\frac{y^2}{3}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{53}"
} | {
"index": 10622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x = \\frac{y^2}{2}.$"
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{17}"
} | {
"index": 10623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the shortest distance from a point on the curve $y=x^{2}-\\ln x$ to the line $x-y-2=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 10624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the shortest distance from the line $3 x+4 y=25$ to the circle $x^{2}+y^{2}=6 x-8 y$."
}
] | math | {
"style": "rule",
"ground_truth": "7 / 5"
} | {
"index": 10625,
"solution": "The circle is $(x-3)^{2}+(y+4)^{2}=5^{2}$. The center $(3,-4)$ is a distance of $$ \\frac{|3 \\cdot 3+4 \\cdot-4-25|}{\\sqrt{3^{2}+4^{2}}}=\\frac{32}{5} $$ from the line, so we subtract 5 for the radius of the circle and get $7 / 5$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the simplest method to solve the system of equations using substitution $$\\begin{cases} x=2y\\textcircled{1} \\\\ 2x-y=5\\textcircled{2} \\end{cases}$$"
}
] | math | {
"style": "rule",
"ground_truth": "y = \\frac{5}{3}"
} | {
"index": 10626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, with two vertices lying on the base, is a constant value."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 10627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the slope angle of the tangent line to the curve $f(x)=\\frac{1}{3}{x}^{3}-{x}^{2}+5$ at $x=1$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"index": 10628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the slope of the line $3x+5y=20$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"index": 10629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the slope of the line that is perpendicular to the line $2x + 3y = 6$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 10630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest $n$ such that every subset of $\\{1, 2, 3, . . . , 2004 \\}$ with $n$ elements contains at least two elements that are relatively prime."
}
] | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"index": 10631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest $n$ such that $n!$ ends with 10 zeroes."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 10632,
"solution": "The number of zeroes that $n!$ ends with is the largest power of 10 dividing $n!$. The exponent of 5 dividing $n!$ exceeds the exponent of 2 dividing $n!$, so we simply seek the exponent of 5 dividing $n!$. For a number less than 125, this exponent is just the number of multiples of 5... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest $n$ such that $n$! ends in 290 zeroes."
}
] | math | {
"style": "rule",
"ground_truth": "1170"
} | {
"index": 10633,
"solution": "Each 0 represents a factor of $10=2 \\cdot 5$. Thus, we wish to find the smallest factorial that contains at least 290 2's and 290 5's in its prime factorization. Let this number be $n$!, so the factorization of $n$! contains 2 to the power $p$ and 5 to the power $q$, where $$p=\\left... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest 6-digit palindrome in base 2, that can be expressed as a 4-digit palindrome in a different base. Provide your response in base 2."
}
] | math | {
"style": "rule",
"ground_truth": "100001_2"
} | {
"index": 10634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest \\( n > 4 \\) for which we can find a graph on \\( n \\) points with no triangles and such that for every two unjoined points we can find just two points joined to both of them."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest \\( n \\) such that whenever the elements of the set \\(\\{1, 2, \\ldots, n\\}\\) are colored red or blue, there always exist \\( x, y, z, w \\) (not necessarily distinct) of the same color such that \\( x + y + z = w \\)."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest \\(k\\) such that for any arrangement of 3000 checkers in a \\(2011 \\times 2011\\) checkerboard, with at most one checker in each square, there exist \\(k\\) rows and \\(k\\) columns for which every checker is contained in at least one of these rows or column... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"index": 10637,
"solution": "If there is a chip in every square along a main diagonal, then we need at least 1006 rows and columns to contain all these chips. We are left to show that 1006 is sufficient. Take the 1006 rows with greatest number of chips. Assume without loss of generality they are the first 1006 ro... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest composite number that has no prime factors less than 10."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 10638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest composite number that has no prime factors less than 15."
}
] | math | {
"style": "rule",
"ground_truth": "289"
} | {
"index": 10639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest composite number that has no prime factors less than 20."
