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-28c9a37907e49c4fd2c636ac03dbc95b1f79b77610c83593f8322f6cbcc19c5d | What are the last three digits of $3^{3^{3^{3^{3}}}}$? | [
"387"
] | [
"What are the last three digits of $3^{3^{3^{3^{3}}}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d1b3b7ea9aabf7ee8fbbe36ecc03797538599956c8a510de2365801e6051015 | What are the last three digits of $3^{3^{3^{3}}}$? | [
"387"
] | [
"What are the last three digits of $3^{3^{3^{3}}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6c48507cd17372940edb3ed5acb82d8b9d341a1ffc12b2d467ceda1be54e27d | What are the last three digits of $49^{303} \cdot 3993^{202} \cdot 39^{606}$? | [
"561"
] | [
"What are the last three digits of $49^{303} \\cdot 3993^{202} \\cdot 39^{606}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-006584a450144bf2a705d926de17861325fff798d581c634837cad88bbd8ffa0 | What are the last three digits of $5^{5^{5^5}}$? | [
"125"
] | [
"What are the last three digits of $5^{5^{5^5}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-192a183355778913e5da852a7861a8d00e5f47d6bfa7adf0972f7b6381375bfb | What are the last three digits of $7^{999}$? | [
"143"
] | [
"What are the last three digits of $7^{999}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-44e13ddf36f5eded8f5e8dca8a158bc6e42a8e34aa897f8c5e2d4544275390bc | What are the last three digits of $\left \lfloor{\dfrac{10^{36}}{10^{12}+7}}\right \rfloor$? | [
"048"
] | [
"What are the last three digits of $\\left \\lfloor{\\dfrac{10^{36}}{10^{12}+7}}\\right \\rfloor$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-79106b88eb7652cf05143986eaab09f4411aa39d2c91d0a3581f957b0089a207 | What are the last three digits of \( \left\lfloor \sqrt{2009^2+1} + \sqrt{2009^2+2} + \sqrt{2009^2+3} + \cdots + \sqrt{2009^2+4018} \right\rfloor \)? | [
"171"
] | [
"What are the last three digits of \\( \\left\\lfloor \\sqrt{2009^2+1} + \\sqrt{2009^2+2} + \\sqrt{2009^2+3} + \\cdots + \\sqrt{2009^2+4018} \\right\\rfloor \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d28b971ee59deec836752b52750c8a6c0abc0f75ae669c65171e4c9047f4c2d1 | What are the last three digits of \(2003^N\), where \(N = 2002^{2001}\)? | [
"241"
] | [
"What are the last three digits of \\(2003^N\\), where \\(N = 2002^{2001}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-83e068c5d7c1d7bab39649d206922ee32b97a83908acadb413ef361a76a9e7c7 | What are the last three digits of \(2003^{2002^{2001}}\)? | [
"241"
] | [
"What are the last three digits of \\(2003^{2002^{2001}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-33fa20fb6015d263a4b1b8946d97d078f75cb9ab97d57d074d0e594b6159488c | What are the last three digits of \(2007^{100} - 2003^{100}\)? | [
"000"
] | [
"What are the last three digits of \\(2007^{100} - 2003^{100}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11a110abd8a165e3abdb79089db7e63e0777b0839498fd6545f6da6086084dfd | What are the last three digits of \(2007^{100} \times 2003^{100}\)? | [
"001"
] | [
"What are the last three digits of \\(2007^{100} \\times 2003^{100}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-5deb545750ad0defd6588cf3e9cbc4efd85ffb185324ec0ac30a796678a24ac5 | What are the last three digits of \(3^{4798}\)? | [
"889"
] | [
"What are the last three digits of \\(3^{4798}\\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-cf2fdac4cd0bf8b155f7a3ac7b9421f7f65d8b1b7fd9d357bf66ab8a81539c9d | What are the last three digits of \(49^{303} \cdot 3993^{202} \cdot 39^{606}\)? | [
"561"
] | [
"What are the last three digits of \\(49^{303} \\cdot 3993^{202} \\cdot 39^{606}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d2d96789d61db010a84d64ee18b3be28183a1fed3f068f45c074c88e0bdd805 | What are the last three digits of \(7^{1051}\)? | [
"743"
] | [
"What are the last three digits of \\(7^{1051}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-3ae7f76f78585d7774e32de9f61c4f2111b5240b1e8e4ec975ad07f577022c4b | What are the last three digits of the $2020$th term in the Fibonacci sequence? | [
"600"
] | [
"What are the last three digits of the $2020$th term in the Fibonacci sequence?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77df8a486ff76c04e5590a926af0f715963b67d44d90632de94c8397abb96a90 | What are the last three digits of the base-4 representation of $10 \cdot 3^{195} \cdot 49^{49}$? | [
"232"
] | [
"What are the last three digits of the base-4 representation of $10 \\cdot 3^{195} \\cdot 49^{49}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-a0e0ee952347903ec2ce1c7934c9972aca4e66645cc211b509e17cf0e047fe3e | What are the last three digits of the coefficient of \( x^{10} \) in the expansion of \( (1 + x + x^2 + x^3 + x^4 + x^5)^6 \)? | [
