data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Compute \\(\\sqrt[4]{506250000}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 6500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\n\\left\\lfloor \\frac{2007! + 2004!}{2006! + 2005!}\\right\\rfloor.\n\\](Note that $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$.)"
}
] | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"index": 6501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\n\\left\\lfloor \\frac{2017! + 2014!}{2016! + 2015!}\\right\\rfloor.\n\\] (Note that $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$.)"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 6502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[ \\left\\lfloor \\dfrac {1007^3}{1005 \\cdot 1006} - \\dfrac {1005^3}{1006 \\cdot 1007} + 5 \\right\\rfloor,\\] where $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[ \\left\\lfloor \\dfrac {2005^3}{2003 \\cdot 2004} - \\dfrac {2003^3}{2004 \\cdot 2005} \\right\\rfloor,\\]where $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\dfrac{2^3-1}{2^3+1}\\cdot\\dfrac{3^3-1}{3^3+1}\\cdot\\dfrac{4^3-1}{4^3+1}\\cdot\\dfrac{5^3-1}{5^3+1}\\cdot\\dfrac{6^3-1}{6^3+1}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{43}{63}"
} | {
"index": 6505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "373"
} | {
"index": 6506,
"solution": "In both the numerator and the denominator, each factor is of the form $N^4+324=N^4+18^2$ for some positive integer $N.$\n We factor $N^4+18^2$ by completing the square, then applying the difference of squares: \\begin{align*} N^4+18^2&=\\left(N^4+36N^2+18^2\\right)-36N^2 \\\\ &=\\left(... |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\frac{(N-1)!(N)}{(N+1)!}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{N+1}"
} | {
"index": 6507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\lfloor 1 \\rfloor + \\lfloor 1.6 \\rfloor + \\lfloor 2.2 \\rfloor + \\lfloor 2.8 \\rfloor + \\dots + \\lfloor 99.4 \\rfloor + \\lfloor 100 \\rfloor,\\]where the arguments of the floor functions are in arithmetic progression."
}
] | math | {
"style": "rule",
"ground_truth": "8317"
} | {
"index": 6508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 6509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\sum_{k=2}^{31} \\log_3\\left(1 + \\frac{2}{k}\\right) \\log_k 3 \\log_{k+1} 3.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{0.5285}{3}"
} | {
"index": 6510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\sum_{k=2}^{63} \\log_2\\left(1 + \\frac{1}{k}\\right) \\log_k 2 \\log_{k+1} 2.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac56"
} | {
"index": 6511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\sum_{n=1}^{1000} \\frac{1}{n^2 + n}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1000}{1001}"
} | {
"index": 6512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \\[\\sum_{n=1}^{500} \\frac{1}{n^2 + 2n}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1499}{2008}"
} | {
"index": 6513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations: $$\\begin{aligned} x y+z & =40 \\\\ x z+y & =51 \\\\ x+y+z & =19 \\end{aligned}$$"
}
] | math | {
"style": "rule",
"ground_truth": "(12,3,4),(6,5.4,7.6)"
} | {
"index": 6514,
"solution": "Solution 1: By adding the first two equations, we can get $$x y+z+x z+y=(x+1)(y+z)=91$$ From the third equation we have $$(x+1)+(y+z)=19+1=20$$ so $x+1$ and $y+z$ are the two roots of $t^{2}-20 t+91=0$ by Vieta's theorem. As the quadratic equation can be decomposed into $$(t-7)(t-13)=0... |
DeepScaleR | [
{
"role": "user",
"content": "Compute all values of $b$ for which the following system has a solution $(x,y)$ in real numbers:\n\\begin{align*}\n\\sqrt{xy} &= b^b, \\\\\n\\log_b (x^{\\log_b y}) + \\log_b (y^{\\log_b x}) &= 4b^4.\n\\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 0, \\frac{1}{\\sqrt{2}} \\right]"
} | {
"index": 6515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute and simplify the following expressions:\n1. $(1)(1 \\frac{1}{2})^{0}-(1-0.5^{-2})÷(\\frac{27}{8})^{\\frac{2}{3}}$\n2. $\\sqrt{2 \\sqrt{2 \\sqrt{2}}}$"
}
] | math | {
"style": "rule",
"ground_truth": "2^{\\frac{7}{8}}"
} | {
"index": 6516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the argument of the sum:\n\\[ e^{2\\pi i/40} + e^{6\\pi i/40} + e^{10\\pi i/40} + e^{14\\pi i/40} + e^{18\\pi i/40} + e^{22\\pi i/40} + e^{26\\pi i/40} + e^{30\\pi i/40} + e^{34\\pi i/40} + e^{38\\pi i/40} \\]\nand express it in the form \\( r e^{i \\theta} \\), where \... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 6517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the circumradius of cyclic hexagon $A B C D E F$, which has side lengths $A B=B C=$ $2, C D=D E=9$, and $E F=F A=12$."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6518,
