data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{\\lfloor \\sqrt{1} \\rfloor \\cdot \\lfloor \\sqrt{2} \\rfloor \\cdot \\lfloor \\sqrt{3} \\rfloor \\cdot \\lfloor \\sqrt{5} \\rfloor \\dotsm \\lfloor \\sqrt{15} \\rfloor}{\\lfloor \\sqrt{2} \\rfloor \\cdot \\lfloor \\sqrt{4} \\rfloor \\cdot \\lfloor \\sqrt{6}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 6300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{\\tan^2 20^\\circ - \\sin^2 20^\\circ}{\\tan^2 20^\\circ \\sin^2 20^\\circ}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\left( 1 + \\cos \\frac {\\pi}{8} \\right) \\left( 1 + \\cos \\frac {3 \\pi}{8} \\right) \\left( 1 + \\cos \\frac {5 \\pi}{8} \\right) \\left( 1 + \\cos \\frac {7 \\pi}{8} \\right).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 6302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\left( 1 - \\frac{1}{\\cos 23^\\circ} \\right) \\left( 1 + \\frac{1}{\\sin 67^\\circ} \\right) \\left( 1 - \\frac{1}{\\sin 23^\\circ} \\right) \\left( 1 + \\frac{1}{\\cos 67^\\circ} \\right).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\prod_{k = 1}^{12} \\prod_{j = 1}^{10} (e^{2 \\pi ji/11} - e^{2 \\pi ki/13}).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\prod_{n = 1}^{15} \\frac{n + 4}{n}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "11628"
} | {
"index": 6305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\prod_{n = 1}^{20} \\frac{n + 3}{n}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1771"
} | {
"index": 6306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sin^2 4^\\circ + \\sin^2 8^\\circ + \\sin^2 12^\\circ + \\dots + \\sin^2 176^\\circ.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{45}{2}"
} | {
"index": 6307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sin^2 6^\\circ + \\sin^2 12^\\circ + \\sin^2 18^\\circ + \\dots + \\sin^2 174^\\circ.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "15.5"
} | {
"index": 6308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{1 \\le a < b < c} \\frac{1}{2^a 3^b 5^c}.\\](The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \\le a < b < c.$)"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1624}"
} | {
"index": 6309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{1 \\le a < b < c} \\frac{1}{3^a 5^b 7^c}.\\]\n(The sum is taken over all triples \\((a,b,c)\\) of positive integers such that \\(1 \\le a < b < c\\).)"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{21216}"
} | {
"index": 6310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{j = 0}^\\infty \\sum_{k = 0}^\\infty 2^{-3k - j - (k + j)^2}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 6311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{k = 1}^\\infty \\frac{6^k}{(3^k - 2^k)(3^{k + 1} - 2^{k + 1})}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 6312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 1}^\\infty \\frac{1}{n(n + 2)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 6313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 1}^\\infty \\frac{2n + 1}{n(n + 1)(n + 2)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 6314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 1}^\\infty \\frac{2n - 1}{n(n + 1)(n + 2)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 6315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 2}^\\infty \\frac{4n^3 - n^2 - n + 1}{n^6 - n^5 + n^4 - n^3 + n^2 - n}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 2}^{10000} \\frac{1}{n \\sqrt{n - 1} + (n - 1) \\sqrt{n}}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{99}{100}"
} | {
