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 minimum value of\n\\[(12 - x)(10 - x)(12 + x)(10 + x).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-484"
} | {
"index": 10100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[(15 - x)(8 - x)(15 + x)(8 + x).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-6480.25"
} | {
"index": 10101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[(\\sin x + \\csc x)^2 + (\\cos x + \\sec x)^2\\]for $0 < x < \\frac{\\pi}{2}.$"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 10102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[2 \\cos \\theta + \\frac{1}{\\sin \\theta} + \\sqrt{2} \\tan \\theta\\]for $0 < \\theta < \\frac{\\pi}{2}.$"
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{2}"
} | {
"index": 10103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[2 \\sqrt{x} + \\frac{1}{x}\\]for $x > 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 10104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[2x^2 + 2xy + y^2 - 2x + 2y + 4\\]over all real numbers $x$ and $y.$"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 10105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[3x^2 + 3xy + y^2 - 3x + 3y + 9\\]\nover all real numbers $x$ and $y.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{45}{8}"
} | {
"index": 10106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[4x + \\frac{1}{x^4}\\]for $x > 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 10107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[9y + \\frac{1}{y^6}\\] for \\(y > 0\\)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\frac{\\sin^6 x + \\cos^6 x + 1}{\\sin^4 x + \\cos^4 x + 1}\\]over all real values $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 10109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\frac{x^2 + 7}{\\sqrt{x^2 + 3}}\\]over all real numbers $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 10110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\frac{x^2}{x - 8}\\]for $x > 8.$"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 10111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\frac{x^2}{y - 1} + \\frac{y^2}{x - 1}\\]for real numbers $x > 1$ and $y > 1.$"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sin^4 x + \\frac{3}{2} \\cos^4 x,\\]as $x$ varies over all real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 10113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sqrt{x^2 + (1 + 2x)^2} + \\sqrt{(x - 1)^2 + (x - 1)^2}\\]over all real numbers $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} \\times \\sqrt{5}"
} | {
"index": 10114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sqrt{x^2 + (1 - x)^2} + \\sqrt{(1 - x)^2 + (1 + x)^2}\\]over all real numbers $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 10115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sqrt{x^2 + (2 - x)^2} + \\sqrt{(2 - x)^2 + (2 + x)^2}\\]over all real numbers $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 10116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sqrt{x^2 + (x-2)^2} + \\sqrt{(x-2)^2 + (x+2)^2}\\] over all real numbers $x$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 10117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[f(x) = x + \\frac{1}{x} + \\frac{1}{x + \\frac{1}{x}}\\]for $x > 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 10118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[f(x) = x + \\frac{x}{x^2 + 1} + \\frac{x(x + 4)}{x^2 + 2} + \\frac{2(x + 2)}{x(x^2 + 2)}\\]for $x > 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 10119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[x^2 + 2xy + 3y^2 - 6x - 2y,\\]over all real numbers $x$ and $y.$"
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"index": 10120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[x^2 + 8x + \\frac{64}{x^3}\\]for $x > 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 10121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[x^2 + xy + y^2\\]over all real numbers $x$ and $y.$"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 10122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of \n\\[(13 - x)(11 - x)(13 + x)(11 + x) + 1000.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "424"
} | {
"index": 10123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of \n\\[x^3 + 12x + \\frac{81}{x^4}\\]\nfor $x > 0$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 10124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $ \\int_0^1 |e^{ \\minus{} x} \\minus{} a|dx\\ ( \\minus{} \\infty < a < \\infty)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1 - 2e^{-1}"
} | {
"index": 10125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $m$ such that any $m$ -element subset of the set of integers $\\{1,2,...,2016\\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\\le 3$ ."
