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-bc90205ca803aa371105380a067ffab9507c4d1578c8ad332867dae4e0f2b45b | What is the ratio of the volume of a regular octahedron formed by joining the centers of adjoining faces of a cube to the volume of the cube? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the ratio of the volume of a regular octahedron formed by joining the centers of adjoining faces of a cube to the volume of the cube?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7aa7c3a226a1dd50ce8bf9789b781ad8ac050c88017120b9e37a25bc9533fd22 | What is the ratio of the volume of a sphere to the maximum volume of a cone inscribed in that sphere? | [
"\\(\\frac{27}{8}\\)"
] | [
"What is the ratio of the volume of a sphere to the maximum volume of a cone inscribed in that sphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-6ef373b312e773659a9b749b22e81e2fa86cd5ded1360b36f27873b95126a3c4 | What is the ratio of width to length of a folded sheet of paper if the rectangle formed when the unfolded sheet is folded in half (so the new longest side is the shortest side of the unfolded paper and the shortest side is half the unfolded longest side) is similar to the unfolded rectangle? | [
"\\(\\sqrt{2}\\)"
] | [
"What is the ratio of width to length of a folded sheet of paper if the rectangle formed when the unfolded sheet is folded in half (so the new longest side is the shortest side of the unfolded paper and the shortest side is half the unfolded longest side) is similar to the unfolded rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 5
} | 7 | 0 | 0 |
omr-ws-67552ce202c7744caf55d5b7cd6c0b7bd5d34d91d50c66b6e7ce232d8d33bfd5 | What is the rational form of $\sqrt{\frac{10+2\sqrt{24}}{11-2\sqrt{30}}}$? | [
"\\(6+2\\sqrt{5}+2\\sqrt{6}+\\sqrt{30}\\)"
] | [
"What is the rational form of $\\sqrt{\\frac{10+2\\sqrt{24}}{11-2\\sqrt{30}}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-e684a38e1a5401bd3ce522a82f3f98bb820509654297ba730933994350c8dfc2 | What is the rational number represented by the infinite sequence $\sqrt{\frac{9}{4} + \sqrt{\frac{9}{4} + \sqrt{\frac{9}{4} + \sqrt{\frac{9}{4} + \cdots}}}}$? | [
"\\(\\frac{1 + \\sqrt{10}}{2}\\)"
] | [
"What is the rational number represented by the infinite sequence $\\sqrt{\\frac{9}{4} + \\sqrt{\\frac{9}{4} + \\sqrt{\\frac{9}{4} + \\sqrt{\\frac{9}{4} + \\cdots}}}}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-f0acd626668ceb53d478772b46b447a9d40a3b56f09c0eb517872c5b6111e0e0 | What is the real part of the conjugate of $4+3i$ multiplied by $i$? | [
"0"
] | [
"What is the real part of the conjugate of $4+3i$ multiplied by $i$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-860f465e69c027c96aaecc8ed14ec4952421d4c3a7a70b33c0e270bf45e83006 | What is the reasonable order of magnitude for the maximal cardinality of a group having $n$ conjugacy classes? | [
"e^{O(n)}"
] | [
"What is the reasonable order of magnitude for the maximal cardinality of a group having $n$ conjugacy classes?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-30178309b43d8d1c07fbeae0ec334626325a1458433c37e5e765b0b38e7795a7 | What is the reciprocal of the common fraction form of $.\overline{2}$? | [
"\\frac92"
] | [
"What is the reciprocal of the common fraction form of $.\\overline{2}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-164bb229fd68310d744e385659ea3d5845538b324175129f8d4b2784edc62d54 | What is the recursion relation for \( x_1^n + x_2^n + x_3^n \), where \( x_1, x_2, x_3 \) are the roots of a cubic equation? | [
"\\( a_n + ba_{n-1} + ca_{n-2} + da_{n-3} = 0 \\)"
] | [
"What is the recursion relation for \\( x_1^n + x_2^n + x_3^n \\), where \\( x_1, x_2, x_3 \\) are the roots of a cubic equation?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 2
} | 7 | 0 | 0 |
omr-ws-91a9051ac3b1e401383bf934027afa9837af229354f4c5061e9604ef3b99e5ac | What is the recursive relation between the number of all possible melodies until the $(n+1)$-th piano and the number of all possible melodies on the $(n+1)$-th piano? | [
"\\( a_{n+1} = \\sum_{i=1}^{n+1} \\binom{n+1}{i} a_{n+1-i} \\)"
] | [
"What is the recursive relation between the number of all possible melodies until the $(n+1)$-th piano and the number of all possible melodies on the $(n+1)$-th piano?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea622e59512e32d83a01f5064b18ced28bc2c6ffbbc1df93fda17a513eec1c67 | What is the reference angle for an angle \( A = \frac{23\pi}{5} \) radians? | [
"\\( \\frac{3\\pi}{5} \\) radians was initially suggested as the reference angle, but since it is not acute, the correct reference angle is \\( 0.4\\pi \\) radians or \\( \\frac{2\\pi}{5} \\) radians."
