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-84659239556c094754bf0eb9bbeb3e804cf82180427d6c1f85b42b194f7dcd9d | Which is greater: \((1+\dfrac{2}{3^3})(1+\dfrac{2}{5^3})\cdots(1+\dfrac{2}{2013^3})\) or \(\sqrt{\dfrac{3}{2}}\)? | [
"\\sqrt{\\dfrac{3}{2}}"
] | [
"Which is greater: \\((1+\\dfrac{2}{3^3})(1+\\dfrac{2}{5^3})\\cdots(1+\\dfrac{2}{2013^3})\\) or \\(\\sqrt{\\dfrac{3}{2}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-26d7daacdf2c63dee25128e15d0bd6415c729fef00df6b1d45874a7e15d7ec39 | Which is greater: \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{1}{199} - \frac{1}{200}\) or \(\frac{1}{101} + \frac{1}{102} + \cdots + \frac{1}{200}\)? | [
"They are the same."
] | [
"Which is greater: \\(1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\cdots + \\frac{1}{199} - \\frac{1}{200}\\) or \\(\\frac{1}{101} + \\frac{1}{102} + \\cdots + \\frac{1}{200}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 28
} | 49 | 0 | 0 |
omr-ws-f3a35980d3173398372e8e8498ca5c49fe9219b944b3a13f732f952e004628ff | Which is greater: \(200!\) or \(100^{200}\)? | [
"\\(100^{200}\\) is greater."
] | [
"Which is greater: \\(200!\\) or \\(100^{200}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2f790bea1e4bf0d1d960a91786ce8c5f161d089d86db7e89d140126159649d25 | Which is greater: \(2^{100!}\) or \(2^{100}!\)? | [
"\\(2^{100!}\\)"
] | [
"Which is greater: \\(2^{100!}\\) or \\(2^{100}!\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-9ec629610245650f19af2fde7faf4798e7913a88b299b71fbac66e21d1f5dfad | Which is greater: \(3^{4^{5^{6^{7^8}}}}\) or \(4^{5^{6^{7^{8^3}}}}\)? | [
"\\(3^{4^{5^{6^{7^8}}}}\\) is greater."
] | [
"Which is greater: \\(3^{4^{5^{6^{7^8}}}}\\) or \\(4^{5^{6^{7^{8^3}}}}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-07e68306aab0c4482b7faa7fd1fa894e83c3d12a855559c7b0d1499cc48f533a | Which is greater: \(4^{4^{4^{4^{4^{4}}}}}\) or \(5^{5^{5^{5}}}\)? | [
"\\(4^{4^{4^{4^{4^{4}}}}} > 5^{5^{5^{5}}}\\)"
] | [
"Which is greater: \\(4^{4^{4^{4^{4^{4}}}}}\\) or \\(5^{5^{5^{5}}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 4,
"tir": 9
} | 40 | 0 | 0 |
omr-ws-2b591fd638a287321cbbc32148de647d5868aadbbc56688ff43591b6cf514c30 | Which is greater: \[ \frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}} \] or \[ 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}? \] | [
"The first sum is greater."
] | [
"Which is greater:\n\\[\n\\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\\cdots +\\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\n\\]\nor\n\\[\n1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\frac{1}{9}-\\cdots +\\frac{1}{5781}?\n\\]"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-b0dd8068b6909f0c273e6e2b8bf0c0071ea0eef916737906211287cb415fb41c | Which is greater: the number of ways each person at a party of 2024 people could shake hands with 4 others, or the number of ways each person could shake hands with 3 others? | [
"The number of ways each person could shake hands with 4 others is greater."
] | [
"Which is greater: the number of ways each person at a party of 2024 people could shake hands with 4 others, or the number of ways each person could shake hands with 3 others?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 18
} | 30 | 0 | 0 |
omr-ws-be81ceba7a08e2c4a8684f38b6d37aaf96a63b00171d438f60ff25f0e1e89fcc | Which is larger, $100^{117}$ or $117^{100}$? No log tables allowed. | [
"\\(100^{117}\\) is larger."
] | [
"Which is larger, $100^{117}$ or $117^{100}$? No log tables allowed."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-68827150dd5fb3040a26dc3622946264372e8cb5d31cd7051cc4a3f5a3453435 | Which is larger, $100^{300}$ or $300!$? | [
"\\(300!\\) is larger."
] | [
"Which is larger, $100^{300}$ or $300!$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c43548729ed584a08af9e06282f2daff8f5cb88cef9c2520f6eaa6a7747ce96a | Which is larger, $10^{30!}$ or $10^{30}!$? | [
"\\( 10^{30!} \\)"
] | [
"Which is larger, $10^{30!}$ or $10^{30}!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-0e984a577213625ca5b2f6674aa44ef59750c4083cca6256e39daac3945ebdf1 | Which is larger, $2000!$ or $1000^{2000}$? | [
"\\(1000^{2000}\\) is larger."
] | [
"Which is larger, $2000!$ or $1000^{2000}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-a0b842ef4dc59d21660c52fb3cbae7f88be146f36aafdebcdd70aa2c0473d7a5 | Which is larger, $300!$ or $100^{300}$? | [
"300!"
] | [
"Which is larger, $300!$ or $100^{300}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-488a98f69cdbc3f09f18b656fb785fa25a9f7ee4b596d5a15aba1c70d98dd8ad | Which is larger, $50^{100}$ or $100^{50}$? | [
"\\( 50^{100} \\) is larger."
] | [
"Which is larger, $50^{100}$ or $100^{50}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-af77972ddf8c357c1a622ea30cf90c56e22fa737e13d75f4f92fb89ac4377435 | Which is larger, $5^{99346}$ or $3^{62336}$? | [
"5^{99346}"
] | [
"Which is larger, $5^{99346}$ or $3^{62336}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bca6c6e49d3ae751b96c0cb10462190135d1bd1f3cfde9868f618483102b6358 | Which is larger, $\frac{10^{1965}+1}{10^{1966}+1}$ or $\frac{10^{1966}+1}{10^{1967}+1}$? | [
"\\(\\frac{10^{1965}+1}{10^{1966}+1}\\) is larger."