"index": 6317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[e^{2 \\pi i/13} + e^{4 \\pi i/13} + e^{6 \\pi i/13} + \\dots + e^{24 \\pi i/13}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 6318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \n\\[\n\\sin^2 3^\\circ + \\sin^2 6^\\circ + \\sin^2 9^\\circ + \\dots + \\sin^2 177^\\circ.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 6319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute \n\\[\n\\sin^6 0^\\circ + \\sin^6 1^\\circ + \\sin^6 2^\\circ + \\dots + \\sin^6 90^\\circ.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{229}{8}"
} | {
"index": 6320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\left\\lceil\\displaystyle\\sum_{k=2018}^{\\infty}\\frac{2019!-2018!}{k!}\\right\\rceil$ . (The notation $\\left\\lceil x\\right\\rceil$ denotes the least integer $n$ such that $n\\geq x$ .)\n\n*Proposed by Tristan Shin*"
}
] | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"index": 6321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $e^{\\pi}+\\pi^e$ . If your answer is $A$ and the correct answer is $C$ , then your score on this problem will be $\\frac{4}{\\pi}\\arctan\\left(\\frac{1}{\\left|C-A\\right|}\\right)$ (note that the closer you are to the right answer, the higher your score is).\n\... | math | {
"style": "rule",
"ground_truth": "45.5999"
} | {
"index": 6322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$ \\sum_{a_{1}=0}^{\\infty} \\sum_{a_{2}=0}^{\\infty} \\cdots \\sum_{a_{7}=0}^{\\infty} \\frac{a_{1}+a_{2}+\\cdots+a_{7}}{3^{a_{1}+a_{2}+\\cdots+a_{7}}} $$"
}
] | math | {
"style": "rule",
"ground_truth": "15309 / 256"
} | {
"index": 6323,
"solution": "Note that, since this is symmetric in $a_{1}$ through $a_{7}$, $$ \\sum_{a_{1}=0}^{\\infty} \\sum_{a_{2}=0}^{\\infty} \\cdots \\sum_{a_{7}=0}^{\\infty} \\frac{a_{1}+a_{2}+\\cdots+a_{7}}{3^{a_{1}+a_{2}+\\cdots+a_{7}}} =7 \\sum_{a_{1}=0}^{\\infty} \\sum_{a_{2}=0}^{\\infty} \\cdots \\sum_... |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$100^{2}+99^{2}-98^{2}-97^{2}+96^{2}+95^{2}-94^{2}-93^{2}+\\ldots+4^{2}+3^{2}-2^{2}-1^{2}$$"
}
] | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"index": 6324,
"solution": "Note that $(n+3)^{2}-(n+2)^{2}-(n+1)^{2}+n^{2}=4$ for every $n$. Therefore, adding $0^{2}$ to the end of the given sum and applying this identity for every four consecutive terms after $100^{2}$, we see that the given sum is equivalent to $100^{2}+25 \\cdot 4=10100$. Alternatively, we ... |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$2 \\sqrt{2 \\sqrt[3]{2 \\sqrt[4]{2 \\sqrt[5]{2 \\cdots}}}}$$"
}
] | math | {
"style": "rule",
"ground_truth": "2^{e-1}"
} | {
"index": 6325,
"solution": "Taking the base 2 logarithm of the expression gives $$1+\\frac{1}{2}\\left(1+\\frac{1}{3}\\left(1+\\frac{1}{4}(1+\\cdots)\\right)\\right)=1+\\frac{1}{2!}+\\frac{1}{3!}+\\frac{1}{4!}+\\cdots=e-1$$ Therefore the expression is just $2^{e-1}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$\\lim _{A \\rightarrow+\\infty} \\frac{1}{A} \\int_{1}^{A} A^{\\frac{1}{x}} \\mathrm{~d} x$$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6326,
"solution": "We prove that $$\\lim _{A \\rightarrow+\\infty} \\frac{1}{A} \\int_{1}^{A} A^{\\frac{1}{x}} \\mathrm{~d} x=1$$ For $A>1$ the integrand is greater than 1, so $$\\frac{1}{A} \\int_{1}^{A} A^{\\frac{1}{x}} \\mathrm{~d} x>\\frac{1}{A} \\int_{1}^{A} 1 \\mathrm{~d} x=\\frac{1}{A}(A-1)=1-\\fr... |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$\\sum_{\\substack{a+b+c=12 \\\\ a \\geq 6, b, c \\geq 0}} \\frac{a!}{b!c!(a-b-c)!}$$ where the sum runs over all triples of nonnegative integers $(a, b, c)$ such that $a+b+c=12$ and $a \\geq 6$."