] | [
"What is the reference angle for an angle \\( A = \\frac{23\\pi}{5} \\) radians?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-a412b33370744a42d326f6851677118e3f63efe00e8d2be2c78c12646f5195c7 | What is the reflection of the line $\bar{a}z + a\bar{z} = 0$ in the real axis? | [
"\\(\\bar{a}\\bar{z} + a z = 0\\)"
] | [
"What is the reflection of the line $\\bar{a}z + a\\bar{z} = 0$ in the real axis?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-bad9babce24ecc456598bc5a14f4531a916c7a154272d9c72ad3df3623d8642f | What is the reflection of the point (3,6,8) over the line \(\frac{x-1}{2} = \frac{y+2}{3} = \frac{z-3}{4}\)? | [
"\\(\\left\\langle \\frac{163}{29}, \\frac{-2}{29}, \\frac{326}{29} \\right\\rangle\\)"
] | [
"What is the reflection of the point (3,6,8) over the line \\(\\frac{x-1}{2} = \\frac{y+2}{3} = \\frac{z-3}{4}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8c1630aa51f612457443f0038ac988338cd22cd4e80eb19e13afb356c6fe882d | What is the reflection of the vector $\begin{pmatrix}2 \\ 0 \\ 2 \end{pmatrix}$ across a line with direction $\begin{pmatrix}1 \\ -1 \\ 2 \end{pmatrix}$? | [
"\\(\\begin{pmatrix}0 \\\\ -2 \\\\ 2 \\end{pmatrix}\\)"
] | [
"What is the reflection of the vector $\\begin{pmatrix}2 \\\\ 0 \\\\ 2 \\end{pmatrix}$ across a line with direction $\\begin{pmatrix}1 \\\\ -1 \\\\ 2 \\end{pmatrix}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8f649c164c9630d3ae20218f55f9fdc6446e68ea5a2a845fe927308ed42165d5 | What is the relation between $[x+y]^2$ and $([x]+[y])^2$? | [
"[x+y]^2 \\geq ([x]+[y])^2"
] | [
"What is the relation between $[x+y]^2$ and $([x]+[y])^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-502329f36017f4b75a483924bf10962fc82b0433ee98f213865c8acd5da3f942 | What is the relation between \(a_{k+1} + (k-1) \sqrt[k+1]{a_1a_2 \cdots a_{k+1}}\) and \(k\sqrt[k]{a_{k+1} \cdot (\sqrt[k+1]{a_1a_2 \cdots a_{k+1}})^{k-1}}\)? | [
"\\(a_{k+1} + (k-1) \\sqrt[k+1]{a_1a_2 \\cdots a_{k+1}} \\geq k \\sqrt[k]{a_{k+1} \\cdot (\\sqrt[k+1]{a_1a_2 \\cdots a_{k+1}})^{k-1}}\\)"
] | [
"What is the relation between \\(a_{k+1} + (k-1) \\sqrt[k+1]{a_1a_2 \\cdots a_{k+1}}\\) and \\(k\\sqrt[k]{a_{k+1} \\cdot (\\sqrt[k+1]{a_1a_2 \\cdots a_{k+1}})^{k-1}}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-4c396e73ee7f45a2ea73226a5bfdc105b7d3ddbcef5022e5331d33d755303883 | What is the relation between the area \(A\) of a triangle and the radius \(r\) of its incircle or the radius \(R\) of its excircle? | [
"\\( A = rs \\) and \\( A = r_A(s-a) = r_B(s-b) = r_C(s-c) \\)"
] | [
"What is the relation between the area \\(A\\) of a triangle and the radius \\(r\\) of its incircle or the radius \\(R\\) of its excircle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-181c3089baf32dad8579f2835ed86b9bc483242442f403c855732111c0e53f06 | What is the relation between the integers $a$ and $b$ if the number of perfect cubes between $0$ and $a$ is the same as the number of perfect cubes between $a$ and $b$? | [