}
] | math | {
"style": "rule",
"ground_truth": "505"
} | {
"index": 10126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $9^x - 3^x + 1$ over all real numbers $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 10127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $\\frac{9x^2\\sin^2 x + 4}{x\\sin x}$ for $0 < x < \\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10128,
"solution": "We can rewrite the numerator to be a perfect square by adding $-\\dfrac{12x \\sin x}{x \\sin x}$. Thus, we must also add back $12$.\n This results in $\\dfrac{(3x \\sin x-2)^2}{x \\sin x}+12$.\n Thus, if $3x \\sin x-2=0$, then the minimum is obviously $12$. We show this possible with ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $\\sin^4 x + \\cos^4 x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 10129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $\\sqrt{x^2+y^2}$ if $5x+12y=60$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{60}{13}"
} | {
"index": 10130,
"solution": "\n#### Step 1: Understand the problem\nWe need to find the minimum value of $\\sqrt{x^2 + y^2}$ given the constraint $5x + 12y = 60$. This expression $\\sqrt{x^2 + y^2}$ represents the distance from the origin $(0,0)$ to the point $(x,y)$.\n\n#### Step 2: Relate the problem to the dis... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $n (n > 0)$ such that the function \\\\(f(x)= \\begin{vmatrix} \\sqrt {3} & \\sin x \\\\\\\\ 1 & \\cos x\\\\end{vmatrix} \\\\) when shifted $n$ units to the left becomes an even function."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"index": 10131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $| \\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x |$ for real numbers $x$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 1"
} | {
"index": 10132,
"solution": "\\textbf{First solution:} Write \\begin{align*} f(x) &= \\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x \\\\ &= \\sin x + \\cos x + \\frac{1}{\\sin x \\cos x} + \\frac{\\sin x + \\cos x}{\\sin x \\cos x}. \\end{align*} We can write $\\sin x + \\cos x = \\sqrt{2} \\cos(\\pi/4... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the constant \\(\\lambda\\) such that for all positive integers \\(n\\) and any positive real numbers \\(x_{k}\\) \\((1 \\leq k \\leq n)\\) summing to 1, the following inequality holds:\n$$\n\\lambda \\prod_{k=1}^{n}\\left(1-x_{k}\\right) \\geq 1-\\sum... | math | {
"style": "rule",
"ground_truth": "e"
} | {
"index": 10133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the discriminant of a quadratic trinomial whose graph does not intersect the regions below the x-axis and above the graph of the function \\( y = \\frac{1}{\\sqrt{1-x^2}} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 10134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the distance $|AB|$ where point $A$ is the intersection of the line $y=a$ and the line $y=2x+2$, and point $B$ is the intersection of the line $y=a$ and the curve $y=x+\\ln x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 10135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the expression\n\n$$\n\\sqrt{x^{2}-2 \\sqrt{3} \\cdot|x|+4}+\\sqrt{x^{2}+2 \\sqrt{3} \\cdot|x|+12}\n$$\n\nas well as the values of $x$ at which it is achieved."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{7}"
} | {
"index": 10136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the expression \\((\\sqrt{2(1+\\cos 2x)} - \\sqrt{3-\\sqrt{2}} \\sin x + 1) \\cdot (3 + 2\\sqrt{7-\\sqrt{2}} \\cos y - \\cos 2y)\\). If the answer is not an integer, round it to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 10137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the expression \\((\\sqrt{2(1+\\cos 2x)} - \\sqrt{36 - 4\\sqrt{5}} \\sin x + 2) \\cdot (3 + 2\\sqrt{10 - \\sqrt{5}} \\cos y - \\cos 2y)\\). If the answer is not an integer, round it to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "-27"
} | {
"index": 10138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the expression \\(\\frac{13 x^{2}+24 x y+13 y^{2}+16 x+14 y+68}{\\left(9-x^{2}-8 x y-16 y^{2}\\right)^{5 / 2}}\\). Round the answer to the nearest hundredth if needed."
}
] | math | {
"style": "rule",
"ground_truth": "0.26"
} | {
"index": 10139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the expression \\(\\frac{13 x^{2}+24 x y+13 y^{2}-14 x-16 y+61}{\\left(4-16 x^{2}-8 x y-y^{2}\\right)^{7 / 2}}\\). If necessary, round your answer to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "0.44"
} | {
"index": 10140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the expression \\(\\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\\left(9-25 x^{2}+10 x y-y^{2}\\right)^{5 / 2}}\\). If necessary, round the answer to hundredths."
}
] | math | {
"style": "rule",
"ground_truth": "0.19"
} | {
"index": 10141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the expression \\(\\left\\lfloor \\frac{8(a+b)}{c} \\right\\rfloor + \\left\\lfloor \\frac{8(a+c)}{b} \\right\\rfloor + \\left\\lfloor \\frac{8(b+c)}{a} \\right\\rfloor\\), where \\(a\\), \\(b\\), and \\(c\\) are arbitrary natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 10142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the expression:\n\\[(15 - x)(8 - x)(15 + x)(8 + x).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-6480.25"
} | {
"index": 10143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the following function $f(x) $ defined at $0<x<\\frac{\\pi}{2}$ .\r\n\r\n\\[f(x)=\\int_0^x \\frac{d\\theta}{\\cos \\theta}+\\int_x^{\\frac{\\pi}{2}} \\frac{d\\theta}{\\sin \\theta}\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\ln(3 + 2\\sqrt{2})"
} | {
"index": 10144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function\n\n$$\nf(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\\ldots+(x-102)^{2}\n$$\n\nIf you obtain a non-integer number, round the result to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "46852"
} | {
"index": 10145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function\n\n$$\nf(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\\ldots+(x-104)^{2}\n$$\n\nIf the result is a non-integer, round it to the nearest integer."