}
] | math | {
"style": "rule",
"ground_truth": "529"
} | {
"index": 10640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest constant $ C$ such that for all real $ x,y$ \r\n\\[ 1\\plus{}(x\\plus{}y)^2 \\leq C \\cdot (1\\plus{}x^2) \\cdot (1\\plus{}y^2)\\]\r\nholds."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"index": 10641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest constant $C$ such that for all real numbers $x, y, z$ satisfying $x + y + z = -1$, the following inequality holds:\n$$\n\\left|x^3 + y^3 + z^3 + 1\\right| \\leqslant C \\left|x^5 + y^5 + z^5 + 1\\right|.\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"index": 10642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, \\[ \\int_0^1 \\left| P(x) \\right|\\,dx \\leq C \\max_{x \\in [0,1]} \\left| P(x) \\right|. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 10643,
"solution": "We prove that the smallest such value of $C$ is $5/6$. We first reduce to the case where $P$ is nonnegative in $[0,1]$ and $P(0) = 0$. To achieve this reduction, suppose that a given value $C$ obeys the inequality for such $P$. For $P$ general, divide the interval $[0,1]$ into subinte... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest constant $D$ so that\n\\[ 2x^2 + 3y^2 + z^2 + 3 \\ge D(x + y + z) \\]\nfor all real numbers $x$, $y$, and $z$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{\\frac{72}{11}}"
} | {
"index": 10644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest constant $M$, such that for any positive real numbers $x$, $y$, $z$, and $w$,\n\\[\\sqrt{\\frac{x}{y + z + w}} + \\sqrt{\\frac{y}{x + z + w}} + \\sqrt{\\frac{z}{x + y + w}} + \\sqrt{\\frac{w}{x + y + z}} < M.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{\\sqrt{3}}"
} | {
"index": 10645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest constant $N$, such that for any triangle with sides $a, b,$ and $c$, and perimeter $p = a + b + c$, the inequality holds:\n\\[\n\\frac{a^2 + b^2 + k}{c^2} > N\n\\]\nwhere $k$ is a constant."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 10646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest constant \\(C\\) such that\n\\[ x^2 + y^2 + z^2 + 1 \\ge C(x + y + z) \\]\nfor all real numbers \\(x, y,\\) and \\(z.\\)"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{4}{3}}"
} | {
"index": 10647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest digit $d$ so that $437{,}d03$ is divisible by 9."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest exact square with last digit not $0$ , such that after deleting its last two digits we shall obtain another exact square."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 10649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest four-digit number SEEM for which there is a solution to the puzzle MY + ROZH = SEEM. (The same letters correspond to the same digits, different letters - different.)"
}
] | math | {
"style": "rule",
"ground_truth": "2003"
} | {
"index": 10650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest four-digit number that is equal to the square of the sum of the numbers formed by its first two digits and its last two digits."
}
] | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"index": 10651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest four-digit number that satisfies the following system of congruences:\n\\begin{align*}\n3x &\\equiv 6 \\pmod{12} \\\\\n5x + 20 &\\equiv 25 \\pmod{15} \\\\\n3x - 2 &\\equiv 2x \\pmod{35}\n\\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "1274"
} | {
"index": 10652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest four-digit number whose product of all digits equals 512."
}
] | math | {
"style": "rule",
"ground_truth": "1888"
} | {
"index": 10653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest fraction which, when divided by each of the fractions \\(\\frac{21}{25}\\) and \\(\\frac{14}{15}\\), results in natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "42/5"
} | {
"index": 10654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer $k > 1$ for which $n^k-n$ is a multiple of $2010$ for every integer positive $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "133"
} | {
"index": 10655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer $n$ such that an $n\\times n$ square can be partitioned into $40\\times 40$ and $49\\times 49$ squares, with both types of squares present in the partition, if\r\na) $40|n$ ; b) $49|n$ ; c) $n\\in \\mathbb N$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1960"
} | {
"index": 10656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer $k \\geq 2$ such that for every partition of the set $\\{2, 3,\\hdots, k\\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$ and $c$ with $ab = c$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 10657,
"solution": "\nWe are tasked to find the smallest integer \\( k \\geq 2 \\) such that every partition of the set \\( \\{2, 3, \\ldots, k\\} \\) into two parts results in at least one part containing numbers \\( a \\), \\( b \\), and \\( c \\) such that \\( ab = c \\).\n\nTo solve this, we will pro... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer $k$ for which the conditions\n (1) $a_1,a_2,a_3\\cdots$ is a nondecreasing sequence of positive integers\n (2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$\n (3) $a_9=k$\n are satisfied by more than one sequence."