] | [
"Which is larger, $\\frac{10^{1965}+1}{10^{1966}+1}$ or $\\frac{10^{1966}+1}{10^{1967}+1}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-b86c00ab77d77c0bbe608be8f44089758e85ae2924d5bab2aa591a5fcbb22e89 | Which is larger, $\sqrt{1996} + \sqrt{1998}$ or $2\sqrt{1997}$? | [
"\\(2\\sqrt{1997}\\)"
] | [
"Which is larger, $\\sqrt{1996} + \\sqrt{1998}$ or $2\\sqrt{1997}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-62b30cf6134909d25a29e1110e6fa347edf1b08dced08760afab89d2cfdc7229 | Which is larger, \( (\sqrt{6})^{2\sqrt{2}} \) or \( (2\sqrt{2})^{\sqrt{6}} \)? | [
"(2\\sqrt{2})^{\\sqrt{6}}"
] | [
"Which is larger, \\( (\\sqrt{6})^{2\\sqrt{2}} \\) or \\( (2\\sqrt{2})^{\\sqrt{6}} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e7e1222590feb87dfb9e41ec1e95083a04a65403520b0de41c6721d47f558b8 | Which is larger, \( \int_0^{\pi}e^{\sin^2x}\,dx \) or \( \frac{27\pi}{16} \)? (No calculators.) | [
"\\( \\int_0^{\\pi}e^{\\sin^2x}\\,dx \\) is larger."
] | [
"Which is larger, \\( \\int_0^{\\pi}e^{\\sin^2x}\\,dx \\) or \\( \\frac{27\\pi}{16} \\)? (No calculators.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 5
} | 6 | 0 | 0 |
omr-ws-ece3ef9db70fb67b9c614a87417c2b5ee7bd17f808a66bbcbb5217f5b4abb07e | Which is larger, \( \pi^{\sqrt{e}} \) or \( e^{\sqrt{\pi}} \)? | [
"\\( e^{\\sqrt{\\pi}} \\)"
] | [
"Which is larger, \\( \\pi^{\\sqrt{e}} \\) or \\( e^{\\sqrt{\\pi}} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f3bbd6d5b772f05252efddcc3a241ad063317bdb3acda3839aac94e4b1d0154 | Which is larger, \( e^{\pi^{\pi^e}} \) or \( \pi^{e^{e^\pi}} \)? | [
"e^{\\pi^{\\pi^e}}"
] | [
"Which is larger, \\( e^{\\pi^{\\pi^e}} \\) or \\( \\pi^{e^{e^\\pi}} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c5b2f559040636c3eceb2460a73c1bb79fb0db4a9722aef511715d810814a2ea | Which is larger, \( e^{\pi} \) or 21? | [
"e^{\\pi}"
] | [
"Which is larger, \\( e^{\\pi} \\) or 21?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4baa28c55cf9e8d7558cb79a396c0861aafd2931ada18ed25ba212451d56c0c | Which is larger, \(1000^{1000}\) or \(1001^{999}\)? | [
"\\(1000^{1000}\\) is larger."
] | [
"Which is larger, \\(1000^{1000}\\) or \\(1001^{999}\\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a81df6b74c8d9c42090cd66c6e7c69543e7b62882d5c73b7cdbb7a37b50fa516 | Which is larger, \(2^{3^{100}}\) or \(3^{2^{150}}\)? | [
"\\(2^{3^{100}}\\)"
] | [
"Which is larger, \\(2^{3^{100}}\\) or \\(3^{2^{150}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 12
} | 29 | 0 | 0 |
omr-ws-68384e7f6450bb4cc3f8cd2bc6d93f32b65c6ef6dfd06f3caf4bbfff309fe6b9 | Which is larger, \(\int_{0}^{1}(1-x^{n})^{\frac{1}{m}}dx\) or \(\int_{0}^{1}(1-x^{m})^{\frac{1}{n}}dx\), where \(m\) and \(n\) are positive integers? | [
"\\text{They are equal}"
] | [
"Which is larger, \\(\\int_{0}^{1}(1-x^{n})^{\\frac{1}{m}}dx\\) or \\(\\int_{0}^{1}(1-x^{m})^{\\frac{1}{n}}dx\\), where \\(m\\) and \\(n\\) are positive integers?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-eabd3213a6ef8dc1b918ed7c7e19bcc5a3493b6dab493a3aa98c0369b0583ce2 | Which is larger, \[ \frac{\int_{0}^{\frac{\pi}{2}} x^{2014}\sin ^{2014}x\ dx}{\int_{0}^{\frac{\pi}{2}} x^{2013}\sin ^{2013}x\ dx}\ \text{or}\ \frac{\int_{0}^{\frac{\pi}{2}} x^{2011}\sin ^{2011}x\ dx}{\int_{0}^{\frac{\pi}{2}} x^{2012}\sin ^{2012}x\ dx}\ ?\] | [
"\\(\\frac{\\int_{0}^{\\frac{\\pi}{2}} x^{2014}\\sin ^{2014}x\\ dx}{\\int_{0}^{\\frac{\\pi}{2}} x^{2013}\\sin ^{2013}x\\ dx}\\)"
] | [
"Which is larger,\n\\[\n\\frac{\\int_{0}^{\\frac{\\pi}{2}} x^{2014}\\sin ^{2014}x\\ dx}{\\int_{0}^{\\frac{\\pi}{2}} x^{2013}\\sin ^{2013}x\\ dx}\\ \\text{or}\\ \\frac{\\int_{0}^{\\frac{\\pi}{2}} x^{2011}\\sin ^{2011}x\\ dx}{\\int_{0}^{\\frac{\\pi}{2}} x^{2012}\\sin ^{2012}x\\ dx}\\ ?\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-5861b64cd9547d6cf0206c61747414c13b8f965b7e22e7a044eabffda021be83 | Which is larger: $(((5!)!)!)!$ or $5^{5^{5^5}}$? | [
"$(((5!)!)!)!$ is larger."
] | [
"Which is larger: $(((5!)!)!)!$ or $5^{5^{5^5}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 41,
"genselect": 5
} | 46 | 0 | 0 |
omr-ws-fc77db533d4cfa20fafcca4ac9bbb85344e6f791e2a55a3342688509421a589c | Which is larger: $(\sqrt{2}^{\sqrt{3}})^{\sqrt{3}}$ or $(\sqrt{3}^{\sqrt{2}})^{\sqrt{2}}$? | [
"\\((\\sqrt{3}^{\\sqrt{2}})^{\\sqrt{2}}\\) is larger."