"\\(\\left\\lceil\\sqrt[3]{a}\\right\\rceil + \\left\\lfloor\\sqrt[3]{a}\\right\\rfloor = \\left\\lceil\\sqrt[3]{b}\\right\\rceil\\)"
] | [
"What is the relation between the integers $a$ and $b$ if the number of perfect cubes between $0$ and $a$ is the same as the number of perfect cubes between $a$ and $b$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c5eae67a85c26cad3d4951fbdf318e8e5b4992d17d26dde4a27e7240510d673 | What is the relation between the radius \(r\) of the incircle and the sides \(a, b, c\) of a triangle? | [
"\\( rs = \\sqrt{s(s-a)(s-b)(s-c)} \\)"
] | [
"What is the relation between the radius \\(r\\) of the incircle and the sides \\(a, b, c\\) of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-f126d6488a7d6df81d5860d8e350390a5ee295025722dcb7cad55e52d4d40960 | What is the relation given by the equation $ \cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1 $? | [
"\\( A + B + C = \\pi \\)"
] | [
"What is the relation given by the equation $ \\cos^2A+\\cos^2B+\\cos^2C+2\\cos A\\cos B\\cos C=1 $?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-0443e55c39a197407057b2b8f89da8e74f3ad32632be285a28d7cd0a4198ef47 | What is the relationship between \( x \) and \( y \) if \[ x(y - x) \, dy = y^2 \, dx \]? | [
"\\(-2\\ln \\left( \\frac{y}{x} \\right) + \\left( \\frac{y}{x} \\right) = k\\)"
] | [
"What is the relationship between \\( x \\) and \\( y \\) if\n\\[ x(y - x) \\, dy = y^2 \\, dx \\]?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a5d68530067cb5386ae7b8bce1f29c31cba91a7469881534e51b6350fa1cbe93 | What is the relationship between the critical points of $f$ and the critical points of $f^2$? | [
"\\text{The critical points of } f^2 \\text{ are all critical points of } f \\text{ and all zeros of } f."
] | [
"What is the relationship between the critical points of $f$ and the critical points of $f^2$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70c5ec8dc960516f664a22a866d7fb4b0c946c79de1897bcda1078cbfb18df9d | What is the relationship between the function \( f(z) = \sum_{\nu=1}^\infty \left( \frac{\sqrt{\nu}}{z-\nu} + \frac{1}{\sqrt{\nu}} \right) \) and the gamma function? | [
"\\( f(z) \\leq \\gamma + \\frac{\\Gamma'(1-z)}{\\Gamma(1-z)} \\) for \\( 0 < z < 1 \\)"
] | [
"What is the relationship between the function \\( f(z) = \\sum_{\\nu=1}^\\infty \\left( \\frac{\\sqrt{\\nu}}{z-\\nu} + \\frac{1}{\\sqrt{\\nu}} \\right) \\) and the gamma function?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78bf0aa8393816cb766622372923a597c9c36802bc30b415f84ea5066b996f5f | What is the relationship between the perimeters of regular polygons with a fixed area as the number of sides increases? | [
"The perimeter increases as the number of sides increases."