"2247"
] | [
"What are the last three digits of the coefficient of \\( x^{10} \\) in the expansion of \\( (1 + x + x^2 + x^3 + x^4 + x^5)^6 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-a00dfdcd655f89868f7b3728bb2b826bf57d3856a649ca02eb27b297b6a70dc9 | What are the last three digits of the expanded form of $3(9998)^4 + 5(9999)^3 - 73(10001)^3$? | [
"970"
] | [
"What are the last three digits of the expanded form of $3(9998)^4 + 5(9999)^3 - 73(10001)^3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ebf4b80d1bb5b3d194de75c003834ef4788dc18dcd03830c6503f8ce219edb4f | What are the last three digits of the maximum possible value of the product of all labels across every problem in a packet of IMO Shortlist Geometry problems, where each problem is labeled with a positive integer between 1 and 10, inclusive? | [
"001"
] | [
"What are the last three digits of the maximum possible value of the product of all labels across every problem in a packet of IMO Shortlist Geometry problems, where each problem is labeled with a positive integer between 1 and 10, inclusive?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-370852dcbadfc395f9f7cdbab2b841bcfa6849a9645af694003177a8bb1f3b94 | What are the last three digits of the number \(10 \cdot 3^{195} \cdot 49^{49}\) when written in base 4? | [
"232"
] | [
"What are the last three digits of the number \\(10 \\cdot 3^{195} \\cdot 49^{49}\\) when written in base 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-b2a29ff6cb402de32a2c00f415ac24353b8209b00093fbe43d9f126612383ef5 | What are the last three digits of the product $1 \times 3 \times 5 \times \cdots \times 2007 \times 2009 \times 2011$? | [
"875"
] | [
"What are the last three digits of the product $1 \\times 3 \\times 5 \\times \\cdots \\times 2007 \\times 2009 \\times 2011$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-cf748a68e362858a8b18a89e5478a3fcda4c1711cd64a6e43478cf80d7f67da4 | What are the last three digits of the smallest integer greater than $ (\sqrt{11} + \sqrt{10})^{100}$? | [
"002"
] | [
"What are the last three digits of the smallest integer greater than $ (\\sqrt{11} + \\sqrt{10})^{100}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 15
} | 37 | 0 | 0 |
omr-ws-893ec4d8f64fcb39c8b675934f76fd63abd9b4b5bfe5e2855c30c1f3b96e0431 | What are the last three digits of the sum $11! + 12! + 13! + \ldots + 2006!$? | [
"400"
] | [
"What are the last three digits of the sum $11! + 12! + 13! + \\ldots + 2006!$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-278f68d6f715349e0db3f6fdbb89b56d705a0b6fd7e1427b181bd3870eeb05f0 | What are the last three digits of the sum $9 + 99 + 999 + 9999 + 99999 + 999999 + \ldots + 999\ldots999$ where the last addend has 2017 nines? | [
"093"
] | [
"What are the last three digits of the sum $9 + 99 + 999 + 9999 + 99999 + 999999 + \\ldots + 999\\ldots999$ where the last addend has 2017 nines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-df5db75de18fe5c0680054e1fd6cb3726311f63b483d2b5168fde38053294adf | What are the last three elements in the pattern A, 2, B, 0, C, 2, D, 0, E, 3, F, 3, G, 2, H, 4, I, J, 2, K, 4, L, 2, M, 2, N, 2, O, 0, P, 1, Q, 0, R, 2, S, 2, T, 3, U, 2, V, 2, W, 2, X, 4, Y, ?, ?, ? | [
"3, Z, 2"
] | [
"What are the last three elements in the pattern A, 2, B, 0, C, 2, D, 0, E, 3, F, 3, G, 2, H, 4, I, J, 2, K, 4, L, 2, M, 2, N, 2, O, 0, P, 1, Q, 0, R, 2, S, 2, T, 3, U, 2, V, 2, W, 2, X, 4, Y, ?, ?, ?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8526296fa0cd8dd339ce0d8f7ed20c9936628b5253ef8f16b50ddec2fb59d93c | What are the last two digits in the sum of factorials of the first 100 positive integers? | [
"13"
] | [
"What are the last two digits in the sum of factorials of the first 100 positive integers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4fd8a8c774784bd1b529dc1e5d0417c25e3c96ab57938769e2ed8090084578f | What are the last two digits of $103^{102^{101}}$? | [
"41"
] | [
"What are the last two digits of $103^{102^{101}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-fb9bdc42c34629bdc4142a7adaf10babe18434fd02dbb134d0173f68ba3006bf | What are the last two digits of $123^{789^{456}}$? | [
"23"
] | [
"What are the last two digits of $123^{789^{456}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cad10485390f81c52f5ac1bccf02a13517a933551589d8bad62b5fe849742c2b | What are the last two digits of $173^{3^{321}}$? | [
"17"
] | [
"What are the last two digits of $173^{3^{321}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7fde196443da8014d2b717a6a60128b505d3cf1004ed01688d532b66f6620cd | What are the last two digits of $21^{2006}$? | [