}
] | math | {
"style": "rule",
"ground_truth": "49608"
} | {
"index": 10146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function $f(x)=27x-x^{3}$ in the interval $[-4,2]$."
}
] | math | {
"style": "rule",
"ground_truth": "-54"
} | {
"index": 10147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function $f(x)=\\log_{2}^{\\sqrt{x}} \\cdot \\log_{\\sqrt{2}}^{(2x)}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"index": 10148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function $f(x)=\\sum_{n=1}^{19}{|x-n|}$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 10149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = 3^x - 9^x \\) for real numbers \\( x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 10150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = \\operatorname{tg}^{2} x - 4 \\operatorname{tg} x - 12 \\operatorname{ctg} x + 9 \\operatorname{ctg}^{2} x - 3 \\) on the interval \\(\\left( -\\frac{\\pi}{2}, 0 \\right)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 8\\sqrt{3}"
} | {
"index": 10151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = \\tan^2 x - 4 \\tan x - 8 \\cot x + 4 \\cot^2 x + 5 \\) on the interval \\( \\left( \\frac{\\pi}{2}, \\pi \\right) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "9 - 8\\sqrt{2}"
} | {
"index": 10152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = x^{2} + 2x + \\frac{6}{x} + \\frac{9}{x^{2}} + 4 \\) for \\( x > 0 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "10 + 4 \\sqrt{3}"
} | {
"index": 10153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x)=\\cos 3x + 4 \\cos 2x + 8 \\cos x \\) for \\( x \\in \\mathbb{R} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 10154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x)=\\cos 4x + 6\\cos 3x + 17\\cos 2x + 30\\cos x \\) for \\( x \\in \\mathbb{R} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-18"
} | {
"index": 10155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the maximum of \\( |x^2 - 2xy| \\) over \\( 0 \\leq x \\leq 1 \\) for \\( y \\) in \\( \\mathbb{R} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"index": 10156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 10157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value, for \\(a, b > 0\\), of the expression\n\n\\[\n\\frac{|a + 3b - b(a + 9b)| + |3b - a + 3b(a - b)|}{\\sqrt{a^{2} + 9b^{2}}}\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{5}"
} | {
"index": 10158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the modular inverse of $27$, modulo $28$.\n\nExpress your answer as an integer from $0$ to $27$, inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 10159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the modular inverse of $4$, modulo $21$.\n\nExpress your answer as an integer from $0$ to $20$, inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the modular inverse of \\( 31 \\), modulo \\( 45 \\).\n\nExpress your answer as an integer from \\( 0 \\) to \\( 44 \\), inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 10161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $-2 - i \\sqrt{5}$ as a root."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 + 4x + 9"
} | {
"index": 10162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $1 - i$ as a root."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 - 2x + 2"
} | {
"index": 10163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the natural number $A$ such that there are $A$ integer solutions to $x+y\\geq A$ where $0\\leq x \\leq 6$ and $0\\leq y \\leq 7$ . \n\n*Proposed by David Tang*"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the natural number \\( N \\) such that it is divisible by 5 and 49, and it has exactly 10 divisors, including 1 and \\( N \\)."
}
] | math | {
"style": "rule",
"ground_truth": "12005"
} | {
"index": 10165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the natural number that is divisible by 9 and 5 and has 14 distinct divisors."
}
] | math | {
"style": "rule",
"ground_truth": "3645"
} | {
"index": 10166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the next two smallest juicy numbers after 6, and show a decomposition of 1 into unit fractions for each of these numbers."