] | [
"What is the relationship between the perimeters of regular polygons with a fixed area as the number of sides increases?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62c708a4b0e9c141c044ef8fd1800f745b609ce4c82df2ca30a19b95ce8f65e6 | What is the relationship between the sets $S=\{15a+9b+3c\}$ and $T=\{7a+15b+9c\}$, where $a$, $b$, and $c$ are integers? | [
"\\( S \\subset T \\) and there is a bijection from \\( S \\) to \\( T \\) given by \\( x \\rightarrow \\frac{x}{3} \\)."
] | [
"What is the relationship between the sets $S=\\{15a+9b+3c\\}$ and $T=\\{7a+15b+9c\\}$, where $a$, $b$, and $c$ are integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-c673d74f4c77a942014359f5a872958eace5868fe7324ab6766da22526f6281b | What is the relationship between the solutions of the differential equations $\ddot{y} + y = e^{-x}\sin^2 x$ and $\ddot{y} + y = e^{x}\sin^2 x$? | [
"If \\( y(x) \\) is a solution to \\( \\ddot{y} + y = e^{-x}\\sin^2 x \\), then \\( y(-x) \\) is a solution to \\( \\ddot{y} + y = e^{x}\\sin^2 x \\)."
] | [
"What is the relationship between the solutions of the differential equations $\\ddot{y} + y = e^{-x}\\sin^2 x$ and $\\ddot{y} + y = e^{x}\\sin^2 x$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-e9933c2f4c03589df5aac9a6ec1399c024219cd14009f194695ebe11ea1cdf87 | What is the remainder if $10^{2018}$ is divided by $999$? | [
"100"
] | [
"What is the remainder if $10^{2018}$ is divided by $999$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-f2944828605d002adc1e7f2fa45f59c7b468abaab41e17a3f7192185cfadb646 | What is the remainder of $10^{2017}$ when divided by $42$? | [
"10"
] | [
"What is the remainder of $10^{2017}$ when divided by $42$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42c004ed5cad878887ed3ec479905c860dc6608fbfe96aa26d58b29f9d4a19c2 | What is the remainder of $17^{2019}$ when divided by $12$? | [
"5"
] | [
"What is the remainder of $17^{2019}$ when divided by $12$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b1f65908ee0c9cef420f892015192f2a7fb4a11624715916bdf9eaba0c7c76d | What is the remainder of $17^{76} - 8^{76} + 1 - 9^{76}$ when divided by 72? | [
"1"
] | [
"What is the remainder of $17^{76} - 8^{76} + 1 - 9^{76}$ when divided by 72?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73eeb21a3075766e450c4730571a42c8a442793488422da76179dba7429f5e79 | What is the remainder of $19^{1999}$ divided by 25? | [
"4"
] | [
"What is the remainder of $19^{1999}$ divided by 25?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-361f8bba0b4d1ae7e5800a4a615ef383e4de5e20c0b49d2f9817b5f051e3a0f6 | What is the remainder of $1^{32} + 2^{32} + 3^{32} + \ldots + 2024^{32}$ when divided by 32? | [
"20"
] | [
"What is the remainder of $1^{32} + 2^{32} + 3^{32} + \\ldots + 2024^{32}$ when divided by 32?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f30193fbea7b63805391fc7a74d66da206cb0169c68309136f56db5bf456ab47 | What is the remainder of $5^{2010}$ when it is divided by 7? | [
"1"
] | [
"What is the remainder of $5^{2010}$ when it is divided by 7?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ab58e556d5db659f8d328c5effed4b0ed0cb96968fb94d84be69f58361d1961 | What is the remainder of $8^6 + 7^7 + 6^8$ divided by 5? | [
"3"
] | [
"What is the remainder of $8^6 + 7^7 + 6^8$ divided by 5?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9016ea0ddb9a794bafdf311df884fa306682fb056e7c0f49de884398e8f76647 | What is the remainder of $8^6+7^7+6^8$ when divided by 5? | [