"21"
] | [
"What are the last two digits of $21^{2006}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-98cecb629c443c15d3825cd214f69757a107e9bf141ce6adc797695ce7bfd7f9 | What are the last two digits of $3^{7^{2008}}$? | [
"03"
] | [
"What are the last two digits of $3^{7^{2008}}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-75e28b7567e2c4aa429b41a815d5e11ffbe18a21ee8d947d85a449a8413eaeb5 | What are the last two digits of $3^{{3}^{3}...}$ where it is a tower of 2007 threes? Note that $3^{{3}^{3}} = 3^{27}$, not $(27)^3$. | [
"87"
] | [
"What are the last two digits of $3^{{3}^{3}...}$ where it is a tower of 2007 threes? Note that $3^{{3}^{3}} = 3^{27}$, not $(27)^3$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-a6ed26a78b0636276c281162a2b47d93f25dfb541dc8143de0f33a8ff694faa2 | What are the last two digits of $\lfloor (\sqrt{5} + 2)^{2016} \rfloor$? | [
"21"
] | [
"What are the last two digits of $\\lfloor (\\sqrt{5} + 2)^{2016} \\rfloor$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-392bf20527f6c1189bba21f0841054d9828cf397b92e64b58e5ae87aa3d0e8bc | What are the last two digits of $\lfloor{(3 + \sqrt{5})^{34}}\rfloor$? (With $\lfloor x \rfloor$ being the floor function, the greatest integer less than or equal to $x$.) | [
"47"
] | [
"What are the last two digits of $\\lfloor{(3 + \\sqrt{5})^{34}}\\rfloor$? (With $\\lfloor x \\rfloor$ being the floor function, the greatest integer less than or equal to $x$.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5b0e5bd134d9d1226f5a32d833eaa4ea02d7cfc147f219766f81578800519d29 | What are the last two digits of \( 13^{2005} \times 9999^4 \times 3^{24} \)? | [
"33"
] | [
"What are the last two digits of \\( 13^{2005} \\times 9999^4 \\times 3^{24} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-521cd60aa1a1164ca0e466c25bd10747459b80317981c1a4a427573aceb7dc2f | What are the last two digits of \( 13^{2009} \times 9999^6 \times 3^{12} \)? | [
"93"
] | [
"What are the last two digits of \\( 13^{2009} \\times 9999^6 \\times 3^{12} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-dd804933a0bf6e59ad2b562dfc550a314232cd24507b6cb90ac54d9ecb5f962e | What are the last two digits of \(17^{19^{23^{29^{\cdots^{97}}}}}\)? | [
"53"
] | [
"What are the last two digits of \\(17^{19^{23^{29^{\\cdots^{97}}}}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-d76e74421857b65d4ff716df38a9235aaa65e6e1fd119c71352eaba9dde950fa | What are the last two digits of \(17^{19^{23^{29^{\cdots}}}}\), where the exponents are the prime numbers up to 100? | [
"53"
] | [
"What are the last two digits of \\(17^{19^{23^{29^{\\cdots}}}}\\), where the exponents are the prime numbers up to 100?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-11f4c8355f6b2f1d4a4e57bfbf0536b1f3e07aab19c2f358965cb8bec82d91be | What are the last two digits of \(1996^{1996}\)? | [
"96"
] | [
"What are the last two digits of \\(1996^{1996}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-468124660aab5352f71deddf337f11ddca2ba70e82f58f9a5fdab6bf1b9730ce | What are the last two digits of \(2^{3^{4^{\cdots^{2019}}}}\)? | [
"52"
] | [
"What are the last two digits of \\(2^{3^{4^{\\cdots^{2019}}}}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-706aa305da9279f8f928d019424dd57fb9f51773fe07f1f71d28f2fd5b5e2ead | What are the last two digits of \(3^{44}\)? | [
"81"
] | [
"What are the last two digits of \\(3^{44}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50e39480b4a16104e5fdfedf991581b8e2a099b21c7c849109d87a7a3036188b | What are the last two digits of \(7^{2018} - 3^{2018}\)? | [
"60"
] | [
"What are the last two digits of \\(7^{2018} - 3^{2018}\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-2d44ce71afd6f5bd8c639b5e42d5430ebd0bddf0abeb0944cd2ff2b7f1c9ce84 | What are the last two digits of \(7^{7^{7^{\cdots^7}}}\) if there are 1000 sevens as exponents and only one seven in the middle? | [
"07"
] | [
"What are the last two digits of \\(7^{7^{7^{\\cdots^7}}}\\) if there are 1000 sevens as exponents and only one seven in the middle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c05014e58c75ee1a29193f602e89bb9f4e009dcf4f14aac849d8081ac998bec9 | What are the last two digits of the base-$3$ representation of $2005^{2003^{2004}+3}$? | [
"21"
] | [
"What are the last two digits of the base-$3$ representation of $2005^{2003^{2004}+3}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-1626762b65af7674eae4f91a75fa07ea1e63440827b842bab76e281227af3257 | What are the last two digits of the decimal representation of $21^{2006}$? | [