] | [
"Which is larger: $(\\sqrt{2}^{\\sqrt{3}})^{\\sqrt{3}}$ or $(\\sqrt{3}^{\\sqrt{2}})^{\\sqrt{2}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-197d6f95a0e89e3175893ddca3d8efee0227f79a556dc4c2f9f9ec74c60d36c8 | Which is larger: $1000^{1000}$ or $1001^{999}$? | [
"1000^{1000}"
] | [
"Which is larger: $1000^{1000}$ or $1001^{999}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-64b76e703e1067bbe19bb46dbc7afa1eb3c68ef66ac7490c2f602458e297486b | Which is larger: $100^{300}$ or $300!$? | [
"300!"
] | [
"Which is larger: $100^{300}$ or $300!$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3f1e075f9b85586eb7438276c54ab3767e2762b8cbb5518cd4a9983e7cc817e | Which is larger: $99^n + 100^n$ or $101^n$ for a natural number $n$? | [
"101^n"
] | [
"Which is larger: $99^n + 100^n$ or $101^n$ for a natural number $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-58a0493034e92b0aa8c585ae27ed5ed9d89da59e11f332d8d5128baf1bfff17e | Which is larger: $\sqrt{5/12} + \sqrt{1/5}$ or $\sqrt{1/3} + \sqrt{2/7}$? | [
"\\sqrt{\\frac{1}{3}} + \\sqrt{\\frac{2}{7}}"
] | [
"Which is larger: $\\sqrt{5/12} + \\sqrt{1/5}$ or $\\sqrt{1/3} + \\sqrt{2/7}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6add063c4a932fbdd814f1744f5ce8ec66d6c518a54f04fc30bcdbaa5cb4bae8 | Which is larger: $\sqrt{999999} + \sqrt{1000001}$ or $2000$? | [
"2000"
] | [
"Which is larger: $\\sqrt{999999} + \\sqrt{1000001}$ or $2000$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2fe1f18c1540f763b7ba319e8025b5b566b02a4384800e25cbc990fc31260860 | Which is larger: (a) $\sin 1^\circ$ or $\sin 1$, (b) $\tan 1$ or $\tan 2$? | [
"(a) $\\sin 1^\\circ<\\sin 1$, (b) $\\tan 1>\\tan 2$"
] | [
"Which is larger: (a) $\\sin 1^\\circ$ or $\\sin 1$, (b) $\\tan 1$ or $\\tan 2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75afdb97fe100153f3e7937b98a4496c3be7ce19ab2193e0b737a41b5289339b | Which is larger: I. \(\left(\frac{1}{3}\right)^{1000} + \left(\frac{2}{3}\right)^{1000}\) or II. \(\left(\frac{1}{\sqrt{2}}\right)^{1000} + \left(1 - \frac{1}{\sqrt{2}}\right)^{1000}\)? | [
"I"
] | [
"Which is larger: \nI. \\(\\left(\\frac{1}{3}\\right)^{1000} + \\left(\\frac{2}{3}\\right)^{1000}\\)\nor \nII. \\(\\left(\\frac{1}{\\sqrt{2}}\\right)^{1000} + \\left(1 - \\frac{1}{\\sqrt{2}}\\right)^{1000}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d3947686add739c17b8f5b25c4324b2a8c4a9ab1d06d7a5e9a621963522ca9e | Which is larger: \( \frac{2009}{2008} \) or \( \sqrt[2008]{2008} \)? | [
"\\( \\sqrt[2008]{2008} \\)"
] | [
"Which is larger: \\( \\frac{2009}{2008} \\) or \\( \\sqrt[2008]{2008} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-39b74d935b172690b620bce2101d764048588d5468bd4f187c6a0dbeacc0004b | Which is larger: \( a = 1 + 2^2 + 3^3 + \cdots + 2020^{2020} \) or \( b = 2^{2016!} \)? | [
"\\( b \\)"
] | [
"Which is larger: \\( a = 1 + 2^2 + 3^3 + \\cdots + 2020^{2020} \\) or \\( b = 2^{2016!} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-41321e2f4b63e9ea3ac53cf8c4af25792a1bdda1f4908f057afa25dd150b6098 | Which is larger: \( n^n \) or the product of the first \( n \) odd numbers, i.e., \( (2n-1)(2n-3)(2n-5) \cdots 3 \cdot 1 \)? | [
"\\( n^n \\) is larger."
] | [
"Which is larger: \\( n^n \\) or the product of the first \\( n \\) odd numbers, i.e., \\( (2n-1)(2n-3)(2n-5) \\cdots 3 \\cdot 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-57772decd151e779df837152b414aa6954ba2c8676b8d61c6012dd16eeef558b | Which is larger: \( n^{n+1} \) or \( (n+1)^n \) for a positive integer \( n \)? | [
"n^{n+1} \\text{ is larger when } n \\geq 3 \\text{ and } (n+1)^n \\text{ is larger when } n = 1 \\text{ or } 2"
] | [
"Which is larger: \\( n^{n+1} \\) or \\( (n+1)^n \\) for a positive integer \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-add034e06f94fe07b8f9599bd7ee6e50142921d7e033eaf377bdf7acd513fcef | Which is larger: \(((2)^{1/3} - 1)^{1/3}\) or \((1/9)^{1/3} - (2/9)^{1/3} + (4/9)^{1/3}\)? | [
"="
] | [
"Which is larger: \\(((2)^{1/3} - 1)^{1/3}\\) or \\((1/9)^{1/3} - (2/9)^{1/3} + (4/9)^{1/3}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-cf2b91fae0750b50fc211e437de293861a39ba9e1a9846e20ccb77eb03270b37 | Which is larger: \(2^{2^{2^{\cdot^{\cdot^{2}}}}}\) (with 10 twos in the exponentiation tower) or \(3^{3^{3^{\cdot^{\cdot^{3}}}}}\) (with 8 threes in the exponentiation tower)? | [
"\\(2^{2^{2^{\\cdot^{\\cdot^{2}}}}}\\) (with 10 twos) is larger."
] | [
"Which is larger: \\(2^{2^{2^{\\cdot^{\\cdot^{2}}}}}\\) (with 10 twos in the exponentiation tower) or \\(3^{3^{3^{\\cdot^{\\cdot^{3}}}}}\\) (with 8 threes in the exponentiation tower)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 4
} | 9 | 0 | 0 |
omr-ws-703d73dc8d2119ba6c2478c1e1f88c198dfbb03fb4afe963fb3c2d27813668dd | Which is larger: \(\frac{10^{1999} + 1}{10^{2000} + 1}\) or \(\frac{10^{1998} + 1}{10^{1999} + 1}\)? | [
"\\(\\frac{10^{1998} + 1}{10^{1999} + 1}\\) is larger."