"solution": "Construct point $E^{\\prime}$ on the circumcircle of $A B C D E F$ such that $D E^{\\prime}=E F=12$ and $E^{\\prime} F=D E=9$; then $\\overline{B E^{\\prime}}$ is a diameter. Let $B E^{\\prime}=d$. Then $C E^{\\prime}=\\sqrt{B E^{\\prime 2}-B C^{2}}=\\sqrt{d^{2}-4}$ and $B D=\\sqrt{B E... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the decimal expansion of \\sqrt{\\pi}$. Your score will be \\min (23, k)$, where $k$ is the number of consecutive correct digits immediately following the decimal point in your answer."
}
] | math | {
"style": "rule",
"ground_truth": "1.77245385090551602729816 \\ldots"
} | {
"index": 6519,
"solution": "For this problem, it is useful to know the following square root algorithm that allows for digit-by-digit extraction of \\sqrt{x}$ and gives one decimal place of \\sqrt{x}$ for each two decimal places of $x$. We will illustrate how to extract the second digit after the decimal point of... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{-\\pi}^{0} 2^{8} \\sin ^{6} x \\cos ^{2} x \\, dx\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "10\\pi"
} | {
"index": 6520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{1} \\frac{x^{4}}{\\left(2-x^{2}\\right)^{3 / 2}} \\, dx\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} - \\frac{3\\pi}{4}"
} | {
"index": 6521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{2 \\operatorname{arctg} \\frac{1}{2}} \\frac{(1-\\sin x) dx}{\\cos x(1+\\cos x)}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "2 \\ln \\frac{3}{2} - \\frac{1}{2}"
} | {
"index": 6522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{\\frac{\\pi}{2}} \\frac{\\sin x \\, dx}{(1+\\cos x+\\sin x)^{2}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\ln 2 - \\frac{1}{2}"
} | {
"index": 6523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{\\frac{\\pi}{4}} \\left( x^{2} + 17.5 \\right) \\sin 2x \\, dx\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{68 + \\pi}{8}"
} | {
"index": 6524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\cdot \\sin ^{4} x \\cos ^{4} x \\, dx\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{8}"
} | {
"index": 6525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\cdot \\sin ^{6}\\left(\\frac{x}{2}\\right) \\cos ^{2}\\left(\\frac{x}{2}\\right) d x\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{8}"
} | {
"index": 6526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{\\pi}\\left(x^{2}-3 x+2\\right) \\sin x \\, dx\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\pi^2 - 3\\pi"
} | {
"index": 6527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{\\sqrt{2}} \\frac{x^{4} \\cdot d x}{\\left(4-x^{2}\\right)^{3 / 2}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "5 - \\frac{3\\pi}{2}"
} | {
"index": 6528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{1}^{2} \\frac{x+\\sqrt{3 x-2}-10}{\\sqrt{3 x-2}+7} d x\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{22}{27}"
} | {
"index": 6529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{\\pi / 2}^{\\pi} 2^{4} \\cdot \\sin ^{6} x \\cos ^{2} x \\, dx\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{16}"
} | {
"index": 6530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the determinant of the following matrix:\n\\[\n\\begin{vmatrix} 3 & 1 & 0 \\\\ 8 & 5 & -2 \\\\ 3 & -1 & 6 \\end{vmatrix}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"index": 6531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the determinant of the matrix:\n\\[\n\\begin{vmatrix} 2 & 4 & -2 \\\\ 0 & 3 & -1 \\\\ 5 & -1 & 2 \\end{vmatrix}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 6532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the distance between the parallel lines given by\n\\[\\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix} + t \\begin{pmatrix} 1 \\\\ -7 \\end{pmatrix}\\]and\n\\[\\begin{pmatrix} 1 \\\\ -5 \\end{pmatrix} + s \\begin{pmatrix} 1 \\\\ -7 \\end{pmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\sqrt{2}}{10}"
} | {
"index": 6533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the domain of the real-valued function $$f(x)=\\sqrt{3-\\sqrt{5-\\sqrt{x}}}.$$"
}
] | math | {
"style": "rule",
"ground_truth": "[0, 25]"
} | {
"index": 6534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the exact value of the expression\n\\[|\\pi - |\\pi - 7||.\\]Write your answer using only integers and $\\pi,$ without any absolute value signs."
