id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-af2efce7b894bb9b45d865852001125ec54c5a65eb3fe8a91787c8518a71fbe5 | What is the value of the expression $[ a-(b-c) ] - [(a-b) - c ]$ when $a = 17$, $b=21$ and $c=5$? | [
"10"
] | [
"What is the value of the expression $[ a-(b-c) ] - [(a-b) - c ]$ when $a = 17$, $b=21$ and $c=5$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5abfa1a5844fe57645bae9eeb075146cb9997fb12a47b7c359b4602acbf2fffd | What is the value of the expression $\cos \frac{\pi}{5} + \cos \frac{3\pi}{5}$? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the value of the expression $\\cos \\frac{\\pi}{5} + \\cos \\frac{3\\pi}{5}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-7fe572a246fcc2db743193d4925f15f959efba584817778442c74aa668f36f18 | What is the value of the expression $\frac {x^2-x-6}{x-3}$ for $x=2$? Express your answer in simplest form. | [
"4"
] | [
"What is the value of the expression $\\frac {x^2-x-6}{x-3}$ for $x=2$? Express your answer in simplest form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5647b2372b16f4dafd72135b8476c8eb9172bfd5e20b3bc33f25c4304dc8aa4e | What is the value of the expression $\frac{\pi}{1+\frac{\pi}{2+\frac{\pi}{3+\ldots}}}$? | [
"2"
] | [
"What is the value of the expression $\\frac{\\pi}{1+\\frac{\\pi}{2+\\frac{\\pi}{3+\\ldots}}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-811c779794dc05b71b52741965ef462810f2b8fdfaf5fa25252cc30ac083688c | What is the value of the expression $\sqrt{1+2\sqrt{1+4\sqrt{1+6\sqrt{1+8\ldots}}}}$? | [
"3"
] | [
"What is the value of the expression $\\sqrt{1+2\\sqrt{1+4\\sqrt{1+6\\sqrt{1+8\\ldots}}}}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-8ef54464cb33d86382b0badf3e07e445dd17622c23783f9eccc06b219259d219 | What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}$? | [
"8"
] | [
"What is the value of the expression $\\sqrt{16\\sqrt{8\\sqrt{4}}}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b152dcb3aac05e1dbe2eb8dfb66867dfb7998b807da67c30d4418a9e2506a6c1 | What is the value of the expression $x^2+ 5x-6$, when $x =-1$? | [
"-10"
] | [
"What is the value of the expression $x^2+ 5x-6$, when $x =-1$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-27c14cc5d680e88894eca552effdef3bad312f433d9511cd71405d1510be27d9 | What is the value of the expression \( 2 + 8 \left( -\frac{1}{2} + \frac{i \sqrt{3}}{2} \right)^8 + 2 \left( -\frac{1}{2} + \frac{i \sqrt{3}}{2} \right)^9 \)? | [
"\\(-4i\\sqrt{3}\\)"
] | [
"What is the value of the expression \\( 2 + 8 \\left( -\\frac{1}{2} + \\frac{i \\sqrt{3}}{2} \\right)^8 + 2 \\left( -\\frac{1}{2} + \\frac{i \\sqrt{3}}{2} \\right)^9 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-61ae7c2b7e61e18887b91871879c48ea22067672872e1e1f0f9e240d5673b5e3 | What is the value of the expression \( 3 - \frac{2}{3 - \frac{2}{3 - \frac{2}{3 - \dots}}} \)? | [
"2"
] | [
"What is the value of the expression \\( 3 - \\frac{2}{3 - \\frac{2}{3 - \\frac{2}{3 - \\dots}}} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f823d06b0d0b3dd385cdaec8d3ec0c4d346ab56ea40855a2ea010d9b421d7ec2 | What is the value of the expression \( \frac{6}{2(2+1)} \)? | [
"9"
] | [
"What is the value of the expression \\( \\frac{6}{2(2+1)} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-61e596a4b126bf047ac6458346841a0757a483f1e55b6e447e96a192eeb9f25c | What is the value of the expression \(7-\dfrac{12}{7-\dfrac{12}{7-\dfrac{12}{7-\dfrac{12}{\cdots}}}}\)? | [
"4"
] | [
"What is the value of the expression \\(7-\\dfrac{12}{7-\\dfrac{12}{7-\\dfrac{12}{7-\\dfrac{12}{\\cdots}}}}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-806684b689556ec562cb21a154e8b67925fd279d6bbd44ca29c83b9a8f32a876 | What is the value of the expression \(99^{50} - 99 \cdot 98^{50} + \frac{99 \cdot 98}{2} \cdot 97^{50} - \ldots + 99\)? | [
"0"
] | [
"What is the value of the expression \\(99^{50} - 99 \\cdot 98^{50} + \\frac{99 \\cdot 98}{2} \\cdot 97^{50} - \\ldots + 99\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89e728ce4fad66485d8eba921a90fa2a88af06bb92dd0840c5be95846ec7eaf6 | What is the value of the expression \[ \frac {1}{\log_2 100!} + \frac {1}{\log_3 100!} + \frac {1}{\log_4 100!} + \cdots + \frac {1}{\log_{100} 100!}? \] | [
"1"
] | [
"What is the value of the expression\n\\[ \\frac {1}{\\log_2 100!} + \\frac {1}{\\log_3 100!} + \\frac {1}{\\log_4 100!} + \\cdots + \\frac {1}{\\log_{100} 100!}?\n\\]"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-59bd60104518210d391ad457eed5c9e91871cd5eeaa46b8ea9c7c588ae0db884 | What is the value of the expression \[ \frac {1}{\log_{2}100!} + \frac {1}{\log_{3}100!} + \frac {1}{\log_{4}100!} + \cdots + \frac {1}{\log_{100}100!} \]? | [
"1"
] | [
"What is the value of the expression\n\\[ \\frac {1}{\\log_{2}100!} + \\frac {1}{\\log_{3}100!} + \\frac {1}{\\log_{4}100!} + \\cdots + \\frac {1}{\\log_{100}100!} \\]?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14862cfdb636c4d3689479c6275dc62cb38f634e1b2f933039a9a3daafe7167a | What is the value of the expression \[ \left(\dfrac{2^2}{1+3}-\dfrac{4^2}{3+5}\right)+\left(\dfrac{6^2}{5+7}-\dfrac{8^2}{7+9}\right)+ \cdots + \left(\dfrac{98^2}{97+99}-\dfrac{100^2}{99+101}\right)? \] | [