}
] | math | {
"style": "rule",
"ground_truth": "12, 15"
} | {
"index": 10167,
"solution": "12 and 15: $1=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{6}+\\frac{1}{12}$, $1=\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{10}+\\frac{1}{15}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the non-zero value of $c$ for which there is exactly one positive value of $b$ for which there is one solution to the equation $x^2 + \\left(b + \\frac 1b\\right)x + c = 0$."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the nonconstant polynomial $P(x)$ such that\n\\[P(P(x)) = (x^2 + x + 1) P(x).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "x^2 + x"
} | {
"index": 10169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number by which the three numbers 480608, 508811, and 723217 when divided will give the same remainder."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 10170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations: \\begin{align*} abc&=70,\\\\ cde&=71,\\\\ efg&=72. \\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 10171,
"solution": "We know that any two consecutive numbers are coprime. Using this, we can figure out that $c=1$ and $e=1$. $d$ then has to be $71$. Now we have two equations left. $ab=70$ and $fg=72$. To solve these we just need to figure out all of the factors. Doing the prime factorization of $70$ a... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of $x$-intercepts on the graph of $y = \\sin \\frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.0001, 0.001).$"
}
] | math | {
"style": "rule",
"ground_truth": "2865"
} | {
"index": 10172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of $x$-intercepts on the graph of $y = \\sin \\frac{2}{x}$ (evaluated in terms of radians) in the interval $(0.0002, 0.002).$"
}
] | math | {
"style": "rule",
"ground_truth": "2865"
} | {
"index": 10173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 0-1 binary sequences formed by six 0's and six 1's such that no three 0's are together. For example, 110010100101 is such a sequence but 101011000101 and 110101100001 are not."
}
] | math | {
"style": "rule",
"ground_truth": "357"
} | {
"index": 10174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 10-digit numbers $\\overline{a_{1} a_{2} \\cdots a_{10}}$ which are multiples of 11 such that the digits are non-increasing from left to right, i.e. $a_{i} \\geq a_{i+1}$ for each $1 \\leq i \\leq 9$."
}
] | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"index": 10175,
"solution": "It is well known that $\\overline{a_{1} a_{2} \\cdots a_{10}}$ is divisible by 11 if and only if $S=a_{1}-a_{2}+a_{3}-\\cdots-a_{10}$ is. By the non-increasing condition, we deduce that $$S=\\left(a_{1}-a_{2}\\right)+\\left(a_{3}-a_{4}\\right)+\\cdots+\\left(a_{9}-a_{10}\\right) \\geq... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 10-tuples $(x_1, x_2, \\dots, x_{10})$ of real numbers such that\n\\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \\dots + (x_9 - x_{10})^2 + x_{10}^2 = \\frac{1}{11}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 11-digit positive integers such that the digits from left to right are non-decreasing. (For example, 12345678999, 55555555555, 23345557889.)"
}
] | math | {
"style": "rule",
"ground_truth": "75582"
} | {
"index": 10177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 2's in the factorization of the number $2011 \\cdot 2012 \\cdot 2013 \\cdot \\ldots \\cdot 4020$. Provide the answer in the given field."
}
] | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"index": 10178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 20-tuples of integers $x_{1}, \\ldots, x_{10}, y_{1}, \\ldots, y_{10}$ with the following properties: - $1 \\leq x_{i} \\leq 10$ and $1 \\leq y_{i} \\leq 10$ for each $i$; - $x_{i} \\leq x_{i+1}$ for $i=1, \\ldots, 9$; - if $x_{i}=x_{i+1}$, then $y_{i} \\leq ... | math | {
"style": "rule",
"ground_truth": "\\binom{109}{10}"
} | {
"index": 10179,
"solution": "By setting $z_{i}=10 x_{i}+y_{i}$, we see that the problem is equivalent to choosing a nondecreasing sequence of numbers $z_{1}, z_{2}, \\ldots, z_{10}$ from the values $11,12, \\ldots, 110$. Making a further substitution by setting $w_{i}=z_{i}-11+i$, we see that the problem is equiv... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 3-digit positive integers whose digits multiply to 30."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 5-digit numbers where the ten-thousands place is not 5, the units place is not 2, and all digits are distinct."
}
] | math | {
"style": "rule",
"ground_truth": "21840"
} | {
"index": 10181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 7 -tuples $\\left(n_{1}, \\ldots, n_{7}\\right)$ of integers such that $$\\sum_{i=1}^{7} n_{i}^{6}=96957$$"
}
] | math | {
"style": "rule",
"ground_truth": "2688"
} | {
"index": 10182,
"solution": "Consider the equation in modulo 9. All perfect 6 th powers are either 0 or 1. Since 9 divides 96957, it must be that each $n_{i}$ is a multiple of 3. Writing $3 a_{i}=n_{i}$ and dividing both sides by $3^{6}$, we have $a_{1}^{6}+\\cdots+a_{7}^{6}=133$. Since sixth powers are nonnegati... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of 8-digit numbers where the product of the digits equals 9261. Present the answer as an integer."