"3"
] | [
"What is the remainder of $8^6+7^7+6^8$ when divided by 5?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9fd838a59b66dd576e29af3989a1045df2a48d75ebd1407d6e5b8147fcd64d5 | What is the remainder of $\frac{100!}{1}+\frac{100!}{2}+\frac{100!}{3}+...+\frac{100!}{100}$ when divided by $101$? | [
"0"
] | [
"What is the remainder of $\\frac{100!}{1}+\\frac{100!}{2}+\\frac{100!}{3}+...+\\frac{100!}{100}$ when divided by $101$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9732863298a7e472bcfd0c3757be43316f815bc1c3144661cf935c95a270474a | What is the remainder of $\frac{27}{4} \div 9$? | [
"6"
] | [
"What is the remainder of $\\frac{27}{4} \\div 9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-e14312b351e68da48a15155ad42ffa175c06c4e8ddd1d48d0539d203da4d1f9e | What is the remainder of ${2016^{2017}}^{2018}$ when divided by 11? | [
"4"
] | [
"What is the remainder of ${2016^{2017}}^{2018}$ when divided by 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-64c152223829bb67d23c192a2e2ad5a94492bce2a03c8b2f3ce8818af649c23e | What is the remainder of \( 2012^{2012} + 2014^{2012} \) when divided by \( 2013^2 \)? | [
"2"
] | [
"What is the remainder of \\( 2012^{2012} + 2014^{2012} \\) when divided by \\( 2013^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7001d0f8745301d4f3ee933457800b227fb7100dc3d03630ab9392aa197a49a2 | What is the remainder of \( f(2) \) when \( f(n) = n^{99} + 2n^{98} + 3n^{97} + \cdots + 98n^2 + 99n + 100 \) is divided by 100? | [
"51"
] | [
"What is the remainder of \\( f(2) \\) when \\( f(n) = n^{99} + 2n^{98} + 3n^{97} + \\cdots + 98n^2 + 99n + 100 \\) is divided by 100?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dcfb9af5cfc4795c85850942139e719fb589c1e985524203d75ef37a75c1e674 | What is the remainder of \(1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \ldots + 1000 \cdot 1000!\) when divided by 2002? | [
"2001"
] | [
"What is the remainder of \\(1 \\cdot 1! + 2 \\cdot 2! + 3 \\cdot 3! + \\ldots + 1000 \\cdot 1000!\\) when divided by 2002?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c17ddc30566d14ab97cf1d910574a84fdfe7dc94ef9abcb68aaa2fa270b10c1d | What is the remainder of \(10^{2024!}\) when divided by 2024? | [
"760"
] | [
"What is the remainder of \\(10^{2024!}\\) when divided by 2024?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-9501fe431895e68bfbfb359bfb4e1edaa197073bfb1546bcd3a15f6dc424f7ad | What is the remainder of \(2008^{526}\) when divided by 1005? | [
"754"
] | [
"What is the remainder of \\(2008^{526}\\) when divided by 1005?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-57ba64966ef170bbe365ba11965cf6ccd494e4dc1c953785d0061291f50a00bf | What is the remainder of \(2^{1396}\) when divided by \(3^{100}\)? | [
"1"
] | [
"What is the remainder of \\(2^{1396}\\) when divided by \\(3^{100}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-9890f91997168a56856a5444cf9db4ad907d1d3a7fdbb7335d0102ddce63348b | What is the remainder of \(7^{19}\) when divided by 33? | [
"19"
] | [
"What is the remainder of \\(7^{19}\\) when divided by 33?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-186d1ce0ffb227a12dae855a8000248c9e02fb3d3e80723c2416f5f44cb7fb4f | What is the remainder of \(\frac{x+y}{1000}\) if \(2024^x + 4049 = |2024 - y| + y\)? | [
"37"
] | [