}
] | math | {
"style": "rule",
"ground_truth": "7 - 2 \\pi"
} | {
"index": 6535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the exact value of the expression $\\left|3 - |9 - \\pi^2| \\right|$. Write your answer using only integers, $\\pi$, and necessary mathematical operations, without any absolute value signs."
}
] | math | {
"style": "rule",
"ground_truth": "12 - \\pi^2"
} | {
"index": 6536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the exact value of the expression $\\left|\\pi - | \\pi - 7 | \\right|$. Write your answer using only integers and $\\pi$, without any absolute value signs."
}
] | math | {
"style": "rule",
"ground_truth": "7-2\\pi"
} | {
"index": 6537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the expression: $\\left( \\pi - 1 \\right)^{0} + \\left( \\frac{1}{2} \\right)^{-1} + \\left| 5 - \\sqrt{27} \\right| - 2 \\sqrt{3}$."
}
] | math | {
"style": "rule",
"ground_truth": "8 - 5 \\sqrt{3}"
} | {
"index": 6538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the expressions \n\\[ C = 3 \\times 4 + 5 \\times 6 + 7 \\times 8 + \\cdots + 43 \\times 44 + 45 \\]\nand \n\\[ D = 3 + 4 \\times 5 + 6 \\times 7 + \\cdots + 42 \\times 43 + 44 \\times 45 \\]\nand find the positive difference between integers $C$ and $D$."
}
] | math | {
"style": "rule",
"ground_truth": "882"
} | {
"index": 6539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the following expression:\n\\[\n\\frac{(1 + 15) \\left( 1 + \\dfrac{15}{2} \\right) \\left( 1 + \\dfrac{15}{3} \\right) \\dotsm \\left( 1 + \\dfrac{15}{17} \\right)}{(1 + 17) \\left( 1 + \\dfrac{17}{2} \\right) \\left( 1 + \\dfrac{17}{3} \\right) \\dotsm \\left( 1 + \\d... | math | {
"style": "rule",
"ground_truth": "496"
} | {
"index": 6540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the following expression:\n\\[\n\\frac{(1 + 15) \\left( 1 + \\dfrac{15}{2} \\right) \\left( 1 + \\dfrac{15}{3} \\right) \\dotsm \\left( 1 + \\dfrac{15}{21} \\right)}{(1 + 17) \\left( 1 + \\dfrac{17}{2} \\right) \\left( 1 + \\dfrac{17}{3} \\right) \\dotsm \\left( 1 + \\d... | math | {
"style": "rule",
"ground_truth": "272"
} | {
"index": 6541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the following expression:\n\\[ 2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))) \\]"
}
] | math | {
"style": "rule",
"ground_truth": "510"
} | {
"index": 6542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the following expression:\n\\[ 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4)))))))) \\]"
}
] | math | {
"style": "rule",
"ground_truth": "1398100"
} | {
"index": 6543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the following expressions:\n\n(1) $2 \\sqrt{12} -6 \\sqrt{ \\frac{1}{3}} + \\sqrt{48}$\n\n(2) $(\\sqrt{3}-\\pi)^{0}-\\frac{\\sqrt{20}-\\sqrt{15}}{\\sqrt{5}}+(-1)^{2017}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 2"
} | {
"index": 6544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the following sum:\n\\[\n\\frac{1}{2^{2024}} \\sum_{n = 0}^{1011} (-3)^n \\binom{2024}{2n}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 6545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $105$ and $252$."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 6546,
"solution": "The greatest common divisor of 105 and 252 is $21$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $12$ and $18$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 6547,
"solution": "The greatest common divisor of 12 and 18 is $6$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $13$ and $52$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 6548,
"solution": "The greatest common divisor of 13 and 52 is $13$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $17$ and $68$."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 6549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $23$ and $115$."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 6550,
"solution": "The greatest common divisor of 23 and 115 is $23$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $24$ and $36$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6551,
