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-41bfe7ae0683fe2a6d920f83207b271421a4ff931af3d1aa1b16ba89accabe31 | What is the remainder when \( \binom{169}{13} \) is divided by \( 13^5 \)? | [
"13"
] | [
"What is the remainder when \\( \\binom{169}{13} \\) is divided by \\( 13^5 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-f759daa8571709ef894f32250bf6ec595d18a32f437ccecd52ada6a83c63caab | What is the remainder when \( n! \) is divided by \( n+3 \), where \( n+3 \) is a prime number? | [
"The multiplicative inverse of 2 modulo \\( p \\)"
] | [
"What is the remainder when \\( n! \\) is divided by \\( n+3 \\), where \\( n+3 \\) is a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74f20b3bd96decd0542f329e259e3712b52496d6b4957cf004ff234af39718dc | What is the remainder when \( n^{36} - 1 \) is divided by 33744, where \( n \) is prime to 2, 3, 19, and 37? | [
"0"
] | [
"What is the remainder when \\( n^{36} - 1 \\) is divided by 33744, where \\( n \\) is prime to 2, 3, 19, and 37?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36468eead10944ebb7ab6a3b73517cbc9f8b6f87d3c2c32a25d1e54eb64c466f | What is the remainder when \( x^{2001} - 1 \) is divided by \( (x^2 + 1)(x^2 + x + 1) \)? | [
"\\( (x+1)(x^2+x+1) \\)"
] | [
"What is the remainder when \\( x^{2001} - 1 \\) is divided by \\( (x^2 + 1)(x^2 + x + 1) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-06eccccfda78a0f47b4667b032ecf4b6de791229b18ed90279595536c3ef5a0a | What is the remainder when \( x^{2007} + 2x^{2006} + 3x^{2005} + \cdots + 2006x^2 + 2007x + 2008 \) is divided by \( x-2 \)? | [
"2^{2009} - 2010"
] | [
"What is the remainder when \\( x^{2007} + 2x^{2006} + 3x^{2005} + \\cdots + 2006x^2 + 2007x + 2008 \\) is divided by \\( x-2 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a7264eb72bac16896e44cee84063aca6f6996a126985de779dbc59467450c4b | What is the remainder when \( x^{27} - 3x^{24} + x^{13} + 2x^7 - x + 1 \) is divided by \( x + 1 \)? | [
"-5"
] | [
"What is the remainder when \\( x^{27} - 3x^{24} + x^{13} + 2x^7 - x + 1 \\) is divided by \\( x + 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-11bafb292d52fd58d568c0b7ddf76d11a42aba74bfd5528633d7527dcb180848 | What is the remainder when \(1 + 10 + 100 + \ldots + 100,000,000,000,000,000\) is divided by 3? | [
"0"
] | [
"What is the remainder when \\(1 + 10 + 100 + \\ldots + 100,000,000,000,000,000\\) is divided by 3?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-58f261023f89ac94a25f885054fbf7ff630c749a6435ddbec6b0780bfbf2886e | What is the remainder when \(1 + 2^2 + 3^3 + \cdots + 10^{10}\) is divided by 5? | [
"2"
] | [
"What is the remainder when \\(1 + 2^2 + 3^3 + \\cdots + 10^{10}\\) is divided by 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-208463ae8a0b35dbce547827da3738e0920073291f08434149851637f0ebc8fe | What is the remainder when \(1 + 2^2 + 3^3 + \cdots + 10^{10}\) is divided by 6? | [
"5"
] | [
"What is the remainder when \\(1 + 2^2 + 3^3 + \\cdots + 10^{10}\\) is divided by 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-499b898ffe67ebfa7fe50dc380a543e3eced052490637906ba0806930738e616 | What is the remainder when \(1 + 2^2 + 3^3 + \cdots + 10^{10}\) is divided by 7? | [
"4"
] | [
"What is the remainder when \\(1 + 2^2 + 3^3 + \\cdots + 10^{10}\\) is divided by 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-00f0c31a61aa5fe97a8b45c87a574012d547f18ac233045f59d78b530c1f54c5 | What is the remainder when \(1 + 2^2 + 3^3 + \cdots + 10^{10}\) is divided by 8? | [
"5"
] | [
"What is the remainder when \\(1 + 2^2 + 3^3 + \\cdots + 10^{10}\\) is divided by 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-865cdd22fafd4c3753afcf7dc0bfdbdec692d74bc5f95ff76f280cbfb61df778 | What is the remainder when \(1 + 2^2 + 3^3 + \cdots + 10^{10}\) is divided by 9? | [
"2"
] | [
"What is the remainder when \\(1 + 2^2 + 3^3 + \\cdots + 10^{10}\\) is divided by 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f8abdb1a9416e4be5cfb2656cfbebd3661949793a90398b56f17fe77ab92921 | What is the remainder when \(1 + x^{1111} + x^{2222} + \cdots + x^{8888} + x^{9999}\) is divided by \(1 + x + x^2 + \cdots + x^7 + x^8 + x^9\)? | [
"0"
] | [
"What is the remainder when \\(1 + x^{1111} + x^{2222} + \\cdots + x^{8888} + x^{9999}\\) is divided by \\(1 + x + x^2 + \\cdots + x^7 + x^8 + x^9\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cfa9a9bb93cf63b02440397f7a739da9c3b8b9352367aecc6ad288a7e50e5363 | What is the remainder when \(1! + 2! + 3! + 4! + \cdots + 2006! + 2007!\) is divided by 7? | [
"5"
] | [