"-50"
] | [
"What is the value of the expression\n\\[\n\\left(\\dfrac{2^2}{1+3}-\\dfrac{4^2}{3+5}\\right)+\\left(\\dfrac{6^2}{5+7}-\\dfrac{8^2}{7+9}\\right)+ \\cdots + \\left(\\dfrac{98^2}{97+99}-\\dfrac{100^2}{99+101}\\right)?\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c99f3f4279f9b20afd02fe6b92e243026d661e2a8ec4295dfc3ed2a7e324d07 | What is the value of the following double summation? \[ \sum_{j=2}^{10} \sum_{i=1}^{\infty} \frac{1^i + 2^i + 3^i + \cdots + (j-1)^i}{j^i} \] | [
"\\(\\frac{42131}{504}\\)"
] | [
"What is the value of the following double summation?\n\\[\n\\sum_{j=2}^{10} \\sum_{i=1}^{\\infty} \\frac{1^i + 2^i + 3^i + \\cdots + (j-1)^i}{j^i}\n\\]"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-ffc9d9fe08cf5e25fd9ad64594c98bb4112bc96b91b5ccb962dc62f0134f72ee | What is the value of the following expression: $1 - 4 + 7 - 10 + 13 - \cdots - 46 + 49 - 52 + 55$ ? | [
"28"
] | [
"What is the value of the following expression: $1 - 4 + 7 - 10 + 13 - \\cdots - 46 + 49 - 52 + 55$ ?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cbebc4671cfe71148e195ff5971f56e5eaa548e9199d78142a36d7e655596832 | What is the value of the following expression: $\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}$? Express your answer as a common fraction. | [
"\\frac{61}{243}"
] | [
"What is the value of the following expression: $\\frac{1}{3}-\\frac{1}{9}+\\frac{1}{27}-\\frac{1}{81}+\\frac{1}{243}$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-db51dffa36a47d9f73c889e51f8fd76a7d43f8497fe957d7b44001a00ef929fd | What is the value of the following expression? $$\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}$$ | [
"1"
] | [
"What is the value of the following expression?\n$$\\frac{100^2-7^2}{70^2-11^2} \\cdot \\frac{(70-11)(70+11)}{(100-7)(100+7)}$$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0cc777f28f757a22227fa4ed1616438fcb6af1a879e45ddf75cb39d79e21eec4 | What is the value of the following expression? \[ \frac{1}{{2}^{1990}}\sum_{n=0}^{995}{{(-3)}^{n}}\binom{1990}{2n} \] | [
"\\(-\\frac{1}{2}\\)"
] | [
"What is the value of the following expression?\n\n\\[\n\\frac{1}{{2}^{1990}}\\sum_{n=0}^{995}{{(-3)}^{n}}\\binom{1990}{2n}\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-fa1da64af9ce60a073644110c7c4dcc779d768138678edaf258443bd86bf4dd0 | What is the value of the following expression? \[ \frac{2 + 6}{4^{100}} + \frac{2 + 2 \cdot 6}{4^{99}} + \frac{2 + 3 \cdot 6}{4^{98}} + \cdots + \frac{2 + 100 \cdot 6}{4} \] | [
"200"
] | [
"What is the value of the following expression?\n\\[\n\\frac{2 + 6}{4^{100}} + \\frac{2 + 2 \\cdot 6}{4^{99}} + \\frac{2 + 3 \\cdot 6}{4^{98}} + \\cdots + \\frac{2 + 100 \\cdot 6}{4}\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-6ca9483704a0aeb292c27b40fdaa80a1692629f0653d17df11211c857c975d74 | What is the value of the following integral? \[ \int_{\frac{1}{2014}}^{2014} \frac{\tan^{-1}x}{x} \, dx \] | [
"\\(\\frac{\\pi \\ln 2014}{2}\\)"
] | [
"What is the value of the following integral?\n\n\\[\n\\int_{\\frac{1}{2014}}^{2014} \\frac{\\tan^{-1}x}{x} \\, dx\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 30
} | 44 | 0 | 0 |
omr-ws-45792a6d1a71276e3a06f5bf62b1585c5238197dbd87776a145850c9ce766920 | What is the value of the following product: $\prod_{p \text{ prime}} \frac{p^2+1}{p^2-1}$? | [
"\\(\\frac{5}{2}\\)"
] | [
"What is the value of the following product: $\\prod_{p \\text{ prime}} \\frac{p^2+1}{p^2-1}$?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-60e11605ab5e73e66e59d0a67ad1d2fad5b6f2c565c20423fe2ceb55fedf1a01 | What is the value of the improper integral \(\int_{-\infty}^{\infty} -x \, dx\)? Even if the function is odd/symmetric, can we conclude that the integral is 0? | [
"0"
] | [
"What is the value of the improper integral \\(\\int_{-\\infty}^{\\infty} -x \\, dx\\)? Even if the function is odd/symmetric, can we conclude that the integral is 0?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c082329f24276ee75de7fdb3d090a9e38e3eadc485df1a6a95d886d86fa8f8d1 | What is the value of the infinite continued fraction $7-\frac{12}{7-\frac{12}{\cdots}}$? | [
"4"
] | [
"What is the value of the infinite continued fraction $7-\\frac{12}{7-\\frac{12}{\\cdots}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-256cdf8ae228ffc8a7bd021d2bceee417ea0d9d591c74ed7cd41b33991b0c4bb | What is the value of the infinite nested radical \(\sqrt{1+\sqrt{2+\sqrt{3+\cdots}}}\)? | [
"1.7579"
] | [
"What is the value of the infinite nested radical \\(\\sqrt{1+\\sqrt{2+\\sqrt{3+\\cdots}}}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 27
} | 47 | 0 | 0 |
omr-ws-5f127106477ea991f98ee3129899b454352a56d8e0e41a85d032e91fcd4ec2d1 | What is the value of the infinite nested radical expression $\sqrt{1+2+\sqrt{1+3+\sqrt{1+4+\sqrt{1+5+\cdots}}}}$? | [
"3"
] | [