] | [
"Which is larger: \\(\\frac{10^{1999} + 1}{10^{2000} + 1}\\) or \\(\\frac{10^{1998} + 1}{10^{1999} + 1}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b9cb3a9a553c16774976432d34ec8ad991e384b405c4e55038186ad1de2738e4 | Which is larger: \(\sqrt{83} + \sqrt{85}\) or \(\frac{\sqrt{83 \cdot 85}}{\sqrt{84}} + \sqrt{84}\)? | [
"\\(\\sqrt{83} + \\sqrt{85}\\) is larger."
] | [
"Which is larger: \\(\\sqrt{83} + \\sqrt{85}\\) or \\(\\frac{\\sqrt{83 \\cdot 85}}{\\sqrt{84}} + \\sqrt{84}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-3a1d13c1560b3e11c26ed0d26220ed5186d933086b4db867540ad5c792cc5c73 | Which is larger: \(\sqrt{n+1}^{\sqrt{n}}\) or \(\sqrt{n}^{\sqrt{n+1}}\) for \(n > 8\)? | [
"\\(\\sqrt{n}^{\\sqrt{n+1}}\\) is larger."
] | [
"Which is larger: \\(\\sqrt{n+1}^{\\sqrt{n}}\\) or \\(\\sqrt{n}^{\\sqrt{n+1}}\\) for \\(n > 8\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-61e72d416ee8ce46ccab29a94573948cb343fbda9e7c275cf86d1bce6f717631 | Which is larger: \[ \frac{x}{\sqrt{y+z}} + \frac{y}{\sqrt{x+z}} + \frac{z}{\sqrt{x+y}} \] or \[ \frac{x}{\sqrt{x+y}} + \frac{y}{\sqrt{y+z}} + \frac{z}{\sqrt{z+x}}? \] | [
"\\(\\frac{x}{\\sqrt{y+z}} + \\frac{y}{\\sqrt{x+z}} + \\frac{z}{\\sqrt{x+y}}\\)"
] | [
"Which is larger: \n\\[\n\\frac{x}{\\sqrt{y+z}} + \\frac{y}{\\sqrt{x+z}} + \\frac{z}{\\sqrt{x+y}}\n\\]\nor \n\\[\n\\frac{x}{\\sqrt{x+y}} + \\frac{y}{\\sqrt{y+z}} + \\frac{z}{\\sqrt{z+x}}?\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-5fa4f0e801899505440b62b9252cec2428025197b06b5efc7dca89c4c787fe2e | Which is larger: the number of positive integers not exceeding 1,000,000 that can be represented by the form $2x^2 - 3y^2$ with integral $x$ and $y$, or the number of positive integers not exceeding 1,000,000 that can be represented by the form $10xy - x^2 - y^2$ with integral $x$ and $y$? | [
"The number of positive integers not exceeding 1,000,000 that can be represented by \\(2x^2 - 3y^2\\) is larger."
] | [
"Which is larger: the number of positive integers not exceeding 1,000,000 that can be represented by the form $2x^2 - 3y^2$ with integral $x$ and $y$, or the number of positive integers not exceeding 1,000,000 that can be represented by the form $10xy - x^2 - y^2$ with integral $x$ and $y$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 33,
"genselect": 7,
"tir": 18
} | 58 | 0 | 0 |
omr-ws-6a36d3ddd81a31f273acb6f930be1755fc929058b506518d15d9222d1456e443 | Which is larger: the sum $1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2016}$ or the number $10$? (Do not use a calculator.) | [
"10"
] | [
"Which is larger: the sum $1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{2016}$ or the number $10$? (Do not use a calculator.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-abc6343be26b1f2beb3ffc1ec47a0ec1b5254cacae93df368ff9524ceccc4ca0 | Which is larger? \[ \frac{123456 + 10^{999}}{123457 + 10^{999}} \quad \text{or} \quad \frac{123457 + 10^{999}}{123458 + 10^{999}} \] | [
"\\(\\frac{123457 + 10^{999}}{123458 + 10^{999}}\\) is larger."
] | [
"Which is larger?\n\\[\n\\frac{123456 + 10^{999}}{123457 + 10^{999}} \\quad \\text{or} \\quad \\frac{123457 + 10^{999}}{123458 + 10^{999}}\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-8cf25bfe77021764d61559cce92aedce98ad8fec27d65ccd1729c18e402ae88f | Which is larger? \[ \frac{1}{\sqrt{83}} + \frac{\sqrt{83}}{\sqrt{81 \times 84}} \] or \[ \frac{1}{\sqrt{82}} + \frac{\sqrt{82}}{\sqrt{81 \times 84}} \] | [
"\\(\\frac{1}{\\sqrt{83}} + \\frac{\\sqrt{83}}{\\sqrt{81 \\times 84}}\\) is larger."