"What is the remainder when \\(1! + 2! + 3! + 4! + \\cdots + 2006! + 2007!\\) is divided by 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-c7aa21c0f0f42961c821b09f59d0760098417dd526d971a34cdb29c2a436c12f | What is the remainder when \(10!\) is divided by 13? | [
"6"
] | [
"What is the remainder when \\(10!\\) is divided by 13?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 36
} | 99 | 0 | 0 |
omr-ws-63bb785fd273ad74a2ffd55ff788cb9032f16aee5857ed3a4f9b5108c4555f12 | What is the remainder when \(100!\) is divided by 103? Note: 103 is prime. | [
"51"
] | [
"What is the remainder when \\(100!\\) is divided by 103? Note: 103 is prime."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-91e28d3e27ef5e5d6b087e138364f02b734d0f11ff252bdb40ab5bf605dfadf9 | What is the remainder when \(16^{15} - 8^{15} - 4^{15} - 2^{15} - 1^{15}\) is divided by 96? | [
"31"
] | [
"What is the remainder when \\(16^{15} - 8^{15} - 4^{15} - 2^{15} - 1^{15}\\) is divided by 96?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-b5241599cb0cea319e30340266a9b09e9b8aab5f28c8116108c6321e9edd9369 | What is the remainder when \(1^2 + 2^2 + 3^2 + \cdots + 2018^2 + 2019^2\) is divided by 8? | [
"6"
] | [
"What is the remainder when \\(1^2 + 2^2 + 3^2 + \\cdots + 2018^2 + 2019^2\\) is divided by 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ddc73b0e29cdbe91612719f13204d4891567180b5d6b961478f52017b4a793cc | What is the remainder when \(1^{111} + 2^{111} + 3^{111} + \ldots + 10^{111}\) is divided by 11? | [
"0"
] | [
"What is the remainder when \\(1^{111} + 2^{111} + 3^{111} + \\ldots + 10^{111}\\) is divided by 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7025424d47071e46b1a0f49a27930d1c8bc3ae6d06c4a8957010b2c4f25e911f | What is the remainder when \(2000!\) is divided by 2003? | [
"1001"
] | [
"What is the remainder when \\(2000!\\) is divided by 2003?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-4da5c01a2c94e69b27f4b5fcd8819530ae3ad8bcc833f23163cbb671126a30ac | What is the remainder when \(2005^{2002} + 2002^{2005}\) is divided by 200? | [
"57"
] | [
"What is the remainder when \\(2005^{2002} + 2002^{2005}\\) is divided by 200?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-977683892b76eac72d435e13d4875db2d541a648d441602fb490443763402d84 | What is the remainder when \(2008^{2007} - 2007\) is divided by \(2008^2 - 2007\)? | [
"\\(2007^2\\)"
] | [
"What is the remainder when \\(2008^{2007} - 2007\\) is divided by \\(2008^2 - 2007\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 26
} | 49 | 0 | 0 |
omr-ws-c48992c0879372fbadced3a0e2c19357eff24d72935a16fd3cd7d9086daf3433 | What is the remainder when \(2008^{2008}\) is divided by 2009? | [
"44"
] | [
"What is the remainder when \\(2008^{2008}\\) is divided by 2009?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1ee04d4f9cf946f86c6624b0b8ec4992a9657f68135596508b6f70f5b80571f | What is the remainder when \(2014!\) is divided by 2017? | [
"1008"
] | [
"What is the remainder when \\(2014!\\) is divided by 2017?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-eaa87900873745ee7f9ed35c5774809190fc04c7805158ba5423fa1337b5f636 | What is the remainder when \(20232023...\) [2023 times] is divided by 825? | [
"298"
] | [
"What is the remainder when \\(20232023...\\) [2023 times] is divided by 825?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eba474d8758b5e86cbee656937863c6e9fd3d83612a19ea35b20358aa8e3eb1d | What is the remainder when \(21^{970}\) is divided by 666? | [
"567"
] | [
"What is the remainder when \\(21^{970}\\) is divided by 666?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0da25ad98eade85ececefc2700c67d4b6fa1d22858efb27b462226b8aed87029 | What is the remainder when \(22222^{5555} - 55555^{2222}\) is divided by 7? | [
"0"
] | [
"What is the remainder when \\(22222^{5555} - 55555^{2222}\\) is divided by 7?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-4e6ab890dcad80b10f32f6665feba97045da0523db6890e1671203ba88961a67 | What is the remainder when \(3053^{456} - 2417^{333}\) is divided by 9? | [
"2"
] | [
"What is the remainder when \\(3053^{456} - 2417^{333}\\) is divided by 9?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b145595680a595208ce36eda9eca002cc65170738d8405be0298d22f2d3de1b6 | What is the remainder when \(3^{2006}\) is divided by 11? | [
"3"
] | [
"What is the remainder when \\(3^{2006}\\) is divided by 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d70cd2a3bf5bff5c2e26f65d1ec8fdac45fbf0407cb11c5758e44c5be2b578bf | What is the remainder when \(3^{2^n}\) is divided by \(2^n + 1\)? | [
"1"
] | [