"21"
] | [
"What are the last two digits of the decimal representation of $21^{2006}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53f669b4a512ef39f5862f5a1b9026e0956f1b38068521dce3a1367898b52de2 | What are the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$? | [
"21"
] | [
"What are the last two digits of the decimal representation of $9^{8^{7^{\\cdot^{\\cdot^{\\cdot^{2}}}}}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-cc30ea3e252694d26363ad73f8b4220945cbd8bbd418f65928699a70e37270de | What are the last two digits of the decimal representation of \(9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}\)? | [
"21"
] | [
"What are the last two digits of the decimal representation of \\(9^{8^{7^{\\cdot^{\\cdot^{\\cdot^{2}}}}}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-b2c6cd647b2ac30e59e0b260d34b3d68b62abb1849912fc42952d10b8d783f94 | What are the last two digits of the greatest integer less than $\frac{10^{93}}{10^{31} + 3}$? | [
"08"
] | [
"What are the last two digits of the greatest integer less than $\\frac{10^{93}}{10^{31} + 3}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-8a554ff8d17854581a2afe096a796738a5ccf397d00bff01156fdd378ca28f68 | What are the last two digits of the integer $\left\lfloor \frac{10^{1995}}{10^{95} + 3} \right\rfloor$? | [
"01"
] | [
"What are the last two digits of the integer $\\left\\lfloor \\frac{10^{1995}}{10^{95} + 3} \\right\\rfloor$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-285a12504d8a4a2d2dc1c74cd372850acc5c3936c3aa86334aeeeaff15ce1433 | What are the last two digits of the number \((11 + 12 + 13 + \ldots + 2006)^2\)? | [
"56"
] | [
"What are the last two digits of the number \\((11 + 12 + 13 + \\ldots + 2006)^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-de47c8fcd63f666e3dbd7e275dc6742f019b9f9c8cb1dbc1d61c833326ea0b98 | What are the last two digits of the number \((11^2 + 15^2 + 19^2 + \cdots + 2007^2)^2\)? | [
"00"
] | [
"What are the last two digits of the number \\((11^2 + 15^2 + 19^2 + \\cdots + 2007^2)^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d25eacc3543b4cce8646c697eda29513396603b9d1b48ce5c1fab5ca2840ebf1 | What are the last two digits of the number \((3 + 7 + 11 + \ldots + 2007)^2\)? | [
"00"
] | [
"What are the last two digits of the number \\((3 + 7 + 11 + \\ldots + 2007)^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bdacd539005ace53a9d896f4009093ed1062c525076412ae9824b02fb92fdbf7 | What are the last two digits of the number \(1^{81} + 2^{81} + 3^{81} + \cdots + 100^{81}\)? | [
"00"
] | [
"What are the last two digits of the number \\(1^{81} + 2^{81} + 3^{81} + \\cdots + 100^{81}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-e0206e0aa8ad49b5157f44acf4e33d7988a649cf3b84d79bd236195b2cdf2f5d | What are the last two digits of the product $3 \times 7 \times 11 \times \cdots \times 95 \times 99$? | [
"75"
] | [
"What are the last two digits of the product $3 \\times 7 \\times 11 \\times \\cdots \\times 95 \\times 99$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-2e427ec07c9f7c7da43416a212947981068053226110b21c1d18d0d23c079764 | What are the last two digits of the product \( P = (13)(17)(23) \cdots (97) \), where \( P \) is the product of all two-digit numbers ending with either a 3 or a 7? | [
"81"
] | [
"What are the last two digits of the product \\( P = (13)(17)(23) \\cdots (97) \\), where \\( P \\) is the product of all two-digit numbers ending with either a 3 or a 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-c6c50dc5753d19b8886149e0c6114cebde5c77f4732f6b64fe9edee0bf0a0b94 | What are the last two digits of the simplified value of $1! + 3! + 5! + \cdots + 2009! + 2011!$? | [
"47"
] | [
"What are the last two digits of the simplified value of $1! + 3! + 5! + \\cdots + 2009! + 2011!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-45f5356101e9f5c551ae5e99ca9850c2749d6f45a9724ab399f5e2dbcb3ca634 | What are the last two digits of the sum \(2005^{11} + 2005^{12} + \ldots + 2005^{2006}\)? | [
"00"
] | [
"What are the last two digits of the sum \\(2005^{11} + 2005^{12} + \\ldots + 2005^{2006}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c17a522e6289b730c6736b5dc304cf9ac853442ddfdaa7ac26a51bbd5c2ed81a | What are the last two nonzero digits of \(90!\)? | [
"12"
] | [
"What are the last two nonzero digits of \\(90!\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 25,
"genselect": 3
} | 28 | 0 | 0 |
omr-ws-239d990f8a0b1b1bdff9db99d31c388c1124244a0b7c68bdb71d9d0b0520fd7e | What are the last two nonzero digits of \(90!\)? Let this number be \(n\). What is \(n\)? | [
"12"
] | [