"solution": "The greatest common divisor of 24 and 36 is $12$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $31$ and $217$."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 6552,
"solution": "The greatest common divisor of 31 and 217 is $31$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $35$ and $64$."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6553,
"solution": "The greatest common divisor of 35 and 64 is $1$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $48$ and $180$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $4^{8}-1$ and $8^{12}-1$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6555,
"solution": "Let $d=\\operatorname{gcd}(a, b)$ for some $a, b \\in \\mathbb{Z}^{+}$. Then, we can write $d=a x-b y$, where $x, y \\in \\mathbb{Z}^{+}$, and $$\\begin{align*} & 2^{a}-1 \\mid 2^{a x}-1 \\tag{1}\\\\ & 2^{b}-1 \\mid 2^{b y}-1 \\tag{2} \\end{align*}$$ Multiplying the right-hand side of... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the greatest common divisor of $84$ and $126$."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 6556,
"solution": "The greatest common divisor of 84 and 126 is $42$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the integer $k > 2$ for which\n\\[\\log_{10} (k - 2)! + \\log_{10} (k - 1)! + 2 = 2 \\log_{10} k!.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 6557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the integer $k > 3$ for which\n\\[\\log_{10} (k - 3)! + \\log_{10} (k - 2)! + 3 = 2 \\log_{10} k!.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the integer $m > 3$ for which\n\\[\\log_{10} (m - 3)! + \\log_{10} (m - 1)! + 3 = 2 \\log_{10} m!.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the integral: \\(\\int_{0}^{\\pi / 2}\\left(\\sin ^{2}(\\sin x) + \\cos ^{2}(\\cos x)\\right) \\,dx\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 6560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the integrals:\n1) \\(\\int_{0}^{5} \\frac{x \\, dx}{\\sqrt{1+3x}}\\)\n2) \\(\\int_{\\ln 2}^{\\ln 9} \\frac{dx}{e^{x}-e^{-x}}\\)\n3) \\(\\int_{1}^{\\sqrt{3}} \\frac{(x^{3}+1) \\, dx}{x^{2} \\sqrt{4-x^{2}}}\\)\n4) \\(\\int_{0}^{\\frac{\\pi}{2}} \\frac{dx}{2+\\cos x}\\)"
... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3 \\sqrt{3}}"
} | {
"index": 6561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the largest integer $k$ such that $2004^k$ divides $2004!$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the largest integer $k$ such that $2025^k$ divides $(2025!)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "505"
} | {
"index": 6563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the largest integer \\( k \\) such that \\( 2010^k \\) divides \\( 2010! \\)."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 6564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the largest integer that can be expressed in the form $3^{x(3-x)}$ for some real number $x$ .\n\n[i]Proposed by James Lin"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the largest positive integer such that $\\frac{2007!}{2007^{n}}$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 6566,
"solution": "Note that $2007=3^{2} \\cdot 223$. Using the fact that the number of times a prime $p$ divides $n!$ is given by $$\\left\\lfloor\\frac{n}{p}\\right\\rfloor+\\left\\lfloor\\frac{n}{p^{2}}\\right\\rfloor+\\left\\lfloor\\frac{n}{p^{3}}\\right\\rfloor+\\cdots$$ it follows that the answer i... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the least positive value of $t$ such that\n\\[\\arcsin (\\sin \\alpha), \\ \\arcsin (\\sin 2 \\alpha), \\ \\arcsin (\\sin 7 \\alpha), \\ \\arcsin (\\sin t \\alpha)\\]is a geometric progression for some $\\alpha$ with $0 < \\alpha < \\frac{\\pi}{2}.$"
}
] | math | {
"style": "rule",
"ground_truth": "9 - 4 \\sqrt{5}"
} | {
"index": 6567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the least possible value of $ABCD - AB \\times CD$ , where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$ , $B$ , $C$ , and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.)"