"What is the remainder when \\(3^{2^n}\\) is divided by \\(2^n + 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-47ca7d560c0f899dc2445687cf23361e93e6468d9b6eea789c61e499bef90cc4 | What is the remainder when \(3^{2^n}\) is divided by \(2^{n+3}\)? | [
"\\( r_n = \\begin{cases} \n 3 & \\text{if } n = 0 \\\\\n 2^{n+2} + 1 & \\text{if } n \\geq 1 \n \\end{cases} \\)"
] | [
"What is the remainder when \\(3^{2^n}\\) is divided by \\(2^{n+3}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-86d5c8b01363d207937151ff6b7149b75af0dbd8f2faaaa22b0ebe5b8adfcd18 | What is the remainder when \(3^{48}\) is divided by 91? | [
"1"
] | [
"What is the remainder when \\(3^{48}\\) is divided by 91?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81f03587207762a87eaceb31c06448943af0226949fd282b24dcdca1ba14f08b | What is the remainder when \(3x \cdot 2^{n+3} - 4x \cdot 2^{n+2} + 5x \cdot 2^{n+1} - 8\) is divided by \(x + 1\), where \(n\) is a non-negative integer? | [
"-9 \\cdot 2^{n+1} - 8"
] | [
"What is the remainder when \\(3x \\cdot 2^{n+3} - 4x \\cdot 2^{n+2} + 5x \\cdot 2^{n+1} - 8\\) is divided by \\(x + 1\\), where \\(n\\) is a non-negative integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a749fb23ee7cc484c1fcf1824927fa4966f6fa73bc187ca73586f84a374b9622 | What is the remainder when \(47^{2005} + 29^{2005}\) is divided by 19? | [
"0"
] | [
"What is the remainder when \\(47^{2005} + 29^{2005}\\) is divided by 19?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20d90bbd6277ecfa0478177c6bcadd8293e30bf752d319bac86fd7eac1ad82e9 | What is the remainder when \(5^{5^{5^5}}\) is divided by 13? | [
"5"
] | [
"What is the remainder when \\(5^{5^{5^5}}\\) is divided by 13?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b73d83d2147e9d4e906ac605bf8498c479df703c74c068f037298cc9e06b88e6 | What is the remainder when \(6^{83} + 8^{83}\) is divided by 49? | [
"35"
] | [
"What is the remainder when \\(6^{83} + 8^{83}\\) is divided by 49?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 15
} | 47 | 0 | 0 |
omr-ws-00fdd7c4a5d7e8ef0a561fa395ad635c02341a1a38f4f4b40b31a33ed5d5b77a | What is the remainder when \(7^{8^9}\) is divided by 1000? | [
"801"
] | [
"What is the remainder when \\(7^{8^9}\\) is divided by 1000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-e075ab84c8791e0968268ea66fd4b0ce2dea4db968c1f08bfba49ccdfca2270c | What is the remainder when \(8^6 + 7^7 + 6^8\) is divided by 5? | [
"3"
] | [
"What is the remainder when \\(8^6 + 7^7 + 6^8\\) is divided by 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5242c3a7c8956ce9ee1bd208541b338e8874b43f83deb6a8494ce00df8770407 | What is the remainder when \(9^9 + 9^{99} + 9^{999} + \ldots + 9^{999\ldots600 \text{ 9s}}\) is divided by 63? | [
"54"
] | [
"What is the remainder when \\(9^9 + 9^{99} + 9^{999} + \\ldots + 9^{999\\ldots600 \\text{ 9s}}\\) is divided by 63?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-c00f885331bb56ec8eecb6b384937d15a41768cc46ca8e1299b5a047c337896c | What is the remainder when \(\frac{100!}{1} + \frac{100!}{2} + \frac{100!}{3} + \cdots + \frac{100!}{99} + \frac{100!}{100}\) is divided by 101? | [
"0"
] | [
"What is the remainder when \\(\\frac{100!}{1} + \\frac{100!}{2} + \\frac{100!}{3} + \\cdots + \\frac{100!}{99} + \\frac{100!}{100}\\) is divided by 101?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef6f893677458bcb0a893e23503275afffc31748175dc4e18b0c9cc13143c72b | What is the remainder when a googolplex $(10^{(10^{100})})$ is divided by $23$? | [
"13"
] | [
"What is the remainder when a googolplex $(10^{(10^{100})})$ is divided by $23$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-c0047d4ce569a762b81084559e8fcaf4ee3663e7914b936826ad8ff9440dad0d | What is the remainder when the $2009$-th number in the Fibonacci sequence is divided by $8$? | [
"1"
] | [
"What is the remainder when the $2009$-th number in the Fibonacci sequence is divided by $8$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c382d20273779f43044bf8a434ed25c95d7fc07b2d88c82a637eccaec2905b4 | What is the remainder when the $2020$-digit number $202020 \ldots 20$ is divided by $275$? | [
"70"
] | [
"What is the remainder when the $2020$-digit number $202020 \\ldots 20$ is divided by $275$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-79910c4c4eb9cf853198d7f7191262fe1b46d1489c31072c76ace37d1440a5f5 | What is the remainder when the $2022$-nd term of the sequence of positive multiples of $3$ that are one less than a perfect square is divided by 1000? | [
"155"
] | [