"What is the value of the infinite nested radical expression $\\sqrt{1+2+\\sqrt{1+3+\\sqrt{1+4+\\sqrt{1+5+\\cdots}}}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-4afc3f871d29a48332ace4d5c0cd1d3e1bde77ad6b183039e2013f1a09bc9c26 | What is the value of the infinite nested radical expression $\sqrt{3 \cdot \sqrt{9 \cdot \sqrt{27 \cdot \ldots}}}$? | [
"9"
] | [
"What is the value of the infinite nested radical expression $\\sqrt{3 \\cdot \\sqrt{9 \\cdot \\sqrt{27 \\cdot \\ldots}}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-805514eb172f574ba2247ed4ff5c66ee33b4c6f34bf82eb4247e13dcf391c83d | What is the value of the infinite product $$\sqrt[3]{10}\cdot\sqrt[3]{\sqrt[3]{10}}\cdot\sqrt[3]{\sqrt[3]{\sqrt[3]{10}}}\dots?$$ | [
"\\(\\sqrt{10}\\)"
] | [
"What is the value of the infinite product \n$$\\sqrt[3]{10}\\cdot\\sqrt[3]{\\sqrt[3]{10}}\\cdot\\sqrt[3]{\\sqrt[3]{\\sqrt[3]{10}}}\\dots?$$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-a619f4bbfbca4061855e745085f15419e53e49afe74acc4aa329dd9225c59fb6 | What is the value of the infinite series $\sum_{i=1}^{\infty}\frac{1}{i}$? | [
"\\infty"
] | [
"What is the value of the infinite series $\\sum_{i=1}^{\\infty}\\frac{1}{i}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f9c00ea0ec63a5fb06e7e3e8c53778ba7434bec9db36ac4c4214c20dcb1c223 | What is the value of the infinite series: \[ \frac{-\pi}{3} + \frac{\pi^3}{162} - \frac{\pi^5}{29160} + \frac{\pi^7}{11022480} + \cdots \] | [
"\\(-\\frac{\\sqrt{3}}{2}\\)"
] | [
"What is the value of the infinite series:\n\\[\n\\frac{-\\pi}{3} + \\frac{\\pi^3}{162} - \\frac{\\pi^5}{29160} + \\frac{\\pi^7}{11022480} + \\cdots\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-0346d9fb5efa4cd544e1fd4a5c8e3759a487c3deec45822f5b8993caa1091f62 | What is the value of the infinite sum $$\sum_{n=1}^{\infty} \frac{F_n}{10^{n+2}}$$ where $F_n$ is the nth Fibonacci number with $F_1=1$, $F_2=1$, and for $n>2$, $F_n=F_{n-1}+F_{n-2}$? | [
"\\(\\frac{1}{890}\\)"
] | [
"What is the value of the infinite sum $$\\sum_{n=1}^{\\infty} \\frac{F_n}{10^{n+2}}$$ where $F_n$ is the nth Fibonacci number with $F_1=1$, $F_2=1$, and for $n>2$, $F_n=F_{n-1}+F_{n-2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-5907d005a1162686ccdf0df4251d48120324e5559b1f0437f6139f409960e51e | What is the value of the infinite sum $\sum_{i=1}^\infty \left(\frac{1}{i} - \frac{2}{2i+1}\right)$? | [
"\\(2 - \\ln 4\\)"
] | [
"What is the value of the infinite sum $\\sum_{i=1}^\\infty \\left(\\frac{1}{i} - \\frac{2}{2i+1}\\right)$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e917a9fe816dd459152491f1af3b838916d3868ee128298f86d55695361d8cfb | What is the value of the infinite sum \[ \frac{1}{2}+\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{5}{32}+\frac{8}{64}+\frac{13}{128}+\frac{21}{256}+\cdots \] | [
"2"
] | [
"What is the value of the infinite sum\n\\[\n\\frac{1}{2}+\\frac{1}{4}+\\frac{2}{8}+\\frac{3}{16}+\\frac{5}{32}+\\frac{8}{64}+\\frac{13}{128}+\\frac{21}{256}+\\cdots\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6dd88770df0ed9a2fe0b267c083aca2e73540512c925cacf4d9c903f64eb083d | What is the value of the inradius of a pyramid with a regular octagon base of side length $2$ and a height of $1 + \sqrt{2}$? | [
"1"
] | [
"What is the value of the inradius of a pyramid with a regular octagon base of side length $2$ and a height of $1 + \\sqrt{2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e2ac03f8d7a728f8b0341d7e4de7a07d3f3206dea1816ad99094256fcd072cd3 | What is the value of the integer $k$ (where $1 \le k \le 100$) such that $$\frac{\displaystyle \prod_{i=1}^{100} i!}{k!}$$ is a perfect square? | [
"50"
] | [
"What is the value of the integer $k$ (where $1 \\le k \\le 100$) such that \n$$\\frac{\\displaystyle \\prod_{i=1}^{100} i!}{k!}$$\nis a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 11
} | 22 | 0 | 0 |
omr-ws-d43417860c65eb5edb67dd90cfe1d2bab8d9c12597c5e3938133909453978b15 | What is the value of the integral $$\frac{1}{2\pi}\int_0^{2\pi}(1+2\cos\theta)^2(\cos\theta+i\sin\theta)~d\theta?$$ | [
"2"
] | [
"What is the value of the integral $$\\frac{1}{2\\pi}\\int_0^{2\\pi}(1+2\\cos\\theta)^2(\\cos\\theta+i\\sin\\theta)~d\\theta?$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-9b24bd5bbb14b7377d36a8eefe5270ab5ea1ae4119bc234b2c344fa257059480 | What is the value of the integral $\int (x-1) \sqrt{\frac{x^4 + 2x^3 - x^2 + 2x + 1}{x^3 + x^2}} \, dx$? | [
"no_valid_answer_found"
] | [
"What is the value of the integral $\\int (x-1) \\sqrt{\\frac{x^4 + 2x^3 - x^2 + 2x + 1}{x^3 + x^2}} \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c04241087b83ed618136d44a361a32bf2a56e4498500266c085c5f4e60522d4 | What is the value of the integral $\int_0 ^{\tan^{-1}3} [\tan{x}] \, dx$ where $[k] \equiv \lfloor k \rfloor$? | [
"\\(2\\tan^{-1}3 - \\tan^{-1}2 - \\tan^{-1}1\\) or approximately \\(0.552\\)"
] | [
"What is the value of the integral $\\int_0 ^{\\tan^{-1}3} [\\tan{x}] \\, dx$ where $[k] \\equiv \\lfloor k \\rfloor$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 4
} | 15 | 0 | 0 |