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"index": 6568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the length of the segment tangent from the origin to the circle that passes through the points $(3,4),$ $(6,8),$ and $(5,13).$"
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{2}"
} | {
"index": 6569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the length of the segment tangent from the origin to the circle that passes through the points $(4,5)$, $(8,10)$, and $(10,25)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{82}"
} | {
"index": 6570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the length of the segment tangent from the point $(1,1)$ to the circle that passes through the points $(4,5),$ $(7,9),$ and $(6,14).$"
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"index": 6571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.\n\n$$\ny = e^{x} + e, \\ln \\sqrt{3} \\leq x \\leq \\ln \\sqrt{15}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{1}{2} \\ln \\left( \\frac{9}{5} \\right)"
} | {
"index": 6572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the limit of the function:\n\n$$\n\\lim _{x \\rightarrow 0} \\frac{4^{5 x}-9^{-2 x}}{\\sin x-\\operatorname{tg}(x^{3})}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\ln (1024 \\cdot 81)"
} | {
"index": 6573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the limit of the function:\n\n$$\n\\lim _{x \\rightarrow \\pi} \\frac{\\ln (2+\\cos x)}{\\left(3^{\\sin x}-1\\right)^{2}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2 \\ln^2 3}"
} | {
"index": 6574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the limit of the function:\n\n$$\\lim _{x \\rightarrow \\frac{\\pi}{2}} \\frac{e^{\\sin 2 x}-e^{\\tan 2 x}}{\\ln \\left(\\frac{2 x}{\\pi}\\right)}$$"
}
] | math | {
"style": "rule",
"ground_truth": "-2\\pi"
} | {
"index": 6575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the limit of the function:\n\n\\[\n\\lim _{x \\rightarrow \\frac{\\pi}{4}}(\\tan x)^{1 / \\cos \\left(\\frac{3 \\pi}{4}-x\\right)}\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "e^2"
} | {
"index": 6576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the maximum number of sides of a polygon that is the cross-section of a regular hexagonal prism."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6577,
"solution": "Note that since there are 8 faces to a regular hexagonal prism and a cross-section may only intersect a face once, the upper bound for our answer is 8. Indeed, we can construct a cross-section of the prism with 8 sides. Let $ABCDEF$ and $A'B'C'D'E'F'$ be the two bases of the prism, wit... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the maximum real value of $a$ for which there is an integer $b$ such that $\\frac{ab^2}{a+2b} = 2019$ . Compute the maximum possible value of $a$ ."
}
] | math | {
"style": "rule",
"ground_truth": "30285"
} | {
"index": 6578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the multiplicative inverse of $101$ modulo $401$. Express your answer as an integer from $0$ to $400$."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 6579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the multiplicative inverse of $185$ modulo $341$. Express your answer as an integer from $0$ to $340$."
}
] | math | {
"style": "rule",
"ground_truth": "74466"
} | {
"index": 6580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the multiplicative inverse of $201$ modulo $299$. Express your answer as an integer from $0$ to $298$."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 6581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the multiplicative inverse of $217$ modulo $397$. Express your answer as an integer from $0$ to $396$."