"What is the remainder when the $2022$-nd term of the sequence of positive multiples of $3$ that are one less than a perfect square is divided by 1000?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-f0dee5255c2d0bfb40a421c26851d220717a11bcfc55fb2dc274471c45f515d8 | What is the remainder when the $n$-th number in the Fibonacci sequence is divided by $8$? | [
"The remainder when the $n$-th number in the Fibonacci sequence is divided by 8 is given by the $n \\mod 12$ position in the sequence $1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1, 0$."
] | [
"What is the remainder when the $n$-th number in the Fibonacci sequence is divided by $8$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-6ddf473e856ade79b079e91ee1785c7b7bb04eacc99747c0c9a88d0119c35a3c | What is the remainder when the 100th term of the Fibonacci sequence is divided by 4? | [
"3"
] | [
"What is the remainder when the 100th term of the Fibonacci sequence is divided by 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-05ac8cdd2b9b803dedf7d3d441101d77686bad8b4fc9a53d45f0c9faa39c75ea | What is the remainder when the 1492nd Fibonacci number is divided by 25? (The Fibonacci sequence is defined with the first two numbers being 1.) | [
"4"
] | [
"What is the remainder when the 1492nd Fibonacci number is divided by 25? (The Fibonacci sequence is defined with the first two numbers being 1.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf8dd6593d8e11f9fc2ab7f83844b8bc9a5e30d93b4bddaae9d66b60ba31919f | What is the remainder when the 1994th term of the sequence of positive multiples of 3 that are one less than a perfect square is divided by 1000? | [
"063",
"63"
] | [
"What is the remainder when the 1994th term of the sequence of positive multiples of 3 that are one less than a perfect square is divided by 1000?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 5
} | 44 | 0 | 0 |
omr-ws-7f5dfbcc48dd151cbaa5370d4f56c6a199456ccacca8e91916ba02c36775a69a | What is the remainder when the 224-digit number consisting entirely of 1's is divided by 41? | [
"4"
] | [
"What is the remainder when the 224-digit number consisting entirely of 1's is divided by 41?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-238b6201b9479ef2f052341c83262aa5215851ba24b7d45908b0c128643cba87 | What is the remainder when the 94-digit number 1010101... is divided by 375? | [
"260"
] | [
"What is the remainder when the 94-digit number 1010101... is divided by 375?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 9
} | 32 | 0 | 0 |
omr-ws-c33fbb907142fba85abd585505c9772a2a68a851a21ea2069fe99e31ed2b0aa5 | What is the remainder when the first 100 digits of the number \( N = 123456789\ldots96979899100 \) are divided by 9? | [
"5"
] | [
"What is the remainder when the first 100 digits of the number \\( N = 123456789\\ldots96979899100 \\) are divided by 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1c1dd5b8d0a5d8f108234bac067dab5dbb84a58f9d707b362fb2c6976b1d5d62 | What is the remainder when the integer $ (\sqrt{3}+5)^{103} - (\sqrt{3}-5)^{103} $ is divided by 9? | [
"1"
] | [
"What is the remainder when the integer $ (\\sqrt{3}+5)^{103} - (\\sqrt{3}-5)^{103} $ is divided by 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-02b0ea494e65368313f9e85dfcd73f11642fbb4426be8c48cb39ddd3344462c9 | What is the remainder when the integer \( N = 123456789101112\ldots9899100 \) is divided by 11? | [
"8"
] | [
"What is the remainder when the integer \\( N = 123456789101112\\ldots9899100 \\) is divided by 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-c3c5ba5a902fd72cef4c011f505f3620a3eec44854b63e4bf161f6676f759453 | What is the remainder when the number $$12 + 11(13^1) + 10(13^2) + 9(13^3) + 8(13^4) + 7(13^5) + 6(13^6) + 5(13^7) + 4(13^8) + 3(13^9) + 2(13^{10}) + 13^{11}$$ is divided by $6$? | [
"0"
] | [
"What is the remainder when the number\n$$12 + 11(13^1) + 10(13^2) + 9(13^3) + 8(13^4) + 7(13^5) + 6(13^6) + 5(13^7) + 4(13^8) + 3(13^9) + 2(13^{10}) + 13^{11}$$\nis divided by $6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e28e030da2139d744ae11eff962fc02a39204d3c9aab357e94501576416ac976 | What is the remainder when the number $N$, whose base-nine representation is $27{,}006{,}000{,}052_{\rm nine}$, is divided by 5? | [
"3"
] | [
"What is the remainder when the number $N$, whose base-nine representation is $27{,}006{,}000{,}052_{\\rm nine}$, is divided by 5?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e69d2720cd1f1513d3faac99bc9bda391f5a87714d5a30932f9bb2ed5fb5ff7c | What is the remainder when the number $a=123456789101112...998999$ (formed by writing the numbers from $1$ to $999$ consecutively) is divided by $11$? | [
"0"
] | [