}
] | math | {
"style": "rule",
"ground_truth": "2731"
} | {
"index": 6327,
"solution": "We tile a $1 \\times 12$ board with red $1 \\times 1$ pieces, blue $1 \\times 2$ pieces, and green $1 \\times 2$ pieces. Suppose we use $a$ total pieces, $b$ blue pieces, and $c$ green pieces. Then we must have $a+b+c=12$, and the number of ways to order the pieces is $$\\binom{a}{b, c... |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$\\sum_{n=0}^{\\infty} \\frac{n}{n^{4}+n^{2}+1}$$"
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 6328,
"solution": "Note that $$n^{4}+n^{2}+1=\\left(n^{4}+2 n^{2}+1\\right)-n^{2}=\\left(n^{2}+1\\right)^{2}-n^{2}=\\left(n^{2}+n+1\\right)\\left(n^{2}-n+1\\right)$$ Decomposing into partial fractions, we find that $$\\frac{n}{n^{4}+n^{2}+1}=\\frac{1}{2}\\left(\\frac{1}{n^{2}-n+1}-\\frac{1}{n^{2}+n+1}\\r... |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$\\sum_{n=1}^{\\infty} \\frac{2 n+5}{2^{n} \\cdot\\left(n^{3}+7 n^{2}+14 n+8\\right)}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{137}{24}-8 \\ln 2"
} | {
"index": 6329,
"solution": "First, we manipulate using partial fractions and telescoping: $$\\begin{aligned} \\sum_{n=1}^{\\infty} \\frac{2 n+5}{2^{n} \\cdot\\left(n^{3}+7 n^{2}+14 n+8\\right)} & =\\frac{1}{2} \\cdot \\sum_{n=1}^{\\infty} \\frac{1}{2^{n}}\\left(\\frac{2}{n+1}-\\frac{1}{n+2}-\\frac{1}{n+4}\\right)... |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$\\sum_{n=1}^{\\infty} \\frac{3n-1}{2^n}.$$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 6330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $$\\sum_{n_{60}=0}^{2} \\sum_{n_{59}=0}^{n_{60}} \\cdots \\sum_{n_{2}=0}^{n_{3}} \\sum_{n_{1}=0}^{n_{2}} \\sum_{n_{0}=0}^{n_{1}} 1$$"
}
] | math | {
"style": "rule",
"ground_truth": "1953"
} | {
"index": 6331,
"solution": "The given sum counts the number of non-decreasing 61-tuples of integers $\\left(n_{0}, \\ldots, n_{60}\\right)$ from the set $\\{0,1,2\\}$. Such 61-tuples are in one-to-one correspondence with strictly increasing 61-tuples of integers $\\left(m_{0}, \\ldots, m_{60}\\right)$ from the se... |
DeepScaleR | [
{
"role": "user",
"content": "Compute $(-64)\\div (-32)$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 6332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $(1 + i)^4.$"
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 6333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $(2 \\cos 20^\\circ + 2i \\sin 20^\\circ)^6.$ Enter your answer in rectangular form."
}
] | math | {
"style": "rule",
"ground_truth": "-32 + 32i \\sqrt{3}"
} | {
"index": 6334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $(4+8)^2+(4^2+8^2)$."
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"index": 6335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $(5+7)^3+(5^3+7^3)$."
}
] | math | {
"style": "rule",
"ground_truth": "2196"
} | {
"index": 6336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $(\\cos 185^\\circ + i \\sin 185^\\circ)^{54}.$"
}
] | math | {
"style": "rule",
"ground_truth": "-i"
} | {
"index": 6337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $({11011_{(2)}} - {101_{(2)}} = )$\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_$(.$ (represented in binary)"
}
] | math | {
"style": "rule",
"ground_truth": "10110_{(2)}"
} | {
"index": 6338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $-8\\cdot 4-(-6\\cdot -3)+(-10\\cdot -5)$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $0.18\\div0.003.$"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 6340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $1+6\\cdot2-3+5\\cdot4\\div2$"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 6341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $1-2+3-4+ \\dots -100+101$."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 6342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $1-2+3-4+ \\dots -98+99$ ."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 6343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $1-2+3-4+\\dots+100-101$."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 6344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $1011_2 + 101_2 - 1100_2 + 1101_2$. Express your answer in base 2."