}
] | math | {
"style": "rule",
"ground_truth": "161"
} | {
"index": 6582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the nearest integer to $$100 \\sum_{n=1}^{\\infty} 3^{n} \\sin ^{3}\\left(\\frac{\\pi}{3^{n}}\\right)$$"
}
] | math | {
"style": "rule",
"ground_truth": "236"
} | {
"index": 6583,
"solution": "Note that we have $$\\sin 3 x=3 \\sin x-4 \\sin ^{3} x \\Longrightarrow \\sin ^{3} x=\\frac{1}{4}(3 \\sin x-\\sin 3 x)$$ which implies that $$\\frac{\\sin ^{3} x}{3 x}=\\frac{1}{4}\\left(\\frac{\\sin x}{x}-\\frac{\\sin 3 x}{3 x}\\right)$$ Substituting $x=\\frac{\\pi}{3^{n}}$ and simpli... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of complex numbers $z$ with $|z|=1$ that satisfy $$1+z^{5}+z^{10}+z^{15}+z^{18}+z^{21}+z^{24}+z^{27}=0$$"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6584,
"solution": "Let the polynomial be $f(z)$. One can observe that $$f(z)=\\frac{1-z^{15}}{1-z^{5}}+z^{15} \\frac{1-z^{15}}{1-z^{3}}=\\frac{1-z^{20}}{1-z^{5}}+z^{18} \\frac{1-z^{12}}{1-z^{3}}$$ so all primitive 15th roots of unity are roots, along with -1 and $\\pm i$. To show that there are no more, ... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of dates in the year 2023 such that when put in MM/DD/YY form, the three numbers are in strictly increasing order. For example, $06 / 18 / 23$ is such a date since $6<18<23$, while today, $11 / 11 / 23$, is not."
}
] | math | {
"style": "rule",
"ground_truth": "186"
} | {
"index": 6585,
"solution": "January contains 21 such dates, February contains 20, and so on, until December contains 10. The answer is $$21+20+\\cdots+10=186$$"
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of degrees in the smallest positive angle $x$ such that\n\\[8 \\sin x \\cos^5 x - 8 \\sin^5 x \\cos x = 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "7.5^\\circ"
} | {
"index": 6586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of digits is $2015!$ . Your score will be given by $\\max\\{\\lfloor125(\\min\\{\\tfrac{A}{C},\\tfrac{C}{A}\\}-\\tfrac{1}{5})\\rfloor,0\\}$ , where $A$ is your answer and $C$ is the actual answer."
}
] | math | {
"style": "rule",
"ground_truth": "5787"
} | {
"index": 6587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of distinct pairs of the form (first three digits of $x$, first three digits of $x^{4}$ ) over all integers $x>10^{10}$. For example, one such pair is $(100,100)$ when $x=10^{10^{10}}$."
}
] | math | {
"style": "rule",
"ground_truth": "4495"
} | {
"index": 6588,
"solution": "Graph these points on an $x, y$-plane. We claim that there are integers $100=a_{0}<a_{1}<$ $a_{2}<a_{3}<a_{4}=999$, for which the locus of these points is entirely contained in four taxicab (up/right movement by 1 unit) paths from $\\left(a_{i}, 100\\right)$ to $\\left(a_{i+1}, 999\\ri... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of even positive integers $n \\leq 2024$ such that $1,2, \\ldots, n$ can be split into $\\frac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of 3."