}
] | math | {
"style": "rule",
"ground_truth": "748"
} | {
"index": 10658,
"solution": "From $(2)$, $a_9=$ $a_8+a_7=2a_7+a_6=3a_6+2a_5=5a_5+3a_4=8a_4+5a_3=13a_3+8a_2=21a_2+13a_1$ $=k$\n Suppose that $a_1=x_0$ is the smallest possible value for $a_1$ that yields a good sequence, and $a_2=y_0$ in this sequence. So, $13x_0+21y_0=k$.\n Since $\\gcd(13,21)=1$, the next smalle... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer $n \\geq 5$ for which there exists a set of $n$ distinct pairs $\\left(x_{1}, y_{1}\\right), \\ldots,\\left(x_{n}, y_{n}\\right)$ of positive integers with $1 \\leq x_{i}, y_{i} \\leq 4$ for $i=1,2, \\ldots, n$, such that for any indices $r, s \\in\\{1... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10659,
"solution": "In other words, we have a set $S$ of $n$ pairs in $(\\mathbb{Z} / 4 \\mathbb{Z})^{2}$ closed under addition. Since $1+1+1+1 \\equiv 0(\\bmod 4)$ and $1+1+1 \\equiv-1(\\bmod 4),(0,0) \\in S$ and $S$ is closed under (additive) inverses. Thus $S$ forms a group under addition (a subgroup ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer $n$ such that $(x^2+y^2+z^2)^2 \\le n(x^4+y^4+z^4)$ for all real numbers $x,y$, and $z$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 10660,
"solution": "To find the smallest integer $n$ such that $(x^2+y^2+z^2)^2\\le n(x^4+y^4+z^4)$ holds for all real numbers $x, y, z$, we can utilize the Cauchy-Schwarz inequality in its sum of squares form.\n\n1. **Applying Cauchy-Schwarz Inequality:**\n The Cauchy-Schwarz inequality states that fo... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer $n$ such that $\\sqrt{n+99}-\\sqrt{n}<1$."
}
] | math | {
"style": "rule",
"ground_truth": "2402"
} | {
"index": 10661,
"solution": "This is equivalent to $$\\begin{aligned} \\sqrt{n+99} & <\\sqrt{n}+1 \\\\ n+99 & <n+1+2 \\sqrt{n} \\\\ 49 & <\\sqrt{n} \\end{aligned}$$ So the smallest integer $n$ with this property is $49^{2}+1=2402$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer $n$ such that each subset of $\\{1,2,\\ldots, 2004\\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$."
}
] | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"index": 10662,
"solution": "\nTo solve the problem of finding the smallest integer \\( n \\) such that each subset of \\(\\{1, 2, \\ldots, 2004\\}\\) with \\( n \\) elements has two distinct elements \\( a \\) and \\( b \\) for which \\( a^2 - b^2 \\) is a multiple of \\( 2004 \\), we start by analyzing the stru... |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer \\( k \\) such that when \\( 5 \\) is multiplied by a number consisting of \\( k \\) digits of \\( 7 \\) (i.e., \\( 777\\ldots7 \\) with \\( k \\) sevens), the resulting product has digits summing to \\( 800 \\).\nA) 86\nB) 87\nC) 88\nD) 89\nE) 90"
}... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 10663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer \\( n > 1 \\) such that \\(\\frac{1^2 + 2^2 + 3^2 + \\ldots + n^2}{n}\\) is a square."
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 10664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer \\( n \\) such that the expanded form of \\( (xy - 7x - 3y + 21)^n \\) has 2012 terms."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 10665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer satisfying the following conditions:\n\n$\\bullet$ I. The sum of the squares of its digits is $85$.\n\n$\\bullet$ II. Each digit is larger than the one on its left.\n\nWhat is the product of the digits of this integer?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 10666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest integer starting with the digit 7 that becomes three times smaller when this digit is moved to the end. Find all such numbers."
}
] | math | {
"style": "rule",
"ground_truth": "7241379310344827586206896551"
} | {
"index": 10667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest multiple of 9 that does not contain any odd digits."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 10668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural decimal number \\(n\\) whose square starts with the digits 19 and ends with the digits 89."
}
] | math | {
"style": "rule",
"ground_truth": "1383"
} | {
"index": 10669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number $A$ that satisfies the following conditions:\n\na) Its notation ends with the digit 6;\n\nb) By moving the digit 6 from the end of the number to its beginning, the number increases fourfold."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"index": 10670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number $m$ for which there exists a natural number $n$ such that the sets of the last 2014 digits in the decimal representation of the numbers $a=2015^{3 m+1}$ and $b=2015^{6 n+2}$ are identical, with the condition that $a<b$."