}
] | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 10183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \\[\\begin{tabular}{@{\\hspace{3pt}}c@{\\hspace{3pt}}}2342\\hline 65\\end{tabular}\\,,\\qquad\\begin{tabular}{@{\\hs... | math | {
"style": "rule",
"ground_truth": "3240"
} | {
"index": 10184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of all natural numbers in which each subsequent digit is less than the previous one."
}
] | math | {
"style": "rule",
"ground_truth": "1013"
} | {
"index": 10185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"index": 10186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of arrangements of 4 beads (2 red, 2 green, 2 blue) in a circle such that the two red beads are not adjacent."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10187,
"solution": "We divide this problem into cases based on the relative position of the two red beads: - They are adjacent. Then, there are 4 possible placements of the green and blue beads: GGBB, GBBG, GBGB, BGGB. - They are 1 bead apart. Then, there are two choices for the bead between then and 2 c... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of collections of $16$ distinct subsets of $\\{1,2,3,4,5\\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X \\cap Y \\not= \\emptyset.$"
}
] | math | {
"style": "rule",
"ground_truth": "081"
} | {
"index": 10188,
"solution": "Denote the $A$ as $\\{ 1,2,3,4,5 \\}$ and the collection of subsets as $S$.\nCase 1: There are only sets of size $3$ or higher in $S$: Any two sets in $S$ must have at least one element common to both of them (since $3+3>5$). Since there are $16$ subsets of $A$ that have size $3$ or h... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of complex numbers $z$ satisfying $|z| = 1$ and\n\\[\\left| \\frac{z}{\\overline{z}} + \\frac{\\overline{z}}{z} \\right| = 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of complex solutions to\n\\[\\frac{z^3 - 1}{z^2 + z - 2} = 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 10190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of counter examples to the statement:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 10191,
"solution": "We are tasked with finding the number of counterexamples to the statement that a number $N$ with the sum of its digits equal to $4$ and none of its digits being $0$ is prime. We will analyze each possible set of digits that sum to $4$ and check if the resulting numbers are prime.\n\n1... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of cubic centimeters in the volume of the cylinder formed by rotating a rectangle with side lengths 8 cm and 16 cm about its longer side. Express your answer in terms of \\(\\pi\\)."
}
] | math | {
"style": "rule",
"ground_truth": "256\\pi"
} | {
"index": 10192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of cubic centimeters in the volume of the cylinder formed by rotating a square with side length 14 centimeters about its vertical line of symmetry. Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "686\\pi"
} | {
"index": 10193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c,$ where $a, b,$ and $c$ are integers in $\\{-20,-19,-18,\\ldots,18,19,20\\},$ such that there is a unique integer $m \\not= 2$ with $p(m) = p(2).$"
}
] | math | {
"style": "rule",
"ground_truth": "738"
} | {
"index": 10194,
"solution": "$p(x)-p(2)$ is a cubic with at least two integral real roots, therefore it has three real roots, which are all integers.\n There are exactly two distinct roots, so either $p(x)=p(2)+(x-2)^2(x-m)$ or $p(x)=p(2)+(x-2)(x-m)^2$, with $m\\neq 2$.\n In the first case $p(x)=x^3-(4+m)x^2+(4+4... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of degrees in the measure of angle $x$.\n\n[asy]\nimport markers;\nsize (5cm,5cm);\npair A,B,C,D,F,H;\n\nA=(0,0);\nB=(5,0);\nC=(9,0);\nD=(3.8,7);\n\nF=(2.3,7.2);\nH=(5.3,7.2);\n\ndraw((4.2,6.1){up}..{right}(5.3,7.2));\ndraw((3.6,6.1){up}..{left}(2.3,7.2));\n\ndr... | math | {
"style": "rule",
"ground_truth": "82^\\circ"
} | {
"index": 10195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of diagonals in a polygon with 120 sides, and calculate the length of one diagonal of this polygon assuming it is regular and each side length is 5 cm."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of diagonals of a polygon with 150 sides and determine if 9900 represents half of this diagonal count."
}
] | math | {
"style": "rule",
"ground_truth": "11025"
} | {
"index": 10197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of different arrangements for a class to select 6 people to participate in two volunteer activities, with each activity accommodating no more than 4 people."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 10198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of different complex numbers $z$ such that $|z|=1$ and $z^{7!}-z^{6!}$ is a real number."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 10199,
"solution": ""
} |
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