}
] | math | {
"style": "rule",
"ground_truth": "675"
} | {
"index": 6589,
"solution": "There have to be an even number of multiples of 3 at most $n$, so this means that $n \\equiv 0,2 \\pmod{6}$. We claim that all these work. We know there are an even number of multiples of 3, so we can pair them; then we can pair $3k+1$ and $3k+2$ for all $k$. This means the answer is $... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations (the permutations do not necessarily have to be valid five-digit integers).\n\n*Proposed by David Sun*"
}
] | math | {
"style": "rule",
"ground_truth": "9720"
} | {
"index": 6590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of functions $f:\\{1,2, \\ldots, 9\\} \\rightarrow\\{1,2, \\ldots, 9\\}$ which satisfy $f(f(f(f(f(x)))))=$ $x$ for each $x \\in\\{1,2, \\ldots, 9\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "3025"
} | {
"index": 6591,
"solution": "All cycles lengths in the permutation must divide 5 , which is a prime number. Either $f(x)=x$ for all $x$, or there exists exactly one permutation cycle of length 5 . In the latter case, there are $\\binom{9}{5}$ ways to choose which numbers are in the cycle and 4 ! ways to create the... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$ ."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 6592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of increasing sequences of positive integers $b_1 \\le b_2 \\le b_3 \\le \\cdots \\le b_{15} \\le 3005$ such that $b_i-i$ is odd for $1\\le i \\le 15$. Express your answer as ${p \\choose q}$ for some integers $p > q$ and find the remainder when $p$ is divide... | math | {
"style": "rule",
"ground_truth": "509"
} | {
"index": 6593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of integers \\(n \\in\\{1,2, \\ldots, 300\\}\\) such that \\(n\\) is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with integer side lengths."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 6594,
"solution": "Let \\(n=p \\cdot q\\) for primes \\(p<q\\). If \\(n\\) is the second largest side of a right triangle there exist integers \\(c, a\\) such that \\(a<p q\\) and \\((p q)^{2}=c^{2}-a^{2}=(c-a)(c+a)\\). Since \\(c-a<c+a\\) there are three cases for the values of \\(c-a, c+a\\), and in ea... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of intersection points of the graphs of\n\\[(x - \\lfloor x \\rfloor)^2 + y^2 = x - \\lfloor x \\rfloor\\]and $y = \\frac{1}{5} x.$"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of labelings $f:\\{0,1\\}^{3} \\rightarrow\\{0,1, \\ldots, 7\\}$ of the vertices of the unit cube such that $$\\left|f\\left(v_{i}\\right)-f\\left(v_{j}\\right)\\right| \\geq d\\left(v_{i}, v_{j}\\right)^{2}$$ for all vertices $v_{i}, v_{j}$ of the unit cube,... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 6596,
"solution": "Let $B=\\{0,1\\}^{3}$, let $E=\\{(x, y, z) \\in B: x+y+z$ is even $\\}$, and let $O=\\{(x, y, z) \\in B$ : $x+y+z$ is odd $\\}$. As all pairs of vertices within $E$ (and within $O$ ) are $\\sqrt{2}$ apart, is easy to see that $\\{f(E), f(O)\\}=\\{\\{0,2,4,6\\},\\{1,3,5,7\\}\\}$. - Ther... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of nonempty subsets $S \\subseteq\\{-10,-9,-8, \\ldots, 8,9,10\\}$ that satisfy $|S|+\\min (S)$. $\\max (S)=0$."
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"index": 6597,
"solution": "Since $\\min (S) \\cdot \\max (S)<0$, we must have $\\min (S)=-a$ and $\\max (S)=b$ for some positive integers $a$ and $b$. Given $a$ and $b$, there are $|S|-2=a b-2$ elements left to choose, which must come from the set $\\{-a+1,-a+2, \\ldots, b-2, b-1\\}$, which has size $a+b-1$. The... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(a, b)$, with $2 \\leq a, b \\leq 2021$, that satisfy the equation $$a^{\\log _{b}\\left(a^{-4}\\right)}=b^{\\log _{a}\\left(b a^{-3}\\right)}.$$"
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 6598,
"solution": "Taking $\\log _{a}$ of both sides and simplifying gives $$-4 \\log _{b} a=\\left(\\log _{a} b\\right)^{2}-3 \\log _{a} b$$ Plugging in $x=\\log _{a} b$ and using $\\log _{b} a=\\frac{1}{\\log _{a} b}$ gives $$x^{3}-3 x^{2}+4=0$$ We can factor the polynomial as $(x-2)(x-2)(x+1)$, meanin... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x, y)$ such that $x^{2}+y^{2}<2019$ and $$x^{2}+\\min (x, y)=y^{2}+\\max (x, y)$$"
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 6599,
"solution": "We have $$x^{2}-y^{2}=\\max (x, y)-\\min (x, y)=|x-y|$$ Now if $x \\neq y$, we can divide by $x-y$ to obtain $x+y= \\pm 1$. Thus $x=y$ or $x+y= \\pm 1$. If $x=y$, we see that $2019>x^{2}+y^{2}=2 x^{2}$, so we see that $-31 \\leq x \\leq 31$. There are 63 ordered pairs in this case. In ... |
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