omr-ws-d9cb3bd0df24df7abd47d5a88c81a4f8c22f0d9640ae550b42906ab8735e95ac | What is the value of the integral $\int_0^\infty \frac{(x \cos x - \sin x)^2}{x^6} \, dx$? | [
"$\\frac{\\pi}{15}$"
] | [
"What is the value of the integral $\\int_0^\\infty \\frac{(x \\cos x - \\sin x)^2}{x^6} \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 28
} | 37 | 0 | 0 |
omr-ws-7375241c587a8c702003a82537f0a9bf8ad151e779ae0361e88a4d85c0bc363d | What is the value of the integral $\int_0^a x^3 \sqrt{\frac{x}{a-x}} \, dx$? | [
"\\( 70\\left( \\frac{a}{4}\\right)^4 \\pi \\)"
] | [
"What is the value of the integral $\\int_0^a x^3 \\sqrt{\\frac{x}{a-x}} \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-0474400bc1f8695067f967accdc0e0c7ce998f7f5691e854eda2faf7962bbd2a | What is the value of the integral $\int_0^{2\pi} \frac{x \sin^{100}x}{\sin^{100}x + \cos^{100}x} \, dx$? | [
"\\(\\pi^2\\)"
] | [
"What is the value of the integral $\\int_0^{2\\pi} \\frac{x \\sin^{100}x}{\\sin^{100}x + \\cos^{100}x} \\, dx$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-b09c4a0428bf57f147df402b182f6e01d65e5ce0831bea3d2b71ee6afa5b4a59 | What is the value of the integral $\int_0^{\pi/2} \sin^{n}x \, dx$? | [
"\\dfrac{\\sqrt{\\pi}}{2} \\cdot \\dfrac{\\Gamma\\left( \\dfrac{n + 1}{2} \\right)}{\\Gamma\\left( \\dfrac{n}{2} + 1 \\right)}"
] | [
"What is the value of the integral $\\int_0^{\\pi/2} \\sin^{n}x \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-64d50cb2b45f3ca3bdfac5caa1a7b472d5d6a99d71f6e1ed457356838c917ad9 | What is the value of the integral $\int_0^{\pi/3} \frac{\sin^{n}x}{\sin^{n}x+\cos^{n}x} \, dx$? | [
"\\(\\frac{\\pi}{6}\\)"
] | [
"What is the value of the integral $\\int_0^{\\pi/3} \\frac{\\sin^{n}x}{\\sin^{n}x+\\cos^{n}x} \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 5
} | 13 | 0 | 0 |
omr-ws-c755fbde49c3ab12169cabc6c16ebc35546d9edb51c76f25a38d2644b3789b8a | What is the value of the integral $\int_{-\infty }^{\infty }\frac{e^{x}x^{2}}{(1+e^x)^{2}}dx$? | [
"\\(\\frac{\\pi^2}{3}\\)"
] | [
"What is the value of the integral $\\int_{-\\infty }^{\\infty }\\frac{e^{x}x^{2}}{(1+e^x)^{2}}dx$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-c526e539b9aeadc0165a9e69d396a6cc2856089fa91f740bc8fe4a9572765a67 | What is the value of the integral $\int_{0}^{1} \frac{x^4 (1-x)^4}{1 + x^2} \, dx$? | [
"\\(\\frac{59}{21} - \\pi\\)"
] | [
"What is the value of the integral $\\int_{0}^{1} \\frac{x^4 (1-x)^4}{1 + x^2} \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c952185437af2fe1a9c5912711b244d98fa2b863732e3e6fd9ec305ea613c536 | What is the value of the integral $\int_{0}^{\pi} \frac{\sin(n + \frac{1}{2})x }{\sin \frac{1}{2}x}\ dx$ for negative integers $n$? | [
"-\\pi"
] | [
"What is the value of the integral $\\int_{0}^{\\pi} \\frac{\\sin(n + \\frac{1}{2})x }{\\sin \\frac{1}{2}x}\\ dx$ for negative integers $n$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-731b13bcbfadd42c696aeac3f72e68557ce0d033d09795c988cc7ea0959bae11 | What is the value of the integral $\int_{\Delta} \text{Im}(z) \, dz$ where $\Delta$ is a triangle with vertices $(0,0)$, $(8,0)$, and $(5,4)$ traversed anticlockwise? | [
"-16"
] | [
"What is the value of the integral $\\int_{\\Delta} \\text{Im}(z) \\, dz$ where $\\Delta$ is a triangle with vertices $(0,0)$, $(8,0)$, and $(5,4)$ traversed anticlockwise?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8c49c21e9bcca532cadea1a6280c77c78e29d7aab6e031994999f02d0e546c4c | What is the value of the integral $\int{\exp(Ax_1)dS^{n-1}(x)}$, where $S^{n-1}$ is the surface of the $1$-ball in $\mathbb{R}^n$? | [
"(2\\pi)^{n/2} \\frac{I_{n/2 -1}(A)}{A^{n/2 -1}}"
] | [
"What is the value of the integral $\\int{\\exp(Ax_1)dS^{n-1}(x)}$, where $S^{n-1}$ is the surface of the $1$-ball in $\\mathbb{R}^n$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-f6f5d39a9c04eb945c2d9cabb5a152512edc2953dfb8a1887e1b6b9e46c9cfe3 | What is the value of the integral $\int{\sqrt{\tan{x}} \; dx}$? | [
"\\(\\sin^{-1}(\\sin x - \\cos x) - \\ln|(\\sin x + \\cos x) + \\sqrt{\\sin 2x}| + C\\)"
] | [
"What is the value of the integral $\\int{\\sqrt{\\tan{x}} \\; dx}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e5431e6107c14f787dce5437a08418a669ba03fb22a980b127a269fa20c4e25 | What is the value of the integral \( I = \int_{-1}^{2} | \sin(\pi x) | \, dx \)? | [
"\\(\\frac{6}{\\pi}\\)"
] | [
"What is the value of the integral \\( I = \\int_{-1}^{2} | \\sin(\\pi x) | \\, dx \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-693118d48234933ebb384c21117e885c558a54afa4f445454329dc2598c65af2 | What is the value of the integral \( \int_{|z|=\rho} \frac{|dz|}{|z - a|^2} \), where \( |a| \neq \rho \)? | [
"\\dfrac{2\\pi\\rho}{|\\rho^2 - |a|^2|}"
] | [
"What is the value of the integral \\( \\int_{|z|=\\rho} \\frac{|dz|}{|z - a|^2} \\), where \\( |a| \\neq \\rho \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de83c6a3858638faa4f97053f11668799283afc7b4cf99228c12835199ada205 | What is the value of the integral \[ \frac{n!}{2\pi i} \oint_{|z|=3} \frac{z^n + az^{n-2} + bz}{(z-3)^{n+1}} \, dz \] when it is integrated over \( |z|=3 \)? | [
"n!"