"What are the last two nonzero digits of \\(90!\\)? Let this number be \\(n\\). What is \\(n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-d3bc510120088820980019c6f0271c6308ea4e80451c25f7de05033ec186483d | What are the least and greatest number of regions into which a triangle and a square can divide a piece of paper? | [
"Least = 3, Greatest = 8"
] | [
"What are the least and greatest number of regions into which a triangle and a square can divide a piece of paper?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-a8eac43be5a846a9b8be239d4984ae14b089d22ae1f2e54d35a20d258ba3efda | What are the lengths of the sides of the triangle with the least area such that all of its sides and all of its altitudes are integer lengths? | [
"15, 20, 25"
] | [
"What are the lengths of the sides of the triangle with the least area such that all of its sides and all of its altitudes are integer lengths?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 29
} | 30 | 0 | 0 |
omr-ws-64b0f116063518cec2de862d702e8abe1aa8cb7186581a4d600b02e11c71341a | What are the limit points for the set \( A = \{1, 2, 3\} \) with the set of integers equipped with the cofinite topology? | [
"\\( 1, 2, 3 \\)"
] | [
"What are the limit points for the set \\( A = \\{1, 2, 3\\} \\) with the set of integers equipped with the cofinite topology?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f015b4b7ae4f8938d61766b71161f7a6fd2e4f904247e52c599858e79a4c3fe9 | What are the limitations on the value of \( k \) such that the polynomial \( x^4 + 4x^3 + 4x^2 + k \) has 4 real roots? | [
"-1 ≤ k ≤ 0"
] | [
"What are the limitations on the value of \\( k \\) such that the polynomial \\( x^4 + 4x^3 + 4x^2 + k \\) has 4 real roots?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-ef6298ab34b5cda54149a41f4086d842c21af9d4b93880bbe61371ae8e0834e2 | What are the linear maps $f: M_n(K) \to M_n(K)$ such that if $A$ and $B$ commute, then $f(A)$ and $f(B)$ also commute? | [
"f(X) = c \\cdot PX^T P^{-1} + \\varphi(X) I"
] | [
"What are the linear maps $f: M_n(K) \\to M_n(K)$ such that if $A$ and $B$ commute, then $f(A)$ and $f(B)$ also commute?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-533b909060a890217021538d98159a155f3e8be0ed826e3549d3c81e7e0f1574 | What are the linear maps \( f: M_n(K) \to M_n(K) \) such that if \( f(A) \) and \( f(B) \) commute, then \( A \) and \( B \) also commute? | [
"f(A) = c P A P^{-1} \\text{ or } f(A) = c P A^\\top P^{-1}"
] | [
"What are the linear maps \\( f: M_n(K) \\to M_n(K) \\) such that if \\( f(A) \\) and \\( f(B) \\) commute, then \\( A \\) and \\( B \\) also commute?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-f10beea9872512ae211cd459298a4a8c0621ff2f64b81ee25568e06c77884d54 | What are the lower and upper deciles of the standard normal distribution? | [
"1.28"
] | [
"What are the lower and upper deciles of the standard normal distribution?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-c312c08b1740aecbb1a711d849032ac81c213e6e006f3fce0a6c87ba99689e8c | What are the masses of the individual hippos, given the masses of the pairs are 312 kg, 356 kg, 378 kg, 444 kg, and 466 kg? | [
"\\( m_{1} = 112 \\) kg, \\( m_{2} = 200 \\) kg, \\( m_{3} = 244 \\) kg, \\( m_{4} = 266 \\) kg"
] | [
"What are the masses of the individual hippos, given the masses of the pairs are 312 kg, 356 kg, 378 kg, 444 kg, and 466 kg?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-4ef2a718678cb1a0673b682559e80babcf23d71b81e31ed555dbf21de71dbcde | What are the maximal and minimal values of the expression $\frac{|a+b|+|a+c|+|b+c|}{|a|+|b|+|c|}$, where $a$, $b$, and $c$ are arbitrary real numbers (not all zero)? | [
"\\(\\frac{2}{3}\\) and \\(2\\)"
] | [
"What are the maximal and minimal values of the expression $\\frac{|a+b|+|a+c|+|b+c|}{|a|+|b|+|c|}$, where $a$, $b$, and $c$ are arbitrary real numbers (not all zero)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-3a7c4a6b1f7e6b6c5717d77a6bea2c3695262496b269911ad2822e5ee5de413d | What are the maximal commutative subalgebras (for the inclusion) of $\mathcal{M}_n(\mathbb{C})$? | [
"\\text{All maximal commutative subalgebras are conjugate to the diagonal matrices or to the upper triangular Toeplitz matrices.}"
] | [
"What are the maximal commutative subalgebras (for the inclusion) of $\\mathcal{M}_n(\\mathbb{C})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-628f570590fb7e16965c2ade436f850177f5dc6f565b6158074ff92446453182 | What are the maximal ideals in $\mathbb{Z}[x]$? | [
"\\((p, F)\\) where \\(p\\) is a prime number in \\(\\mathbb{Z}\\) and \\(F\\) is an irreducible polynomial in \\(\\mathbb{Z}_p[X]\\)."