] | [
"Which is larger?\n\\[\n\\frac{1}{\\sqrt{83}} + \\frac{\\sqrt{83}}{\\sqrt{81 \\times 84}}\n\\]\nor\n\\[\n\\frac{1}{\\sqrt{82}} + \\frac{\\sqrt{82}}{\\sqrt{81 \\times 84}}\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-66880afcedbef862d58bd23022b7f89ca3d53c34daf442b76aeafbb69cbecffc | Which is more likely: rolling a pair of dice 100 times and getting a double six at least once or rolling four dice 3600 times and getting a quadruple six at least once? | [
"Rolling a pair of dice 100 times and getting a double six at least once"
] | [
"Which is more likely: rolling a pair of dice 100 times and getting a double six at least once or rolling four dice 3600 times and getting a quadruple six at least once?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-29ffb2bad78a86f05456ac4a261a00863b27724d9c228d80975572932de5e205 | Which is the $10^6$th number that is prime and the sum of its digits is also prime? | [
"179424691"
] | [
"Which is the $10^6$th number that is prime and the sum of its digits is also prime?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-711e7842c395f87d3024871cdbc02535c25f47fb81c0ddb2ad604c404d1c6882 | Which is the greatest term in the expansion of $\left(\frac{9}{10}+\frac{1}{10}\right)^{100}$? | [
"The tenth term is the greatest term of the expansion"
] | [
"Which is the greatest term in the expansion of $\\left(\\frac{9}{10}+\\frac{1}{10}\\right)^{100}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eed406e1517a470fe89c57696d93aa5ca0cab96e601d864f3b857aba02a7aa1a | Which is the largest 3-digit palindrome number such that its product with 111 is also a palindrome number? | [
"414"
] | [
"Which is the largest 3-digit palindrome number such that its product with 111 is also a palindrome number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4e6faabc475af2a82a8424338fa28fe80edd26ca66f00f49279c084bbda46429 | Which is the largest factor of \(22451\)? | [
"22451"
] | [
"Which is the largest factor of \\(22451\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 6,
"tir": 8
} | 20 | 0 | 0 |
omr-ws-33914d290b37fb14bc90afc7a29928c0b11f36d05c20fba2a341135f3a01c7e4 | Which is the largest number in the sequence \(1, 2^{1/3}, 3^{1/4}, \ldots, n^{1/(n+1)}, \ldots\)? | [
"\\(4^{1/5}\\)"
] | [
"Which is the largest number in the sequence \\(1, 2^{1/3}, 3^{1/4}, \\ldots, n^{1/(n+1)}, \\ldots\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 27
} | 44 | 0 | 0 |
omr-ws-b37ab1bd9c55e140c03079d96f30f6dc96b259dd64e24dc294328058f97fa992 | Which is the largest positive integer \( n \) satisfying the inequality \( n^{2007} > (2007)^n \)? | [
"2006"
] | [
"Which is the largest positive integer \\( n \\) satisfying the inequality \\( n^{2007} > (2007)^n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-7f974c60c9961c97daa49d35536fff462012fa8270cc8c2d13dbfce923321e45 | Which is the least positive integer $m$ such that for every system $P = \{p_1, p_2, \ldots, p_m\}$ of integers, there exists a $4 \times 4$ matrix $A$ with all elements in $P$ and $\det(A)$ divisible by 4? | [
"1"
] | [
"Which is the least positive integer $m$ such that for every system $P = \\{p_1, p_2, \\ldots, p_m\\}$ of integers, there exists a $4 \\times 4$ matrix $A$ with all elements in $P$ and $\\det(A)$ divisible by 4?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-b8ef46640b2e3453e66906cfde9bc4a5fe7ccc3a30e2e864c6aeb1455ef993f1 | Which is the least positive integer \( m \) such that for every system \( \mathcal{P} := \{p_1, p_2, \ldots, p_m\} \subset \mathbb{Z} \) of integers, there exists at least one \( n \)-"nice" matrix having all elements in \( \mathcal{P} \)? An \( n \)-"nice" matrix \( M \) is defined as an \( n \times n \) matrix with a... | [
"16"
] | [
"Which is the least positive integer \\( m \\) such that for every system \\( \\mathcal{P} := \\{p_1, p_2, \\ldots, p_m\\} \\subset \\mathbb{Z} \\) of integers, there exists at least one \\( n \\)-\"nice\" matrix having all elements in \\( \\mathcal{P} \\)? An \\( n \\)-\"nice\" matrix \\( M \\) is defined as an \\... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ea1bda211ccb08a19f65c16466bb6a30d3a3f7f409f7c616f1c3623a8ab4850 | Which is the only digit that does not occur in the decimal expansion of $2^{29}$? | [
"4"
] | [
"Which is the only digit that does not occur in the decimal expansion of $2^{29}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9cdebfac42cff3b1b5ce46e1ed597ee277acd9ea1ee608dfe2309ea100b40c82 | Which is the smallest positive integer that isn't the difference of two "isoscelable" integers, where a positive integer $n \geq 3$ is called "isoscelable" if a regular polygon of $n$ sides can be divided with $n-3$ diagonals into $n-2$ isosceles triangles? | [
"4"
] | [
"Which is the smallest positive integer that isn't the difference of two \"isoscelable\" integers, where a positive integer $n \\geq 3$ is called \"isoscelable\" if a regular polygon of $n$ sides can be divided with $n-3$ diagonals into $n-2$ isosceles triangles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-a97faeec38213e934a9e544ad083ca17943312aa3d23f51c459deb078ae11afe | Which is the smallest side length of a square that can be formed using 11 square tiles with consecutive integer side lengths, without any spaces or overlaps? | [
"77"
] | [
"Which is the smallest side length of a square that can be formed using 11 square tiles with consecutive integer side lengths, without any spaces or overlaps?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 12
} | 35 | 0 | 0 |
omr-ws-23e501c27a03a1791b377340b69d95a5a523a7e0b79b5e99f12b97ef3566d03d | Which map has the larger scale, given that Map A has a scale of 1cm: 80m and Map B has a scale of 5mm: 110m? (Where 1/10 is a larger scale than 1/20) | [
"Map A has the larger scale."
] | [
"Which map has the larger scale, given that Map A has a scale of 1cm: 80m and Map B has a scale of 5mm: 110m? (Where 1/10 is a larger scale than 1/20)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-58b99675a839c66d0a189ce3da8f97754d4d498f2f834783cb6defcbd5345b35 | Which map shows the most detail of the woodlot, given that Map A has a scale of 1cm: 80m and Map B has a scale of 5mm: 110m? | [
"Map A shows the most detail of the woodlot."
] | [
"Which map shows the most detail of the woodlot, given that Map A has a scale of 1cm: 80m and Map B has a scale of 5mm: 110m?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-41793fbdda3c5f4837e8bba389910605a1f6c545e770810848a64ac0c125ad27 | Which matrices $M \in M_{n}(R)$ are both orthogonal and doubly stochastic? | [
"Permutation matrices"
] | [
"Which matrices $M \\in M_{n}(R)$ are both orthogonal and doubly stochastic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-b81c8c53e9231545b1ec7f8bffc06e6ab7f4492f484f0a181d240c10167896e7 | Which matrices in $M_n(\mathbb{C})$ can be represented as a product of matrices of finite multiplicative order in $M_n(\mathbb{C})$? | [
"\\text{All invertible matrices with determinant a root of unity}"
] | [
"Which matrices in $M_n(\\mathbb{C})$ can be represented as a product of matrices of finite multiplicative order in $M_n(\\mathbb{C})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-8c1d628d916bea7df602380bbe5783ee3a8816a8385ae350ef70c8e5a777ced2 | Which matrices in $M_n(\mathbb{Q})$ can be represented as a product of matrices of finite multiplicative order in $M_n(\mathbb{Q})$? | [
"Matrices in \\( M_n(\\mathbb{Q}) \\) with determinant \\( \\pm 1 \\)."