"What is the remainder when the number $a=123456789101112...998999$ (formed by writing the numbers from $1$ to $999$ consecutively) is divided by $11$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1120f552bf16fe8fb9433824ad5c88a6be346d1e71f50f78c2bfd6aa0a8ff52 | What is the remainder when the number 301302303304...997998999 is divided by 11? | [
"1"
] | [
"What is the remainder when the number 301302303304...997998999 is divided by 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-9e00c65242134a5b62470be1bbf0006f00cd350b1d22caa9ea73e5b2c7b0f5bc | What is the remainder when the number 345345345... (repeating up to 300 digits) is divided by 999,999,999? | [
"396396741"
] | [
"What is the remainder when the number 345345345... (repeating up to 300 digits) is divided by 999,999,999?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-8b33b3d4313267d87b8e69cb750ac405bfd1df577b4b9a54761d9cea706eea9d | What is the remainder when the number 888888....... (100 digits) is divided by 625? | [
"138"
] | [
"What is the remainder when the number 888888....... (100 digits) is divided by 625?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-b67677e5216fb553b00ec4072e237a917fc0a8402a9631c297db82dd89e280e4 | What is the remainder when the number \( N = 12345 \ldots 424344 \) is divided by 45? | [
"9"
] | [
"What is the remainder when the number \\( N = 12345 \\ldots 424344 \\) is divided by 45?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 16,
"genselect": 5
} | 21 | 0 | 0 |
omr-ws-77d0eb628274c01296556936e758e93da7c62f080855ddf444e7d35304821ce8 | What is the remainder when the number \( N = 123456789101112......424344 \) is divided by 45? | [
"9"
] | [
"What is the remainder when the number \\( N = 123456789101112......424344 \\) is divided by 45?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1aab2464c0e960a85ed24d30a3a7475b89a87aa09c0526f34ff3a9f773f0b0bf | What is the remainder when the number \( N = 123456789101112\ldots424344 \) is divided by 45? | [
"9"
] | [
"What is the remainder when the number \\( N = 123456789101112\\ldots424344 \\) is divided by 45?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-80556d027ac95614ceff1c47300bd77098f175f0f42d776aa31d8d4c94bd5ec5 | What is the remainder when the number formed by 1001 consecutive 1's is divided by 1001? | [
"100"
] | [
"What is the remainder when the number formed by 1001 consecutive 1's is divided by 1001?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-2ca34bccd54c26c4cdf16f0fa9e42043ccc28f964df9c21efd11a4becb40a203 | What is the remainder when the number formed by concatenating the digits from 1 to 2016 (i.e., 12345678910111213...2016) is divided by 2016? | [
"0"
] | [
"What is the remainder when the number formed by concatenating the digits from 1 to 2016 (i.e., 12345678910111213...2016) is divided by 2016?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-2486e146a602193d4c403bc3eaacd73b5ef3dba571e96b7e9e53c4738a9e73b8 | What is the remainder when the number formed by concatenating the digits of all integers from 101 to 998 inclusive is divided by 990? | [
"548"
] | [
"What is the remainder when the number formed by concatenating the digits of all integers from 101 to 998 inclusive is divided by 990?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-9e41035721649d5470818afc4c8356dd602d49823d500a655c21c29349c326e7 | What is the remainder when the number formed by concatenating the integers from $1$ to $1979$ is divided by $1980$? | [
"1539"
] | [
"What is the remainder when the number formed by concatenating the integers from $1$ to $1979$ is divided by $1980$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2754de563895523d1eeb5c55eeaecef46f614eb43d7784da31eb31c0010d3cd7 | What is the remainder when the number formed by juxtaposing the natural numbers up to a 1002-digit number is divided by 16? | [
"10"
] | [
"What is the remainder when the number formed by juxtaposing the natural numbers up to a 1002-digit number is divided by 16?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1667e8d5eb803f9405aa1bfea7d6f19ff2331b4942b26db5a35dc9d77ea59c71 | What is the remainder when the number formed by writing the first 44 positive integers in order (123456...41424344) is divided by 45? | [
"9"
] | [
"What is the remainder when the number formed by writing the first 44 positive integers in order (123456...41424344) is divided by 45?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-98505ec331919404f8d99d9b5ffb65d34ce85b559707f324dc0cd08989126b87 | What is the remainder when the number of derangements of n elements is divided by n? | [
"\\( 1 \\) if \\( n \\) is even, \\( n-1 \\) if \\( n \\) is odd."