}
] | math | {
"style": "rule",
"ground_truth": "10001_2"
} | {
"index": 6345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $10^{-1}\\pmod{1001}$. Express your answer as a residue from $0$ to $1000$, inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "901"
} | {
"index": 6346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $11^{-1} \\pmod{1105}$, where $1105=5\\cdot 13 \\cdot 17$. Express your answer as a residue from $0$ to $1104$, inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"index": 6347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $139+27+23+11$."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 6348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $156 + 44 + 26 + 74$."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 6349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $16\\left (\\frac{125}{2}+\\frac{25}{4}+\\frac{9}{16}+1\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "1125"
} | {
"index": 6350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $17^9 \\div 17^7$."
}
] | math | {
"style": "rule",
"ground_truth": "289"
} | {
"index": 6351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $18\\left(\\frac{200}{3} + \\frac{50}{6} + \\frac{16}{18} + 2\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "1402"
} | {
"index": 6352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $2 \\times 2^5 - 8^{58} \\div 8^{56}$"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $2011 \\cdot 2012 \\cdot 2013 \\cdot 2014$ modulo 5."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $20\\left(\\frac{256}{4} + \\frac{64}{16} + \\frac{16}{64} + 2\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "1405"
} | {
"index": 6355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $26\\times33+67\\times26$."
}
] | math | {
"style": "rule",
"ground_truth": "2600"
} | {
"index": 6356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $3 \\begin{pmatrix} 2 \\\\ -8 \\end{pmatrix} - 2 \\begin{pmatrix} 1 \\\\ -7 \\end{pmatrix}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 4 \\\\ -10 \\end{pmatrix}"
} | {
"index": 6357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $35^{1723} - 16^{1723}$ modulo 6."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $54 \\times 46$ in your head."
}
] | math | {
"style": "rule",
"ground_truth": "2484"
} | {
"index": 6359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $55^2 - 45^2$ in your head."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 6360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $63 \\times 57$ in your head."
}
] | math | {
"style": "rule",
"ground_truth": "3591"
} | {
"index": 6361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $7\\left(\\frac{1}{5}\\right)^3$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{125}"
} | {
"index": 6362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $9 \\cdot (-5) - (7 \\cdot -2) + (8 \\cdot -6)$."
}
] | math | {
"style": "rule",
"ground_truth": "-79"
} | {
"index": 6363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $95^2$ in your head."
}
] | math | {
"style": "rule",
"ground_truth": "9025"
} | {
"index": 6364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $97^2$ in your head."
}
] | math | {
"style": "rule",
"ground_truth": "9409"
} | {
"index": 6365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos (\\cos 3).$ All functions are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2\\pi"
} | {
"index": 6366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos (\\cos 7).$ All functions are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "7 - 2 \\pi"
} | {
"index": 6367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos (\\sin 1.5).$ All functions are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2} - 1.5"
} | {
"index": 6368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos (\\sin 2).$ All functions are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\frac{\\pi}{2}"
} | {
"index": 6369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos (\\sin 3)$. All functions are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\frac{\\pi}{2}"
} | {
"index": 6370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos (\\sin 3).$ All functions are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\frac{\\pi}{2}"
} | {
"index": 6371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos \\frac{\\sqrt{3}}{2}.$ Express your answer in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 6372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos(\\cos 9).$ All functions are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "9 - 2\\pi"
} | {
"index": 6373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arccos(\\sin 3)$, where all functions are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\frac{\\pi}{2}"
} | {
"index": 6374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arcsin (-1).$ Express your answer in radians."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{2}"
} | {
"index": 6375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arcsin 0.$ Express your answer in radians."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arcsin 1.$ Express your answer in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 6377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arcsin \\frac{1}{\\sqrt{2}}.$ Express your answer in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 6378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arcsin \\left( -\\frac{\\sqrt{3}}{2} \\right).$ Express your answer in radians."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{3}"
} | {
"index": 6379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arctan ( \\tan 65^\\circ - 2 \\tan 40^\\circ )$. (Express your answer in degrees as an angle between $0^\\circ$ and $180^\\circ$.)"