] | [
"Which matrices in $M_n(\\mathbb{Q})$ can be represented as a product of matrices of finite multiplicative order in $M_n(\\mathbb{Q})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-1b2551a81eafa10c56cd1b9b31d22ed9cb0bddb5b793c81740f25262b2278cfb | Which matrices in $M_n(\mathbb{Z})$ can be represented as a product of matrices of finite multiplicative order in $M_n(\mathbb{Z})$? | [
"Matrices in \\( M_n(\\mathbb{Z}) \\) with determinant \\( \\pm 1 \\)."
] | [
"Which matrices in $M_n(\\mathbb{Z})$ can be represented as a product of matrices of finite multiplicative order in $M_n(\\mathbb{Z})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 3
} | 22 | 0 | 0 |
omr-ws-5fc244b4ae5639b92d90ade3bbdf62ecefea40c5a4351009f2e5299f3f7f43e3 | Which matrix below would produce an anticlockwise rotation of \( \Theta^\circ \) about the origin, followed by a reflection in the x-axis, and finally, a clockwise rotation of \( \Theta^\circ \) about the origin? | [
"\\(\\begin{bmatrix} \\cos 2\\Theta & \\sin 2\\Theta \\\\ \\sin 2\\Theta & -\\cos 2\\Theta \\end{bmatrix}\\)"
] | [
"Which matrix below would produce an anticlockwise rotation of \\( \\Theta^\\circ \\) about the origin, followed by a reflection in the x-axis, and finally, a clockwise rotation of \\( \\Theta^\\circ \\) about the origin?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b572313b7e3d15906723c84b9a9c7c0cea37bbd4c788df4e058101946e946a4f | Which members of the rational numbers are contained in the binomial expansion of $(\sqrt{3}+\sqrt[3]{2})^{120}$? | [
"k \\equiv 0 \\pmod{6}"
] | [
"Which members of the rational numbers are contained in the binomial expansion of $(\\sqrt{3}+\\sqrt[3]{2})^{120}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-2db3f9a99db78a630de06c00a37d8dbdae1ba1b8d197c00d6f2302cf7c08a77c | Which natural numbers $t$ can be represented as \[t = \sum_{k=1}^n x_k\] where $x_k \in \mathbb{N}_0$, with $n = 4 \cdot z$ for some $z \in \mathbb{N}$, and $x_i \cdot x_{i+1} = x_{i+2} \cdot x_{i+3}$ for $i = 4 \cdot q + 1$ with $q \in \mathbb{N}_0$ and $i = 1, \ldots, n-3$? | [
"\\mathbb{N}"
] | [
"Which natural numbers $t$ can be represented as \\[t = \\sum_{k=1}^n x_k\\] where $x_k \\in \\mathbb{N}_0$, with $n = 4 \\cdot z$ for some $z \\in \\mathbb{N}$, and $x_i \\cdot x_{i+1} = x_{i+2} \\cdot x_{i+3}$ for $i = 4 \\cdot q + 1$ with $q \\in \\mathbb{N}_0$ and $i = 1, \\ldots, n-3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c290c269c76a0a9455f23d48a4a2e6e8e96fe0a73d8d0f20dcc5a618681d118 | Which natural numbers can be expressed as the difference of squares of two integers? | [
"all odd numbers and multiples of 4"
] | [
"Which natural numbers can be expressed as the difference of squares of two integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-cb9010621700ef752c3388366a7400dc6de9072bf6303d7aff629e51d4d2b2b3 | Which natural numbers cannot be expressed in the form $[n + \sqrt{n} + \frac{1}{2}]$, where $n \in \mathbb{N}$ and $[y]$ denotes the greatest integer function? | [
"Perfect squares"
] | [
"Which natural numbers cannot be expressed in the form $[n + \\sqrt{n} + \\frac{1}{2}]$, where $n \\in \\mathbb{N}$ and $[y]$ denotes the greatest integer function?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 28
} | 50 | 0 | 0 |
omr-ws-b1285fb02aebc12a2eb49ab38c614489cd5b1dd75f099c6a9c057e4ba32e1732 | Which natural numbers less than \( P \) are coprime to \( P \)? Provide a formula. | [
"\\(\\phi(P) = P \\times \\prod_{x=1}^k \\left(1 - \\frac{1}{p_x}\\right)\\)"
] | [
"Which natural numbers less than \\( P \\) are coprime to \\( P \\)? Provide a formula."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-28fd5429177b3647d8114a06a19e28bfe999e67e32e48b6e1529b2bf9fd424c5 | Which natural numbers less than \( P^2 \) are coprime with \( P \)? Provide a formula and an easy method. | [
"P \\cdot \\phi(P)"
] | [
"Which natural numbers less than \\( P^2 \\) are coprime with \\( P \\)? Provide a formula and an easy method."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-227c591dfca1fc44adbcba48b1894837955561f92369b53c45e89980c1a0ff2a | Which number follows logically in the sequence: 4, 9, 50, 147, 605, 1352, 3757, ...? | [
"7581"
] | [
"Which number follows logically in the sequence: 4, 9, 50, 147, 605, 1352, 3757, ...?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 12
} | 32 | 0 | 0 |
omr-ws-6e1bf1bf5a650747369ffdb4ece2a03056b39493b7b220308736a0c871ddb295 | Which number is bigger, $\sqrt[2012]{2012!}$ or $\sqrt[2013]{2013!}$? | [
"$\\sqrt[2013]{2013!}$ is bigger."
] | [
"Which number is bigger, $\\sqrt[2012]{2012!}$ or $\\sqrt[2013]{2013!}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4aa45d363c1be567c4a07017b9323b4458cf77bfe21659aba82ed6c3f750abca | Which number is bigger, \( M_{2004} = 2^{4^{2^{.^{.^{.^{2^{4}}}}}}} \) or \( N_{2004} = 4^{2^{4^{.^{.^{.^{4^{2}}}}}}} \)? Each exponent has 2004 numbers. | [
"\\( M_{2004} > N_{2004} \\)"
] | [
"Which number is bigger, \\( M_{2004} = 2^{4^{2^{.^{.^{.^{2^{4}}}}}}} \\) or \\( N_{2004} = 4^{2^{4^{.^{.^{.^{4^{2}}}}}}} \\)? Each exponent has 2004 numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 6
} | 26 | 0 | 0 |
omr-ws-4daa58dc1c2ad2f7956f3b62a3dcfeb8699d8bec4e8e4ed60e03ccfe5127aa91 | Which number is bigger: $(1.01)^{1000}$ or $1000$? | [
"\\((1.01)^{1000}\\) is bigger."