] | [
"What is the remainder when the number of derangements of n elements is divided by n?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 26
} | 39 | 0 | 0 |
omr-ws-b96f87cdd744e9644260315adcf3f19d3073bb1db4db0e300dea19caa9ccfd36 | What is the remainder when the numerical expression $1^7 + 2^7 + \ldots + 7^7$ is divided by a given number? | [
"0"
] | [
"What is the remainder when the numerical expression $1^7 + 2^7 + \\ldots + 7^7$ is divided by a given number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87deeb61ddd988e1534cf0b97eac83121eb9b0841232328adbee6fb50f024d53 | What is the remainder when the numerical expression $1^{2013} + 2^{2013} + \ldots + 2013^{2013}$ is divided by a given number? | [
"0"
] | [
"What is the remainder when the numerical expression $1^{2013} + 2^{2013} + \\ldots + 2013^{2013}$ is divided by a given number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-fc3c39a52d5ee01b98315429bf82d722e01a745f513047880e0ef9458d5fb0ad | What is the remainder when the polynomial $x^{19} + x^{17} + x^{13} + x^{11} + x^7 + x^5 + x^3$ is divided by $(x^2 + 1)$? | [
"\\(-x\\)"
] | [
"What is the remainder when the polynomial $x^{19} + x^{17} + x^{13} + x^{11} + x^7 + x^5 + x^3$ is divided by $(x^2 + 1)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-40af9522e5732f6dac93533980054c8ccdba8892620dbe0ee4a7260fab041150 | What is the remainder when the polynomial \(1 - x + 2x^4 - 3x^9 + 4x^{16} - 5x^{25} + 6x^{36} + x^{2005}\) is divided by \(x^2 - 1\)? | [
"\\(13 - 8x\\)"
] | [
"What is the remainder when the polynomial \\(1 - x + 2x^4 - 3x^9 + 4x^{16} - 5x^{25} + 6x^{36} + x^{2005}\\) is divided by \\(x^2 - 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-2fd63da415c6149a1eb44e9fec35f3d3b6e371042db0faf7e8c57ed2e4f7b691 | What is the remainder when the product $1734\times 5389 \times 80,\!607$ is divided by 10? | [
"2"
] | [
"What is the remainder when the product $1734\\times 5389 \\times 80,\\!607$ is divided by 10?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b234e5a70b970f65a84c3162f773ecb6ac78684ddeebec77d8522bac3e612fc2 | What is the remainder when the product \(1 \times 3 \times 5 \times \cdots \times 2005\) is divided by 1000? | [
"375"
] | [
"What is the remainder when the product \\(1 \\times 3 \\times 5 \\times \\cdots \\times 2005\\) is divided by 1000?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-02b615a63a50379e448b95f3120d7587da5afd555ecf9364e0e2c9df6b233744 | What is the remainder when the sum $1 + 7 + 13 + 19 + \cdots + 253 + 259$ is divided by $6$? | [
"2"
] | [
"What is the remainder when the sum $1 + 7 + 13 + 19 + \\cdots + 253 + 259$ is divided by $6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2002321ca293dca096b8e3dc42feb7ed7859f6c5dd592b028d053264ebc56850 | What is the remainder when the sum $1^2 + 2^2 + 3^2 + \dots + 10^2$ is divided by 11? | [
"0"
] | [
"What is the remainder when the sum $1^2 + 2^2 + 3^2 + \\dots + 10^2$ is divided by 11?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ada659e1739cdc7c1640454a580743d0da8024effc73f9f1e2ad00063e97b915 | What is the remainder when the sum $1^{1990} + 2^{1990} + 3^{1990} + \ldots + 1990^{1990}$ is divided by 10? | [
"5"
] | [
"What is the remainder when the sum $1^{1990} + 2^{1990} + 3^{1990} + \\ldots + 1990^{1990}$ is divided by 10?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-9a4a4b4c058df3e6c39abe854cf6179cc9cad82322b82195ff6a107df263c744 | What is the remainder when the sum \(1! + 2! + 3! + \ldots + 98! + 99! + 100!\) is divided by 18? | [
"9"
] | [
"What is the remainder when the sum \\(1! + 2! + 3! + \\ldots + 98! + 99! + 100!\\) is divided by 18?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2b8737543aa6724e11d955458c8dd4e2a22556d61f63c8c930d021aa8a529bda | What is the remainder when the sum \(11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 + \ldots + 20192019\) is divided by 100? | [
"40"
] | [