] | [
"What is the value of the integral \n\\[ \\frac{n!}{2\\pi i} \\oint_{|z|=3} \\frac{z^n + az^{n-2} + bz}{(z-3)^{n+1}} \\, dz \\]\nwhen it is integrated over \\( |z|=3 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ae29194e8584a9cfa39d5dfb2c60d0442d3281494b72432e1e02ede02f9e186 | What is the value of the integral \[ \int_0^1\int_{y^4}^1 y^3e^{x^2}\, \mathrm{d}x\, \mathrm{d}y? \] | [
"\\(\\frac{e-1}{8}\\)"
] | [
"What is the value of the integral\n\\[\n\\int_0^1\\int_{y^4}^1 y^3e^{x^2}\\, \\mathrm{d}x\\, \\mathrm{d}y?\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-630a76518771c85d4b5dd95a7b4d8f6b764b38f386ac0080c3f479e81b54c6cb | What is the value of the integral \[ \int_{0}^{\frac{\pi}{2}} \frac{1}{1+ \cos \alpha \cos \theta}\ d\theta \] when \(\pi < \alpha < 2\pi\)? | [
"\\(\\frac{a - 2\\pi}{\\sin a}\\)"
] | [
"What is the value of the integral \n\n\\[\n\\int_{0}^{\\frac{\\pi}{2}} \\frac{1}{1+ \\cos \\alpha \\cos \\theta}\\ d\\theta\n\\]\n\nwhen \\(\\pi < \\alpha < 2\\pi\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-d06775a1daf17b9de658b65695a6262baa7fb7469e51eb2a12106c18e37a7ab4 | What is the value of the integral of \( f(x,y) = x^3 + y \) over the unit disk \( D = \{(x,y) \mid x^2 + y^2 \leq 1 \} \)? | [
"0"
] | [
"What is the value of the integral of \\( f(x,y) = x^3 + y \\) over the unit disk \\( D = \\{(x,y) \\mid x^2 + y^2 \\leq 1 \\} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4854ec534c782a0fcff1a02c3f472940fa634e2371621f1c852de9d11c292e53 | What is the value of the least positive base ten number which requires six digits for its binary representation? | [
"32"
] | [
"What is the value of the least positive base ten number which requires six digits for its binary representation?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c52e74191cc3eecdd57dff7583f70baeaf6fe1b1b92e070ce5d8d7127dee0f5a | What is the value of the limit \(\lim_{n \to \infty} \frac{1^p + 2^p + \cdots + n^p}{n^{p+1}}\)? | [
"\\(\\frac{1}{p+1}\\)"
] | [
"What is the value of the limit \\(\\lim_{n \\to \\infty} \\frac{1^p + 2^p + \\cdots + n^p}{n^{p+1}}\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-6579c70fcdae369a4d68a781f24660e113f1232bc77bee8095d32bbe0ab555e7 | What is the value of the limit \(\lim_{n\rightarrow\infty}\frac{p_n}{n}\), where \(p_n\) denotes the \(n\)-th prime number? | [
"\\infty"
] | [
"What is the value of the limit \\(\\lim_{n\\rightarrow\\infty}\\frac{p_n}{n}\\), where \\(p_n\\) denotes the \\(n\\)-th prime number?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e3faa938499586a35812bfe6094a2d5eb68af739559335ca54448bab2c456a91 | What is the value of the limit \[ \lim_{n\to\infty} \left( \frac{1}{\sqrt{n} \sqrt{n+1}} + \frac{1}{\sqrt{n} \sqrt{n+2}} + \frac{1}{\sqrt{n} \sqrt{n+3}} + \cdots + \frac{1}{\sqrt{n} \sqrt{2n}} \right)? \] | [
"\\(2\\sqrt{2} - 2\\)"
] | [
"What is the value of the limit\n\\[\n\\lim_{n\\to\\infty} \\left( \\frac{1}{\\sqrt{n} \\sqrt{n+1}} + \\frac{1}{\\sqrt{n} \\sqrt{n+2}} + \\frac{1}{\\sqrt{n} \\sqrt{n+3}} + \\cdots + \\frac{1}{\\sqrt{n} \\sqrt{2n}} \\right)?\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1e0d24ddf28338dda99d5b23f6991b46249fa35cac7386845d8e8ea6be9c61e4 | What is the value of the number \(0.\overline{135}\) when it is expressed as a common fraction? | [
"\\dfrac{5}{37}"
] | [
"What is the value of the number \\(0.\\overline{135}\\) when it is expressed as a common fraction?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f893edda6cb1f33ff5f56d68e8ac59135865b8657d91e2143ac32f6bf2f71c2e | What is the value of the product $$\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?$$ | [
"\\(\\frac{50}{99}\\)"
] | [
"What is the value of the product $$\\left(\\frac{1\\cdot3}{2\\cdot2}\\right)\\left(\\frac{2\\cdot4}{3\\cdot3}\\right)\\left(\\frac{3\\cdot5}{4\\cdot4}\\right)\\cdots\\left(\\frac{97\\cdot99}{98\\cdot98}\\right)\\left(\\frac{98\\cdot100}{99\\cdot99}\\right)?$$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-2aba0a0a5c87cf22ddacbf52de92660328cebff482190a3325827e943d41ad53 | What is the value of the product $$\prod_{n=0}^{100} (1-n^2 +n^4) \pmod{101}?$$ | [
"9"
] | [
"What is the value of the product $$\\prod_{n=0}^{100} (1-n^2 +n^4) \\pmod{101}?$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-b35b57bf0594ab20387f5304781d51b469f864cf9dc289f94fa216942f06b4e6 | What is the value of the product $(\sin{\frac{\pi}{5}})(\sin{\frac{2\pi}{5}})(\sin{\frac{3\pi}{5}})(\sin{\frac{4\pi}{5}})$? | [
"\\(\\frac{5}{16}\\)"
] | [
"What is the value of the product $(\\sin{\\frac{\\pi}{5}})(\\sin{\\frac{2\\pi}{5}})(\\sin{\\frac{3\\pi}{5}})(\\sin{\\frac{4\\pi}{5}})$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-003cf2da75e3b9b023fb60ca32f570725490bd30f1e372d8b979b4f7dfb5ec01 | What is the value of the product $\cos 1^\circ \cdot \cos 3^\circ \cdot \cos 5^\circ \cdots \cos 89^\circ$? | [