] | [
"What are the maximal ideals in $\\mathbb{Z}[x]$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-13241a33472332f9411df60e252b8fca568abd07508ede2bc0b4468365c5f23b | What are the maximal ideals in \( \mathbb{Z}[x] \)? | [
"Maximal ideals in \\( \\mathbb{Z}[x] \\) are generated by a prime \\( p \\) and an irreducible polynomial \\( f \\) modulo \\( p \\)."
] | [
"What are the maximal ideals in \\( \\mathbb{Z}[x] \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-a43567d9a9bb696d8e425069cf6bcb27469d10967c672c602e8c2b33170812de | What are the maximal ideals of the ring $ R$ of continuous real-valued functions on $ [0,1]$? | [
"The maximal ideals of \\( R \\) are the sets of functions that vanish at a particular point \\( x \\in [0,1] \\)."
] | [
"What are the maximal ideals of the ring $ R$ of continuous real-valued functions on $ [0,1]$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-e9f76c86d2694f01afac368befdaed4963a5ee278904e2a5d36dce5b5cb615a2 | What are the maximal ideals of the ring of real-valued continuous functions on $(0,1)$, other than those given by $$\mathcal{M}_{c} := \{ f \in C(0,1) \mid f(c) = 0 \}?$$ | [
"\\mathcal{M}_0 \\text{ and } \\mathcal{M}_1"
] | [
"What are the maximal ideals of the ring of real-valued continuous functions on $(0,1)$, other than those given by\n$$\\mathcal{M}_{c} := \\{ f \\in C(0,1) \\mid f(c) = 0 \\}?$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-17373c8e57b1a71037d3233bd49c6591d46dca4157ff77d221c08c1daf38bc88 | What are the maximally symmetric convex bodies in $\mathbb{R}^n$? Here, a convex body $K$ in $\mathbb{R}^n$ is defined as a compact convex subset with a non-empty interior, and $A(K)$ is the group of affine bijections of $\mathbb{R}^n$ that map $K$ onto itself. A convex body $K$ is maximally symmetric if there is no co... | [
"\\text{Ellipsoids}"
] | [
"What are the maximally symmetric convex bodies in $\\mathbb{R}^n$? \n\nHere, a convex body $K$ in $\\mathbb{R}^n$ is defined as a compact convex subset with a non-empty interior, and $A(K)$ is the group of affine bijections of $\\mathbb{R}^n$ that map $K$ onto itself. A convex body $K$ is maximally symmetric if th... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-98146c9ea0d47e03a12019400ead277c86803f12d3d34ca71e0cd5a1862c0dd3 | What are the maximum and minimum areas of an equilateral triangle inscribed in a unit square? | [
"Maximum area = \\(2\\sqrt3-3\\), Minimum area = \\(\\frac{\\sqrt3}4\\)"
] | [
"What are the maximum and minimum areas of an equilateral triangle inscribed in a unit square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 7
} | 9 | 0 | 0 |
omr-ws-02c1a7041cc2ec5b0d561a13b2ad9e17d4b2748cd9169657c668b4c3961eff29 | What are the maximum and minimum sizes of equivalence relations on the set \([n] = \{1, 2, 3, \ldots, n\}\)? | [
"Minimum size = \\(n\\), Maximum size = \\(n^2\\)"
] | [
"What are the maximum and minimum sizes of equivalence relations on the set \\([n] = \\{1, 2, 3, \\ldots, n\\}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 22
} | 27 | 0 | 0 |
omr-ws-c2ce66b67b17559d67dcf33638336e53a09dac36e1d6a669cb603f8dd22c7b58 | What are the maximum and minimum values of the function \[f(x) = \sin x \cos x \tan x \sec x \csc x \cot x\] for all real $x$ in its domain? | [
"1"
] | [
"What are the maximum and minimum values of the function \\[f(x) = \\sin x \\cos x \\tan x \\sec x \\csc x \\cot x\\] for all real $x$ in its domain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-e3d5753d198d4a30033bf0f4a3a277e3ff8a4a5f3d18056223ad286a6eeb896a | What are the measures of the angles of a triangle with altitudes of 12, 15, and 20? | [
"\\(90^\\circ, \\sin^{-1} \\frac{4}{5}, \\sin^{-1} \\frac{3}{5}\\)"
] | [
"What are the measures of the angles of a triangle with altitudes of 12, 15, and 20?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 28
} | 40 | 0 | 0 |
omr-ws-bbe80760d27623c686fdbb15fe4414a5e400da4a5b6c6cce59cc0962e4f5514b | What are the minimum and maximum values of the expression $\sin x + \cos x + \tan x + \cot x + \sec x + \csc x$? | [
"Minimum value is \\(-\\infty\\), Maximum value is \\(+\\infty\\)."