"What is the remainder when the sum \\(11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 + \\ldots + 20192019\\) is divided by 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-a10d6bdaa0c5ef9fd88c6d6b0981a8b4100d3fef7f677e89f1322867ab196039 | What is the remainder when the sum \(11 + 22 + 33 + \ldots + 1010\) is divided by 5? | [
"0"
] | [
"What is the remainder when the sum \\(11 + 22 + 33 + \\ldots + 1010\\) is divided by 5?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-92c5e5812cf6fdb838bed6ceed574c328b2432745b4114c6bc1eaa637397cfa4 | What is the remainder when the sum \(17 + 177 + 1777 + \ldots + 177777777777777777777\) (where the last number has 21 digits, consisting of 1 followed by 20 sevens) is divided by 8? | [
"4"
] | [
"What is the remainder when the sum \\(17 + 177 + 1777 + \\ldots + 177777777777777777777\\) (where the last number has 21 digits, consisting of 1 followed by 20 sevens) is divided by 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-c213059c1f8d8b7df72c996a77b81a4bdaaa25b229f1e2b883d8b4ec3e5e7571 | What is the remainder when the sum \(17 + 177 + 1777 + \ldots + 1777\ldots7\) (with the last term having 20 sevens) is divided by 8? | [
"4"
] | [
"What is the remainder when the sum \\(17 + 177 + 1777 + \\ldots + 1777\\ldots7\\) (with the last term having 20 sevens) is divided by 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-75ae3c23aa72b47abdd07cc202631b859b952c4b4113a4adc169d88c0247f100 | What is the remainder when the sum \(1^2 + 2^2 + \cdots + 10^2\) is divided by 11? | [
"0"
] | [
"What is the remainder when the sum \\(1^2 + 2^2 + \\cdots + 10^2\\) is divided by 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d90435b21e186349b65c38e9c61ec6d9809defdf07d5f1690b5b74bbf0badb1d | What is the remainder when the sum \(1^{32} + 2^{32} + 3^{32} + \ldots + 2023^{32} + 2024^{32}\) is divided by 32? | [
"20"
] | [
"What is the remainder when the sum \\(1^{32} + 2^{32} + 3^{32} + \\ldots + 2023^{32} + 2024^{32}\\) is divided by 32?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa499a2bd96e6f1f74776e8b4dbfbac804ad29aa5ba43736e98c308cf85afe35 | What is the remainder when the sum \(\binom{2013}{1} + 2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013}\) is divided by 41? | [
"20"
] | [
"What is the remainder when the sum \\(\\binom{2013}{1} + 2013\\binom{2013}{3} + 2013^2\\binom{2013}{5} + \\dots + 2013^{1006}\\binom{2013}{2013}\\) is divided by 41?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 43,
"genselect": 7
} | 50 | 0 | 0 |
omr-ws-e4d9d52fbac3b7ed7c2b3f15a49703be08a0188bb6d9e8a6f5a98764e79f22e9 | What is the remainder when the sum \[ \binom{2013}{1} + 2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013} \] is divided by 41? | [
"20"
] | [
"What is the remainder when the sum \n\\[\n\\binom{2013}{1} + 2013\\binom{2013}{3} + 2013^2\\binom{2013}{5} + \\dots + 2013^{1006}\\binom{2013}{2013}\n\\]\nis divided by 41?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 46,
"genselect": 7
} | 53 | 0 | 0 |
omr-ws-a0d7b9285e1b3c68219d448fd785a432fe0b22c95cbc5cdfa8b35a391fe24f66 | What is the remainder when the sum of all possible numbers created using the digits 1, 2, and 3 is divided by 9? | [
"0"
] | [
"What is the remainder when the sum of all possible numbers created using the digits 1, 2, and 3 is divided by 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-e86b3535c144c70817e27663a63a384a3f5d2a159d8fea7cb6df13c6e76b62eb | What is the remainder when the sum of the factors of \(2015^{2014}\) is divided by 2015? | [
"545"
] | [
"What is the remainder when the sum of the factors of \\(2015^{2014}\\) is divided by 2015?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-d40fcf0162da992844cf60cf5d47b38c3bedf5e0263d3a3dbaf41955cad0b0b0 | What is the remainder when the sum of the first 102 counting numbers is divided by 5250? (The first counting number is 1.) | [
"3"
] | [
"What is the remainder when the sum of the first 102 counting numbers is divided by 5250? (The first counting number is 1.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1cb28055e4c444e91b6e7f2a2db4ff1869ea050331239bbac902453a3853c71 | What is the remainder when the sum of the first five primes is divided by the sixth prime? | [