}
] | math | {
"style": "rule",
"ground_truth": "671"
} | {
"index": 10671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number $n$ such that $\\sin n^{\\circ} = \\sin (2016n^{\\circ})$."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 10672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number $n$ with the following property: in any $n$-element subset of $\\{1, 2, \\cdots, 60\\}$, there must be three numbers that are pairwise coprime."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 10673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number \\( N \\) that is divisible by \\( p \\), ends with \\( p \\), and has a digit sum equal to \\( p \\), given that \\( p \\) is a prime number and \\( 2p+1 \\) is a cube of a natural number."
}
] | math | {
"style": "rule",
"ground_truth": "11713"
} | {
"index": 10674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) for which \\( (n+1)(n+2)(n+3)(n+4) \\) is divisible by 1000."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 10675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) for which the number \\( A = n^3 + 12n^2 + 15n + 180 \\) is divisible by 23."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) such that both \\( n^2 \\) and \\( (n+1)^2 \\) contain the digit 7."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 10677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) such that the sum of the digits of each of the numbers \\( n \\) and \\( n+1 \\) is divisible by 17."
}
] | math | {
"style": "rule",
"ground_truth": "8899"
} | {
"index": 10678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) that satisfies the following conditions:\n1. The units digit of \\( n \\) is 6.\n2. If the units digit 6 is moved to the front of the number, the new number is 4 times \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"index": 10679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number ending in the digit 2 such that it doubles when this digit is moved to the beginning."
}
] | math | {
"style": "rule",
"ground_truth": "105263157894736842"
} | {
"index": 10680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number ending in the digit 4 that becomes 4 times larger when its last digit is moved to the beginning of the number."
}
] | math | {
"style": "rule",
"ground_truth": "102564"
} | {
"index": 10681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number ending in the digit 6, which increases fourfold when its last digit is moved to the beginning of the number."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"index": 10682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number ending with the digit 2, which doubles if this digit is moved to the beginning."
}
] | math | {
"style": "rule",
"ground_truth": "105263157894736842"
} | {
"index": 10683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $ *Gheorghe Iurea*"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 10684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number such that when it is multiplied by 9, the resulting number consists of the same digits in a different order."
}
] | math | {
"style": "rule",
"ground_truth": "1089"
} | {
"index": 10685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number such that when multiplied by 9, the resulting number consists of the same digits but in some different order."
}
] | math | {
"style": "rule",
"ground_truth": "1089"
} | {
"index": 10686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that begins with the digit five, which is reduced by four times if this five is removed from the beginning of its decimal representation and appended to its end."
}
] | math | {
"style": "rule",
"ground_truth": "512820"
} | {
"index": 10687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that can be represented in exactly two ways as \\(3x + 4y\\), where \\(x\\) and \\(y\\) are natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 10688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that cannot be written in the form \\(\\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\\), where \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that consists of identical digits and is divisible by 18."
}
] | math | {
"style": "rule",
"ground_truth": "666"
} | {
"index": 10690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that ends with the digit 6 such that moving this digit to the front increases the number exactly fourfold."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"index": 10691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that has exactly 70 natural divisors (including 1 and the number itself)."
}
] | math | {
"style": "rule",
"ground_truth": "25920"
} | {
"index": 10692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that is a multiple of 80, in which two of its distinct digits can be rearranged so that the resulting number is also a multiple of 80."
}
] | math | {
"style": "rule",
"ground_truth": "1520"
} | {
"index": 10693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1."
}
] | math | {
"style": "rule",
"ground_truth": "11111111100000000"
} | {
"index": 10694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that is greater than the sum of its digits by 1755."
}
] | math | {
"style": "rule",
"ground_truth": "1770"
} | {
"index": 10695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 8."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 10696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that starts with the digit 5, which, when this 5 is removed from the beginning of its decimal representation and appended to its end, becomes four times smaller."
}
] | math | {
"style": "rule",
"ground_truth": "512820"
} | {
"index": 10697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number that starts with the digit five, which becomes four times smaller if this five is removed from the beginning of its decimal representation and appended to the end."
}
] | math | {
"style": "rule",
"ground_truth": "512820"
} | {
"index": 10698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the smallest natural number whose digits sum up to 47."
}
] | math | {
"style": "rule",
"ground_truth": "299999"
} | {
"index": 10699,
"solution": ""
} |
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