] | [
"Which number is bigger: $(1.01)^{1000}$ or $1000$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-d5065165c0e1cf78ba5673646a2c284aa05cf1a31639a7320b3e39c57a597aa9 | Which number is bigger: \(3^{3^{3^{\cdots^3}}}\) (with 2005 threes) or \(2^{2^{2^{\cdots^{2^6}}}}\) (with 2004 twos and one six)? | [
"\\(2^{2^{2^{\\cdots^{2^6}}}}\\) (with 2004 twos and one six)"
] | [
"Which number is bigger: \\(3^{3^{3^{\\cdots^3}}}\\) (with 2005 threes) or \\(2^{2^{2^{\\cdots^{2^6}}}}\\) (with 2004 twos and one six)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 13
} | 61 | 0 | 0 |
omr-ws-32b527f13533e963b08c86abcb0329081fff2c3f0122b1529d48ce9db6d8f03e | Which number is greater, $\frac{1}{20}$ or $\frac{1}{2} \cdot \frac{4}{5} \cdot \frac{7}{8} \cdot \ldots \cdot \frac{7999}{8000}$? | [
"\\(\\frac{1}{20}\\) is greater."
] | [
"Which number is greater, $\\frac{1}{20}$ or $\\frac{1}{2} \\cdot \\frac{4}{5} \\cdot \\frac{7}{8} \\cdot \\ldots \\cdot \\frac{7999}{8000}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-ff1108eb8f6c2ff7f3f731dd1f5b4d10ce57cf8571b3aa6d4d44388dd7ef0446 | Which number is greater: $$\underbrace{888...88}_{19 \, digits} \cdot \underbrace{333...33}_{68 \, digits} \,\,\, \text{or} \,\,\, \underbrace{444...44}_{19 \, digits} \cdot \underbrace{666...67}_{68 \, digits} \, ?$$ | [
"B"
] | [
"Which number is greater: $$\\underbrace{888...88}_{19 \\, digits} \\cdot \\underbrace{333...33}_{68 \\, digits} \\,\\,\\, \\text{or} \\,\\,\\, \\underbrace{444...44}_{19 \\, digits} \\cdot \\underbrace{666...67}_{68 \\, digits} \\, ?$$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-1d080b651c20044683bdf7548998bb0627f04fa20c039d3a9db26c40d12e297a | Which number is greater: \( 4^{4^{4^{4^4}}} \) or \( 5^{5^{5^5}} \)? | [
"\\( 4^{4^{4^{4^4}}} \\) is greater."
] | [
"Which number is greater: \\( 4^{4^{4^{4^4}}} \\) or \\( 5^{5^{5^5}} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 13
} | 34 | 0 | 0 |
omr-ws-5030bdadafbc65758b96186de07178413a488253b22e4ab68b61604e1602ca63 | Which number is greater: \( \sqrt{2003} + \sqrt{2005} \) or \( 2\sqrt{2004} \)? | [
"\\(2\\sqrt{2004}\\) is greater."
] | [
"Which number is greater: \\( \\sqrt{2003} + \\sqrt{2005} \\) or \\( 2\\sqrt{2004} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-91958e39ccb96baa373058e258183e1581dff0c24624acb6b3503190f28c8ad8 | Which number is greater: \(4^{4^{4^{4^{4}}}}\) or \(5^{5^{5^{5}}}\)? | [
"\\(4^{4^{4^{4^{4}}}}\\) is greater."
] | [
"Which number is greater: \\(4^{4^{4^{4^{4}}}}\\) or \\(5^{5^{5^{5}}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 12
} | 35 | 0 | 0 |
omr-ws-be9730b4008ffce9439335e9103d60191bec06150d40cb13fa07e0f27dec0b85 | Which number is greater: \(888\ldots888 \times 3333\ldots333\) or \(444\ldots444 \times 6666\ldots667\)? (Each number has 1989 digits.) | [
"\\(444\\ldots444 \\times 6666\\ldots667\\) is greater."
] | [
"Which number is greater: \\(888\\ldots888 \\times 3333\\ldots333\\) or \\(444\\ldots444 \\times 6666\\ldots667\\)? (Each number has 1989 digits.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-5a1b931e07b19f4ab9c4c474177b7f5a68697f3567fefb975a669c2322581c8e | Which number is greater: \[ A = \frac{2.00\ldots04}{1.00\ldots04^2 + 2.00\ldots04} \quad \text{or} \quad B = \frac{2.00\ldots02}{1.00\ldots02^2 + 2.00\ldots02}, \] where each of the numbers contains 1998 zeros? | [
"B"
] | [
"Which number is greater:\n\\[ A = \\frac{2.00\\ldots04}{1.00\\ldots04^2 + 2.00\\ldots04} \\quad \\text{or} \\quad B = \\frac{2.00\\ldots02}{1.00\\ldots02^2 + 2.00\\ldots02}, \\]\nwhere each of the numbers contains 1998 zeros?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3160c4087912e1449b8c9de9bb7515ca83569cd88d2c7a0ef6a5fd11b4794397 | Which number is greater? $$\frac{\underbrace{1000\ldots001}_{1965\, zeroes}}{\underbrace{1000\ldots001}_{1966\, zeroes}} \quad \text{or} \quad \frac{\underbrace{1000\ldots001}_{1966\, zeroes}}{\underbrace{1000\ldots001}_{1967\, zeroes}}$$ | [
"\\(\\frac{\\underbrace{1000\\ldots001}_{1966\\, zeroes}}{\\underbrace{1000\\ldots001}_{1967\\, zeroes}}\\)"
] | [
"Which number is greater?\n$$\\frac{\\underbrace{1000\\ldots001}_{1965\\, zeroes}}{\\underbrace{1000\\ldots001}_{1966\\, zeroes}} \\quad \\text{or} \\quad \\frac{\\underbrace{1000\\ldots001}_{1966\\, zeroes}}{\\underbrace{1000\\ldots001}_{1967\\, zeroes}}$$"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11e8a4aad20ab21b7754e5afdc04cdc1d7bfd4fff7406e1c65721c17d2ff9b6e | Which number is larger: \( A = \frac{1}{9} \div \sqrt[3]{\frac{1}{2023}} \) or \( B = \log_{2023} 91125 \)? | [
"\\( B \\)"
] | [
"Which number is larger: \\( A = \\frac{1}{9} \\div \\sqrt[3]{\\frac{1}{2023}} \\) or \\( B = \\log_{2023} 91125 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ba150bcba495ca28fc76fe815c1c184739a900bb5cfaa98b75aa25909baa0d23 | Which number is larger: \( \left(1 - \frac{1}{365}\right)\left(1 - \frac{2}{365}\right) \cdots \left(1 - \frac{25}{365}\right) \) or \( \frac{1}{2} \)? | [
"\\(\\frac{1}{2}\\)"
] | [