] | [
"What are the minimum and maximum values of the expression $\\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-2ed1932f47a3b1a412d50cfb22408d81314380c209ed4c7d2c990ed427c909bb | What are the minimum dimensions of a rectangle that can contain a 60° sector of a circle with a diameter of 8? | [
"\\(4 \\times 2\\sqrt{3}\\)"
] | [
"What are the minimum dimensions of a rectangle that can contain a 60° sector of a circle with a diameter of 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 9
} | 17 | 0 | 0 |
omr-ws-7cf75695c51c00e8570cd8d74dde96927209bdef493f7400e391f5e46cff17fb | What are the missing terms in the sequence 1, 2, 4, 6, ___, 12, 16, 18, 22, ___, ___, 36, 40, ___, ___, ___? | [
"10, 28, 30, 46, 52, 58"
] | [
"What are the missing terms in the sequence 1, 2, 4, 6, ___, 12, 16, 18, 22, ___, ___, 36, 40, ___, ___, ___?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f6f7f9aab2fb84cde762f1220689c3b7c3f0bcc2f03a6dfea3b1fb1fba5c4f4 | What are the most economical dimensions for an open cylindrical cup (no top) designed to hold 1000 cm³, assuming there is no waste in the material used? | [
"\\( r = \\frac{10}{\\sqrt[3]{\\pi}}, h = \\frac{10}{\\sqrt[3]{\\pi}} \\)"
] | [
"What are the most economical dimensions for an open cylindrical cup (no top) designed to hold 1000 cm³, assuming there is no waste in the material used?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 28
} | 38 | 0 | 0 |
omr-ws-b10e4792be780e99d7d89712ae9d91ba0f1ed4708c27f6790d60ff7dd9a587b0 | What are the necessary and sufficient conditions for the equation \(a^2 + c^2 = nb^2\) to hold in \(\triangle ABC\) for \(n \ge 3\)? | [
"\\cos B = \\dfrac{n - 1}{n}"
] | [
"What are the necessary and sufficient conditions for the equation \\(a^2 + c^2 = nb^2\\) to hold in \\(\\triangle ABC\\) for \\(n \\ge 3\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-6306abc8b1aa6408856b7123b0ffe89b3efc30ba3400984ea90202bb45bfc0bb | What are the necessary and sufficient conditions for the placement of \( k \) pawns and \( n + k \) dragon kings on an \( n \times n \) board such that no two dragon kings attack each other? | [
"k \\geq 0 \\text{ and } n + 2k \\leq n^2"
] | [
"What are the necessary and sufficient conditions for the placement of \\( k \\) pawns and \\( n + k \\) dragon kings on an \\( n \\times n \\) board such that no two dragon kings attack each other?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0097a20df52c54bf841f3c2cf9da1c91389e2a943a6cf4d2cdd612ac69e523fd | What are the necessary and sufficient conditions on a triangle such that the following method yields the correct result for the third side \(a\) given two sides \(b\) and \(c\) and the included angle \(A\)? The method involves using logarithms as follows: doubling \(\log b\) and \(\log c\), adding these, subtracting th... | [
"\\(A = 90^\\circ\\)"
] | [
"What are the necessary and sufficient conditions on a triangle such that the following method yields the correct result for the third side \\(a\\) given two sides \\(b\\) and \\(c\\) and the included angle \\(A\\)? The method involves using logarithms as follows: doubling \\(\\log b\\) and \\(\\log c\\), adding th... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f6de54406e8a25ea2e7ce4ec4336ac5b2a55e40a3e83e32e5a12b07fc6a2053f | What are the necessary conditions for \( k \) and \( x \) (where \( k \) and \( x \) are natural numbers different from 0) to satisfy the inequality: \[ \ln(x+k) < \ln(x) + \frac{k}{x} \] | [
"\\( k \\) and \\( x \\) are natural numbers (positive integers)."