"2"
] | [
"What is the remainder when the sum of the first five primes is divided by the sixth prime?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5bace4085f47988b04f54a335168dd83be84326c85c96b0e9a37ac191981e5ac | What is the residue of $9^{2010}$, modulo 17? | [
"13"
] | [
"What is the residue of $9^{2010}$, modulo 17?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eda1da6672005aacb8b57504e74acb38b0d7e18fe275a5d07a2e22a01020bd81 | What is the residue of the 2006th Fibonacci number modulo 5? | [
"3"
] | [
"What is the residue of the 2006th Fibonacci number modulo 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 9
} | 29 | 0 | 0 |
omr-ws-6b963b96d2a4e2ea4cc6193d3112a9e98917f1547ac911ec9722a0edb1a17fad | What is the resolvent set of the differential operator $Af = f'$ acting on the Sobolev space $W^{1,2}(\mathbb{R})$? | [
"\\( \\mathbb{C} \\setminus i\\mathbb{R} \\)"
] | [
"What is the resolvent set of the differential operator $Af = f'$ acting on the Sobolev space $W^{1,2}(\\mathbb{R})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-6e0aba98cd9444272bec999af5b4b9be215aca20de934eb1d0281acfaa615420 | What is the result of $\frac{3}{1! + 2! + 3!} + \frac{4}{2! + 3! + 4!} + \frac{5}{3! + 4! + 5!} + \cdots + \frac{8}{6! + 7! + 8!}$? | [
"\\(\\frac{20159}{40320}\\)"
] | [
"What is the result of $\\frac{3}{1! + 2! + 3!} + \\frac{4}{2! + 3! + 4!} + \\frac{5}{3! + 4! + 5!} + \\cdots + \\frac{8}{6! + 7! + 8!}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 28
} | 76 | 0 | 0 |
omr-ws-6e0411729111f0e1ca48b1a1a3a1ac5ca3b69a01a4a77dc5b31a61f90435804e | What is the result of multiplying a 3 by 2 matrix with a 2 by 3 matrix? | [
"3 \\times 3"
] | [
"What is the result of multiplying a 3 by 2 matrix with a 2 by 3 matrix?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-637a629ae6d2e91b1b75916d93d4290eb644fd9c5637e43bbd7306dce24de909 | What is the result when the sequence {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, … } is added up by writing each term as a decimal and placing it in the corresponding decimal place, similar to the sequence {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, … }? | [
"\\(\\frac{10}{9}\\)"
] | [
"What is the result when the sequence {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, … } is added up by writing each term as a decimal and placing it in the corresponding decimal place, similar to the sequence {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, … }?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 6
} | 12 | 0 | 0 |
omr-ws-d51e498e95ca92dca9b801d985b2b16ed228076ce5a4b0a7a5a0b965405037b8 | What is the result when we compute $$1^3 + 2^3 + 3^3 + 4^3 + \dots + 99^3 + 100^3 $$and $$(-1)^3 + (-2)^3 + (-3)^3 + (-4)^3 + \dots + (-99)^3 + (-100)^3,$$and then add the two results? | [
"0"
] | [
"What is the result when we compute $$1^3 + 2^3 + 3^3 + 4^3 + \\dots + 99^3 + 100^3 $$and $$(-1)^3 + (-2)^3 + (-3)^3 + (-4)^3 + \\dots + (-99)^3 + (-100)^3,$$and then add the two results?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9c867afd8c2a5306d2f056a40431667269342a1607faf50e578c68cf11678056 | What is the resulting volume of the figure when the line segment $y=8x+4$, for $-2 \leq x \leq 3$, is rotated about the x-axis? | [
"\\(\\frac{2960\\pi}{3}\\)"
] | [
"What is the resulting volume of the figure when the line segment $y=8x+4$, for $-2 \\leq x \\leq 3$, is rotated about the x-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c38307a85b9c09a6d30eacc9a64383f000b48da61cbcc4d94b002a7b2c78fd8c | What is the rightmost decimal digit of \( N = (3+2\sqrt{2})^{512} + (3-2\sqrt{2})^{512} \)? | [
"6"
] | [
"What is the rightmost decimal digit of \\( N = (3+2\\sqrt{2})^{512} + (3-2\\sqrt{2})^{512} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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