"Which number is larger: \\( \\left(1 - \\frac{1}{365}\\right)\\left(1 - \\frac{2}{365}\\right) \\cdots \\left(1 - \\frac{25}{365}\\right) \\) or \\( \\frac{1}{2} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-81dcdd50cc098386bcd5e3d6e28fdadc1b97ab2a41cc084e090359d30aca4ada | Which numbers $n$ have the property that for all integers $x$, $x^2 \equiv 0, 1, \text{ or } -1 \pmod{n}$? | [
"\\( n = 1, 2, 3, 4, 5 \\)"
] | [
"Which numbers $n$ have the property that for all integers $x$, $x^2 \\equiv 0, 1, \\text{ or } -1 \\pmod{n}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 24
} | 26 | 0 | 0 |
omr-ws-ca6a0155d9d82f794d4824dc324d8ee082e71349a6c2aa19435f873838da9c53 | Which numbers between 1 and 100 have an equal number of even and odd natural divisors? | [
"\\( n \\equiv 2 \\pmod{4} \\)"
] | [
"Which numbers between 1 and 100 have an equal number of even and odd natural divisors?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-89efb881ed6dbe0a5f80600906e5ec19c1e8e614be75f20c45bf39bb2b231a5d | Which numbers can be written in the form $a^b + b$ where $a$ and $b$ are positive integers and $a, b > 1$? | [
"\\text{All integers greater than or equal to }6"
] | [
"Which numbers can be written in the form $a^b + b$ where $a$ and $b$ are positive integers and $a, b > 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-68efbd5a4cd7872be22164270a52140d3351d09857c98dbece992408a04b5775 | Which numbers do not have a square in \(\mathbb{Z}/6\)? | [
"2, 5"
] | [
"Which numbers do not have a square in \\(\\mathbb{Z}/6\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 27
} | 33 | 0 | 0 |
omr-ws-a5182c5732ca1468975ed8e8bfc1b319ace611da181796f903e5b08c3795c7aa | Which numbers do not have square roots in the rings \( \mathbb{Z}/5\mathbb{Z} \) and \( \mathbb{Z}/6\mathbb{Z} \)? | [
"\\( 2, 3 \\) in \\( \\mathbb{Z}/5\\mathbb{Z} \\) and \\( 2, 5 \\) in \\( \\mathbb{Z}/6\\mathbb{Z} \\)"
] | [
"Which numbers do not have square roots in the rings \\( \\mathbb{Z}/5\\mathbb{Z} \\) and \\( \\mathbb{Z}/6\\mathbb{Z} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-34fd59b5f023c6056d8b27a8c4e10eb6ea97ec4259d1aa8d830b72e35e9ecac4 | Which numbers have a factorial that ends in exactly 1000 zeros? | [
"4000, 4001, 4002, 4003, 4004"
] | [
"Which numbers have a factorial that ends in exactly 1000 zeros?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca00cc236b2a7993130f2ac75c77507c808cf4233ec1b7cd43b170c6bfd2fdd9 | Which odd positive integers \( n < 10000 \) are more numerous: those for which the number formed by the four last digits of \( n^9 \) is greater than \( n \) or those for which it is smaller than \( n \)? | [
"\\text{They are equally numerous}"
] | [
"Which odd positive integers \\( n < 10000 \\) are more numerous: those for which the number formed by the four last digits of \\( n^9 \\) is greater than \\( n \\) or those for which it is smaller than \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-f70e3e842bb8700b744138e068760862839978c5ab3a03e3bff0729b1022b107 | Which of the differential equations $\dot{x} = x^n$ determine on an affine line a phase velocity field that can be extended without singular points to the projective line? | [
"\\(n = 0, 1, 2\\)"
] | [
"Which of the differential equations $\\dot{x} = x^n$ determine on an affine line a phase velocity field that can be extended without singular points to the projective line?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 5,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-326a4a0400ad6a3a684da13e249962e4cc8366892d7ce92a83e0a7dff4532148 | Which of the following best describes the linear transformation \( x \rightarrow Ax \), where \( A \) is the 2x2 matrix given by: \[ A = \begin{pmatrix} 5 & 18 \\ 2 & 5 \end{pmatrix} \] A: stretches in two different directions B: stretches in one direction only C: stretches and reflects in one direction only D: stretch... | [
"D: stretches in one direction and reflects in another direction"
] | [
"Which of the following best describes the linear transformation \\( x \\rightarrow Ax \\), where \\( A \\) is the 2x2 matrix given by:\n\n\\[\nA = \\begin{pmatrix}\n5 & 18 \\\\\n2 & 5\n\\end{pmatrix}\n\\]\n\nA: stretches in two different directions \nB: stretches in one direction only \nC: stretches and reflects... | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5f861e8493ede06b0d7f2cd5e371d32bddc73f8005774489116b3357b4b5473e | Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation $$x(x-y)+y(y-z)+z(z-x) = 1?$$ | [
"\\text{Two of the integers are equal and the third differs by 1}",
"\\text{Two of the integers are equal, and the third differs by 1}"
] | [
"Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation\n$$x(x-y)+y(y-z)+z(z-x) = 1?$$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-b9aa03e43dd2355b3ea62870efec1fbf66fbb0e405ab0ff2110b7cf3340f3d57 | Which of the following could be the x-intercept and y-intercept of a line that is perpendicular to the line \(3x + 6y = 0\)? | [
"(2, -4)"
] | [
"Which of the following could be the x-intercept and y-intercept of a line that is perpendicular to the line \\(3x + 6y = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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