] | [
"What are the necessary conditions for \\( k \\) and \\( x \\) (where \\( k \\) and \\( x \\) are natural numbers different from 0) to satisfy the inequality:\n\\[ \\ln(x+k) < \\ln(x) + \\frac{k}{x} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 26
} | 44 | 0 | 0 |
omr-ws-9cfba68c2f075e7ac61e042d4723e7214d2aa38aebf740fefe714772075b8fc4 | What are the necessary conditions such that we may write $$\lim_{x\to a} [f(x)]^{g(x)}=\lim_{x\to a} [f(x)]^{\lim_{x\to a} g(x)}$$ assuming that both $\lim_{x\to a} f(x)$ and $\lim_{x\to a} g(x)$ exist? | [
"\\( f(x) > 0 \\) and \\( (\\lim_{x \\to a} f(x), \\lim_{x \\to a} g(x)) \\neq (0, 0) \\)"
] | [
"What are the necessary conditions such that we may write $$\\lim_{x\\to a} [f(x)]^{g(x)}=\\lim_{x\\to a} [f(x)]^{\\lim_{x\\to a} g(x)}$$ assuming that both $\\lim_{x\\to a} f(x)$ and $\\lim_{x\\to a} g(x)$ exist?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-813777873eac5ad5238375ab600031ee39771cbcc1cb9f9b8b5de9d3a8bcedc8 | What are the next 2 numbers in the sequence -1, 0, 1, 0, 1, 2, 3, 8, 25, ..., ...? | [
"5051"
] | [
"What are the next 2 numbers in the sequence -1, 0, 1, 0, 1, 2, 3, 8, 25, ..., ...?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-ae6e29b0768c8c0cc3e970cbe4d48f057ec42609f95e3b0dd841f32ce4582962 | What are the next 3 numbers in the sequence: 1, 11, 21, 1211? | [
"312211, 13112121, 11132112111211"
] | [
"What are the next 3 numbers in the sequence: 1, 11, 21, 1211?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f343a566683da252dcef728a251f0ac425ac4f6f829591233d6fd4cc4340692a | What are the next five numbers in the sequence: 1, 3, 5, 8, 10, 13, 15, 19, 22, 26, 28, 34, 36, 40, 44, ?, ?, ?, ?, ? | [
"64"
] | [
"What are the next five numbers in the sequence: 1, 3, 5, 8, 10, 13, 15, 19, 22, 26, 28, 34, 36, 40, 44, ?, ?, ?, ?, ?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-19d74e17223c93d8d16bdf6757121a57976b5f2baebab09a5148d2bf983b9f8b | What are the next four numbers in the sequence 3, 5, 6, 10, 9, 11, 17, 18? | [
"26, 27, 37, 38"
] | [
"What are the next four numbers in the sequence 3, 5, 6, 10, 9, 11, 17, 18?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e9642efcd2ab9f16e8aaf0a54a802d3d3cfa7e564341f552613f0d0ce589609 | What are the next three numbers in the sequence $0, 2, 11, 20, 22, 101, 110, 112, \ldots$ and why? | [
"121, 200, 202"
] | [
"What are the next three numbers in the sequence $0, 2, 11, 20, 22, 101, 110, 112, \\ldots$ and why?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-84d22a23853e01ca4b9c1e994ed5db6aaebfd5cf4a61b3dd92941f26fdd69644 | What are the next two numbers in the sequence 125, 135, 145, 155, _, _? The answer is not 165 and 175; it is 163 and 168. Explain why. | [
"168"
] | [
"What are the next two numbers in the sequence 125, 135, 145, 155, _, _? The answer is not 165 and 175; it is 163 and 168. Explain why."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a8a02e16c595ae1dfd2bb6a7360c596788bb024f9b8360b27a33488d190f1fe | What are the next two numbers in the sequence: 11, 31, 71, 91, 32, 92, 13, 73, 14, 34, 74, 35, 95, ...? | [
"16, 76"
] | [
"What are the next two numbers in the sequence: 11, 31, 71, 91, 32, 92, 13, 73, 14, 34, 74, 35, 95, ...?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-1e5f5ba946795be64595fdbf7966f1bcba03690e32aa0db99ae4c50a16ced7a3 | What are the next two numbers in the sequence: 3, 5, 8, 13, 20, 31, __, __? | [
"44, 61"
] | [
"What are the next two numbers in the sequence: 3, 5, 8, 13, 20, 31, __, __?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-8ed0ad721864bf971266e59fed57e149dcbb8da41049de81acf7f77a5121e792 | What are the next two numbers in the sequence: 594, 487, 566, 493, 310, 447, ___, ___? | [
"419"
] | [
"What are the next two numbers in the sequence: 594, 487, 566, 493, 310, 447, ___, ___?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-1800f9100775d3c84e9bf59d77ee5bc4236b0560519a41bf32194170f477c434 | What are the nilpotent elements of the ring $\mathbb{Z}/n\mathbb{Z}$? | [
"$\\nu$ such that $r(n) \\mid \\nu$"
] | [
"What are the nilpotent elements of the ring $\\mathbb{Z}/n\\mathbb{Z}$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 21
} | 26 | 0 | 0 |
omr-ws-5cd26bf70beaeb0f2e70a298b7904e46d22f36617a2fc976f5b87dcfe2a64618 | What are the non-empty parts of space such that the intersection with every plane is either an empty set or a circle? (A set reduced to a point is considered a degenerate circle.) | [
"\\text{Spheres and single points}"
] | [
"What are the non-empty parts of space such that the intersection with every plane is either an empty set or a circle? (A set reduced to a point is considered a degenerate circle.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-6b964f4796d0bb5c3b3cf885d49d123dd14490c9088947572d1228b0b62d31f0 | What are the non-permissible values of $\frac{1}{\cot(x)} = 0$? | [
"\\(x = \\frac{\\pi}{2}n\\) for integer values of \\(n\\)."
] | [
"What are the non-permissible values of $\\frac{1}{\\cot(x)} = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 28
} | 50 | 0 | 0 |
omr-ws-80b701e0a0ba4d5879cd883264e65d91c254e5209aaa7e5ded0159bd35ce4b93 | What are the normal subgroups of the group of permutations of the natural numbers, \( S(\mathbb{N}) \)? | [
"\\{1\\}, \\mathrm{Alt}(\\mathbb{N}), \\mathrm{FSym}(\\mathbb{N}), S(\\mathbb{N})"
] | [
"What are the normal subgroups of the group of permutations of the natural numbers, \\( S(\\mathbb{N}) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
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