}
] | math | {
"style": "rule",
"ground_truth": "25^\\circ"
} | {
"index": 6380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arctan (\\tan 65^{\\circ}-2 \\tan 40^{\\circ})$. (Express your answer in degrees as an angle between $0^{\\circ}$ and $180^{\\circ}$.)"
}
] | math | {
"style": "rule",
"ground_truth": "25^{\\circ}"
} | {
"index": 6381,
"solution": "First Solution: We have $\\tan 65^{\\circ}-2 \\tan 40^{\\circ}=\\cot 25^{\\circ}-2 \\cot 50^{\\circ}=\\cot 25^{\\circ}-\\frac{\\cot ^{2} 25^{\\circ}-1}{\\cot 25^{\\circ}}=\\frac{1}{\\cot 25^{\\circ}}=\\tan 25^{\\circ}$. Therefore, the answer is $25^{\\circ}$. Second Solution: We have $... |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\arctan \\frac{2}{5} + \\arctan \\frac{5}{2}.$ Express your answer in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 6382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} -4 \\\\ -1 \\end{pmatrix} \\cdot \\begin{pmatrix} 6 \\\\ 8 \\end{pmatrix}$."
}
] | math | {
"style": "rule",
"ground_truth": "-32"
} | {
"index": 6383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} 1 & -1 \\\\ 1 & 0 \\end{pmatrix}^3.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -1 & 0 \\\\ 0 & -1 \\end{pmatrix}."
} | {
"index": 6384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} 1 & 0 \\\\ 1 & 1 \\end{pmatrix}^{2018}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 1 & 0 \\\\ 2018 & 1 \\end{pmatrix}"
} | {
"index": 6385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} 2 & - 1 \\\\ - 3 & 4 \\end{pmatrix} \\begin{pmatrix} 3 \\\\ - 1 \\end{pmatrix}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 7 \\\\ -13 \\end{pmatrix}"
} | {
"index": 6386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} 2 & 0 \\\\ 5 & -3 \\end{pmatrix} \\begin{pmatrix} 8 & -2 \\\\ 1 & 1 \\end{pmatrix}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 16 & -4 \\\\ 37 & -13 \\end{pmatrix}"
} | {
"index": 6387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} 2 & 3 \\\\ 7 & -1 \\end{pmatrix} \\begin{pmatrix} 1 & -5 \\\\ 0 & 4 \\end{pmatrix}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 2 & 2 \\\\ 7 & -39 \\end{pmatrix}"
} | {
"index": 6388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} \\sqrt{3} & -1 \\\\ 1 & \\sqrt{3} \\end{pmatrix}^6.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -64 & 0 \\\\ 0 & -64 \\end{pmatrix}"
} | {
"index": 6389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\binom{12}{9}$ and then find the factorial of the result."
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"index": 6390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\binom{17}{9}$. You are told that $\\binom{15}{6} = 5005$ and $\\binom{15}{8} = 6435$."
}
] | math | {
"style": "rule",
"ground_truth": "24310"
} | {
"index": 6391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\binom{18}{6}$."
}
] | math | {
"style": "rule",
"ground_truth": "18564"
} | {
"index": 6392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\cos 0^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\cos 105^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2} - \\sqrt{6}}{4}"
} | {
"index": 6394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\cos 135^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"index": 6395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\cos 150^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"index": 6396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\cos 210^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"index": 6397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\cos 240^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 6398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute $\\cos 270^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6399,
"solution": ""
} |
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