"\\(\\sqrt{2} \\cdot 2^{-45}\\)"
] | [
"What is the value of the product $\\cos 1^\\circ \\cdot \\cos 3^\\circ \\cdot \\cos 5^\\circ \\cdots \\cos 89^\\circ$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-905ba04b5c9dbaebaf920ffd129580ea8eeefef160bf2d899fb7cb713beb4123 | What is the value of the product $\sin{\frac{\pi}{32}}\cos{\frac{\pi}{32}}\cos{\frac{\pi}{16}}\cos{\frac{\pi}{8}}\cos{\frac{\pi}{4}}$? | [
"\\(\\frac{1}{16}\\)"
] | [
"What is the value of the product $\\sin{\\frac{\\pi}{32}}\\cos{\\frac{\\pi}{32}}\\cos{\\frac{\\pi}{16}}\\cos{\\frac{\\pi}{8}}\\cos{\\frac{\\pi}{4}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-e1c91cf2f503885265a60b1ad928c6a44324abe7763b1a45e80e51677d359329 | What is the value of the product \((1 - \frac{1}{4})(1 - \frac{1}{9})(1 - \frac{1}{16}) \cdots (1 - \frac{1}{4012009})\)? | [
"\\(\\frac{1002}{2003}\\)"
] | [
"What is the value of the product \\((1 - \\frac{1}{4})(1 - \\frac{1}{9})(1 - \\frac{1}{16}) \\cdots (1 - \\frac{1}{4012009})\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-badf45f282acb1e9cce1fac4e95acd345f3f391071047edf60f3f85d19afd199 | What is the value of the product \(1 - \cos x \cos 2x \cos 3x \cos 4x \cdots \cos nx\)? | [
"1 - \\cos x \\cos 2x \\cos 3x \\cdots \\cos nx"
] | [
"What is the value of the product \\(1 - \\cos x \\cos 2x \\cos 3x \\cos 4x \\cdots \\cos nx\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6318296db18aa1d46244f3e937b7763677b8a1b1f94eedc63f0ced6f2ae5cca4 | What is the value of the product \(1 \times 2 \times 3 \times \cdots \times 2006\)? | [
"\\(2006!\\)"
] | [
"What is the value of the product \\(1 \\times 2 \\times 3 \\times \\cdots \\times 2006\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-849d7096e533d4e464c56a6ccb3fbb7d996f7233de20e0d0ce0bbe8318ea3ff8 | What is the value of the quotient \(\frac{\frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \cdots}{\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \cdots}\), where the numerator is the sum of the reciprocals of odd numbers and the denominator is the sum of the reciprocals of even numbers? | [
"1"
] | [
"What is the value of the quotient \\(\\frac{\\frac{1}{3} + \\frac{1}{5} + \\frac{1}{7} + \\cdots}{\\frac{1}{2} + \\frac{1}{4} + \\frac{1}{6} + \\cdots}\\), where the numerator is the sum of the reciprocals of odd numbers and the denominator is the sum of the reciprocals of even numbers?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f0381aab0ce2f80ac69d5863d51c18497c749840b7dc8ac2ff4785d1d54fa7b5 | What is the value of the remainder when $\tan^{2005}{\frac{\pi}{12}}+\cot^{2005}{\frac{\pi}{12}}$ is divided by $9$? | [
"7"
] | [
"What is the value of the remainder when $\\tan^{2005}{\\frac{\\pi}{12}}+\\cot^{2005}{\\frac{\\pi}{12}}$ is divided by $9$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93250d827f88af85c336d703334b81e89b56a90d9eb104f089c44367dc16418b | What is the value of the repeating decimal $0.\overline{1337}_8$ when expressed in base 10? | [
"\\dfrac{7}{39}"
] | [
"What is the value of the repeating decimal $0.\\overline{1337}_8$ when expressed in base 10?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf818371dc395e8605030633a45bb678c71b7aa4dad9960b266c5b6795ff37be | What is the value of the series $$\sum_{n=1}^{\infty}\frac{(3-2\sqrt 2)^n}{e^n}?$$ | [
"\\(\\frac{3-2\\sqrt{2}}{e - 3 + 2\\sqrt{2}}\\)"
] | [
"What is the value of the series $$\\sum_{n=1}^{\\infty}\\frac{(3-2\\sqrt 2)^n}{e^n}?$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-0b729cf8f541198e3eda40b161201d6155aad0533a42e4e2ca53a06172dc6858 | What is the value of the series $1*3 - 5*7 + 9*11 - 13*15 + \ldots - 2005*2007 + 2009*2011$? | [
"2024067"
] | [
"What is the value of the series $1*3 - 5*7 + 9*11 - 13*15 + \\ldots - 2005*2007 + 2009*2011$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-80f5eb12c191e7a155c2f3ee81d2c98441c0604ff1bdfd19cd9b57322976d49e | What is the value of the series $\frac{1}{10}+\frac{1}{100}+\frac{2}{1000}+\frac{3}{10000}+\frac{5}{100000}+\frac{8}{1000000}+\ldots$, where the numerators follow the Fibonacci sequence? | [
"\\(\\frac{10}{89}\\)"
] | [
"What is the value of the series $\\frac{1}{10}+\\frac{1}{100}+\\frac{2}{1000}+\\frac{3}{10000}+\\frac{5}{100000}+\\frac{8}{1000000}+\\ldots$, where the numerators follow the Fibonacci sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-47ba4126a324971e42b53be162c0827753594645ef4299e0ad0bdd806a395489 | What is the value of the series $\frac{1}{1} + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \frac{1}{125} + \ldots$? | [
"\\(\\zeta(3)\\)"
] | [
"What is the value of the series $\\frac{1}{1} + \\frac{1}{8} + \\frac{1}{27} + \\frac{1}{64} + \\frac{1}{125} + \\ldots$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-77cc700414019576ccd5f11e574bd9c302aa4c04ee999ce132dc16a528fe0340 | What is the value of the series $\sum_{1 \leq l < m < n} \frac{1}{5^l 3^m 2^n}$? | [
"\\(\\frac{1}{145}\\)"
] | [
"What is the value of the series $\\sum_{1 \\leq l < m < n} \\frac{1}{5^l 3^m 2^n}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-ba3f20e9e112aa27acbd5ac18b7532635eda90273777dc5962daf3a1a031e43d | What is the value of the series \( \dfrac{1}{10} + \dfrac{1}{10^2} + \dfrac{2}{10^3} + \dfrac{3}{10^4} + \dfrac{5}{10^5} + \dfrac{8}{10^6} + \cdots \)? | [
"\\(\\dfrac{10}{89}\\)"
] | [
"What is the value of the series \\( \\dfrac{1}{10} + \\dfrac{1}{10^2} + \\dfrac{2}{10^3} + \\dfrac{3}{10^4} + \\dfrac{5}{10^5} + \\dfrac{8}{10^6} + \\cdots \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-566cb0960ee21d81c1b92627869d77e37ab2a1f37f64b2f403c7898a2302ebfa | What is the value of the series \( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \)? It is known to diverge, but is there a specific number associated with it? | [
"\\gamma"
] | [
"What is the value of the series \\( \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\cdots \\)? It is known to diverge, but is there a specific number associated with it?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-e94348c892c05dbaca72cdd8fbf3ea6c8e4548acb3e21e37f272b1ba9dbb35ad | What is the value of the series \(1 + 2 + 3 - 4 + 5 + 6 + 7 - 8 + \cdots + 197 + 198 + 199 - 200\)? | [
"9900"
] | [
"What is the value of the series \\(1 + 2 + 3 - 4 + 5 + 6 + 7 - 8 + \\cdots + 197 + 198 + 199 - 200\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-f2477289856c866859c723cd3262261602055b834132f4799fdd779553c38d92 | What is the value of the series \(1 - 1 + 1 - 1 + 1 - \ldots\)? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the value of the series \\(1 - 1 + 1 - 1 + 1 - \\ldots\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-5d57778b22a61fa47bdb19562838c25a25e71f8fcd6a8bef13cbe8e1f559c851 | What is the value of the series \(\sum_{n=1}^{\infty}\frac{1}{n^{2}+1}\)? | [
"\\(\\frac{\\pi \\cosh{\\pi}-\\sinh{\\pi}}{2 \\sinh{\\pi}}\\)"
] | [
"What is the value of the series \\(\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}+1}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-25ba5e10d40a3dba8d8a813f166e3507e15b3ac0e5126a463d4df4e27bfa239f | What is the value of the series \[ \frac{2021}{2020} + \frac{2021 \cdot 2019}{2020 \cdot 2018} + \frac{2021 \cdot 2019 \cdot 2017}{2020 \cdot 2018 \cdot 2016} + \cdots + \frac{2021 \cdot 2019 \cdot \cdots \cdot 5 \cdot 3}{2020 \cdot 2018 \cdot \cdots \cdot 4 \cdot 2} - 1? \] | [
"2019"
] | [
"What is the value of the series\n\\[\n\\frac{2021}{2020} + \\frac{2021 \\cdot 2019}{2020 \\cdot 2018} + \\frac{2021 \\cdot 2019 \\cdot 2017}{2020 \\cdot 2018 \\cdot 2016} + \\cdots + \\frac{2021 \\cdot 2019 \\cdot \\cdots \\cdot 5 \\cdot 3}{2020 \\cdot 2018 \\cdot \\cdots \\cdot 4 \\cdot 2} - 1?\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1bb91c4c513cbce5528ba8fb82806f12a21c48c8066f874cfd4fe93cd50a227 | What is the value of the series \[ \sum_{n=1}^{\infty} \frac{d(n)}{n^2}, \] where $d(n)$ denotes the number of divisors of $n$? | [
"\\(\\frac{\\pi^4}{36}\\)"
] | [
"What is the value of the series\n\\[\n\\sum_{n=1}^{\\infty} \\frac{d(n)}{n^2},\n\\]\nwhere $d(n)$ denotes the number of divisors of $n$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 28
} | 46 | 0 | 0 |
omr-ws-bda3facf6dd713077907354716157e9f67936a550af9fa265ec3cac8c8096550 | What is the value of the sum $$\sum_{n=0}^{\infty} \frac{1}{\binom{2n}{n}}$$? | [
"\\(\\frac{2}{27}(18 + \\pi \\sqrt{3})\\)"
] | [
"What is the value of the sum $$\\sum_{n=0}^{\\infty} \\frac{1}{\\binom{2n}{n}}$$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 16
} | 27 | 0 | 0 |
omr-ws-f100255450dbab7ff66b1c6d93f91e64f9107a52751effd2dc3f42cd381e0553 | What is the value of the sum $$\sum_{n=1}^\infty \left(\sqrt{1+\frac{1}{n}} - \sqrt{1+\frac{1}{n+1}}\right)?$$ | [
"\\(\\sqrt{2} - 1\\)"
] | [
"What is the value of the sum $$\\sum_{n=1}^\\infty \\left(\\sqrt{1+\\frac{1}{n}} - \\sqrt{1+\\frac{1}{n+1}}\\right)?$$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-b9b7e14961d55d06798ef8b52a9395468873322c6c4fc13ce93362b1a634c13f | What is the value of the sum $-1 + 2 - 3 + 4 - 5 + 6 - 7 +\dots+ 10,\!000$? | [
"5000"
] | [
"What is the value of the sum $-1 + 2 - 3 + 4 - 5 + 6 - 7 +\\dots+ 10,\\!000$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d42b4beed14916cf11c3568fa6ada1716d3cf90b1f2317943a2fa7dd9bebb61 | What is the value of the sum $1 + i + i^2 + i^3 + \cdots + i^{2004}$, where $i = \sqrt{-1}$? | [
"1"
] | [
"What is the value of the sum $1 + i + i^2 + i^3 + \\cdots + i^{2004}$, where $i = \\sqrt{-1}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12bb2b79f2d2ac5ea430b18a2ba0ce67f18dd1b4bc24139caa6042ca020dedf0 | What is the value of the sum $\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+\frac{3}{(3+1)!}+...+\frac{n}{(n+1)!}$? | [
"\\(1 - \\frac{1}{(n+1)!}\\)"
] | [
"What is the value of the sum $\\frac{1}{(1+1)!}+\\frac{2}{(2+1)!}+\\frac{3}{(3+1)!}+...+\\frac{n}{(n+1)!}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e03bb6a47a67d526b583a6f22e80a86194ee33ed951c93566853505517c97abf | What is the value of the sum $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}$? | [
"\\(1 - \\frac{1}{100!}\\)"
] | [
"What is the value of the sum $\\frac{1}{2!}+\\frac{2}{3!}+\\frac{3}{4!}+...+\\frac{99}{100!}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-746f041b84425e69cfd8765bb2c04655f721f2e7f71a3de2d2dc856e9e1db523 | What is the value of the sum $\frac{2}{3}+\frac{2^2}{3^2}+\frac{2^3}{3^3}+ \ldots +\frac{2^{10}}{3^{10}}$? Express your answer as a common fraction. | [
"\\frac{116050}{59049}"
] | [
"What is the value of the sum $\\frac{2}{3}+\\frac{2^2}{3^2}+\\frac{2^3}{3^3}+ \\ldots +\\frac{2^{10}}{3^{10}}$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce912ef279b9bc823d6e19096e4513b199b5e1a30eaafa47238f3f929cd7f9bb | What is the value of the sum $\sum \frac{1}{p^2}$, where $p$ is a prime number? | [
"0.452247"
] | [
"What is the value of the sum $\\sum \\frac{1}{p^2}$, where $p$ is a prime number?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-696835b824f4316ab72783a57c1519f12427d18a0397372828ff6c3902b52d05 | What is the value of the sum $\sum \left|\begin{array}{cccc}a_{1\alpha_1} & a_{1\alpha_2} & \cdots & a_{1\alpha_n} \\ a_{2\alpha_1} & a_{2\alpha_2} & \cdots & a_{2\alpha_n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n\alpha_1} & a_{n\alpha_2} & \cdots & a_{n\alpha_n}\end{array}\right|$ where the summation is taken over... | [
"0 (for \\(n \\geq 2\\))"
] | [
"What is the value of the sum $\\sum \\left|\\begin{array}{cccc}a_{1\\alpha_1} & a_{1\\alpha_2} & \\cdots & a_{1\\alpha_n} \\\\ a_{2\\alpha_1} & a_{2\\alpha_2} & \\cdots & a_{2\\alpha_n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{n\\alpha_1} & a_{n\\alpha_2} & \\cdots & a_{n\\alpha_n}\\end{array}\\right|$ w... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-8d27d56b96cd05d8c7fa382cfef08dd0441d3634138591e199a93e9cb8880bc9 | What is the value of the sum $\sum_{\text{prime }p} \frac{1}{p^2}$? | [
"0.452247"
] | [
"What is the value of the sum $\\sum_{\\text{prime }p} \\frac{1}{p^2}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-973516b595f29b4670913cdfbfe7d769b63c828205a210ff92013607ed57524d | What is the value of the sum $\sum_{i, j=1}^{n}(i-j)^{2}$? | [
"\\(\\frac{n^2(n+1)(n-1)}{3}\\)"
] | [
"What is the value of the sum $\\sum_{i, j=1}^{n}(i-j)^{2}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42e8bc2c2589aa5bf507ebbc0b193d0e8a0e018d641f7e8b311372e1f7378504 | What is the value of the sum $\sum_{k=1}^{n} k \times 2^k$? | [
"$(n-1)2^{n+1} + 2$"
] | [
"What is the value of the sum $\\sum_{k=1}^{n} k \\times 2^k$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-77e70a35224312bbe82644e103fa301e73b9712f1fc4f0afba54a22b6faace4a | What is the value of the sum $\sum_{k=1}^{n}\frac{1}{k}$? | [
"H_n"
] | [
"What is the value of the sum $\\sum_{k=1}^{n}\\frac{1}{k}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9843ab0dbdcbed4722ae1ffcc176b37a87a19899f5f59241a52e277eae7d23df | What is the value of the sum $\sum_{n=0}^{1947}\frac{1}{2^n + \sqrt{2^{1947}}}$? | [
"\\(\\frac{974}{\\sqrt{2^{1947}}}\\)"
] | [
"What is the value of the sum $\\sum_{n=0}^{1947}\\frac{1}{2^n + \\sqrt{2^{1947}}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 14
} | 28 | 0 | 0 |
omr-ws-b2c7634365d8c9e28a72cd6b8f9b5c03c2a5ccbf93228a4a2261be30c83d222d | What is the value of the sum \( \sum_{k=0}^\infty \frac{1}{{n+k \choose k}} \) for \( n > 1 \)? | [
"\\(\\frac{n}{n-1}\\)"
] | [
"What is the value of the sum \\( \\sum_{k=0}^\\infty \\frac{1}{{n+k \\choose k}} \\) for \\( n > 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-e1f2595f4ceb13dd745d9e69968192d80695919058e631178fd3cfe187c67995 | What is the value of the sum \( nP1 + nP2 + nP3 + nP4 + \ldots + nPn \)? | [
"\\( \\lfloor n! \\cdot e - 1 \\rfloor \\)"
] | [
"What is the value of the sum \\( nP1 + nP2 + nP3 + nP4 + \\ldots + nPn \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 7
} | 15 | 0 | 0 |
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