"What is the remainder of \\(\\frac{x+y}{1000}\\) if \\(2024^x + 4049 = |2024 - y| + y\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-d56a844df34c118b7cbd70a3f283b92c299deeae62e766bc75022c5c6ff342ed | What is the remainder of the division $1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \ldots + 99 \cdot 99! + 100 \cdot 100!$ by $101$? | [
"100"
] | [
"What is the remainder of the division $1 \\cdot 1! + 2 \\cdot 2! + 3 \\cdot 3! + \\ldots + 99 \\cdot 99! + 100 \\cdot 100!$ by $101$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-38cd525c80bd685b43a197d7d79c5f8070a70d0d93280a0085f472b6ffb50db2 | What is the remainder of the division between \( x^{2021} - 1 \) and \( x^4 + x^3 + 2x^2 + x + 1 \)? | [
"\\(-x^3 + x^2\\)"
] | [
"What is the remainder of the division between \\( x^{2021} - 1 \\) and \\( x^4 + x^3 + 2x^2 + x + 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-b3eee2227d3871d790fb39e3ccec7a2e33b6e78243607b22bf6970dc22a9c4cf | What is the remainder of the division of \( 43^{43^{43}} \) by 100? | [
"7"
] | [
"What is the remainder of the division of \\( 43^{43^{43}} \\) by 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a93121b208ca3b2e3165862c565cd284eda462cb2dad64519a40811c3eccb6d8 | What is the remainder of the number $1 \binom{2008}{0} + 2\binom{2008}{1} + \cdots + 2009\binom{2008}{2008}$ when divided by $2008$? | [
"256"
] | [
"What is the remainder of the number $1 \\binom{2008}{0} + 2\\binom{2008}{1} + \\cdots + 2009\\binom{2008}{2008}$ when divided by $2008$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-b9c0717440ae1b0e2368b8c238b7f283f2464e2727a6d37530d253adc85dd781 | What is the remainder when $$\sum_{k=0}^{62}{2014\choose k}$$ is divided by $2017$? | [
"1024"
] | [
"What is the remainder when $$\\sum_{k=0}^{62}{2014\\choose k}$$ is divided by $2017$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 23
} | 46 | 0 | 0 |
omr-ws-fab2989242532b66340ea629e5c9421234d4fdd2f5066452549ca7c061c9419d | What is the remainder when $(1 + x)^{2010}$ is divided by $1 + x + x^2$? | [
"1"
] | [
"What is the remainder when $(1 + x)^{2010}$ is divided by $1 + x + x^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54272a16d276fc0239ef0f1b14ff9e684ed27c36a4782ef37f671f6b127cdac3 | What is the remainder when $(1! + 2! + 3! + 4! + \ldots + 2006! + 2007!)$ is divided by $7$? | [
"5"
] | [
"What is the remainder when $(1! + 2! + 3! + 4! + \\ldots + 2006! + 2007!)$ is divided by $7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-e2fd821f3fcc59ad748ed347fcac01ff5ce3814e137177cd8b7102c8c39ae32e | What is the remainder when $(1)(1+2)(1+2+3) \ldots (1+2+\cdots+45)$ is divided by 47? | [
"45"
] | [
"What is the remainder when $(1)(1+2)(1+2+3) \\ldots (1+2+\\cdots+45)$ is divided by 47?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5b856053ea63b4254884a9bd0cccf7547c9d599e87b014723f9c204240939cbf | What is the remainder when $(1+x)^{2014}$ is divided by $x^2 + x + 1$? | [
"\\(-x - 1\\)"
] | [
"What is the remainder when $(1+x)^{2014}$ is divided by $x^2 + x + 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-01ae1c3c9628124a760b48275389ac8c543c09ee0ba3597bff046ddfb530bcd0 | What is the remainder when $(10^{18} + 11^{18} + 12^{18} + 13^{18} + 14^{18} + 15^{18})$ is divided by 119? | [
"54"
] | [
"What is the remainder when $(10^{18} + 11^{18} + 12^{18} + 13^{18} + 14^{18} + 15^{18})$ is divided by 119?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-1871a1e68a2b6be892713c85419ecc7f45bcdb2ba860febca3cfd33454e934a3 | What is the remainder when $(14)^{40}$ is divided by $100$? | [
"76"
] | [
"What is the remainder when $(14)^{40}$ is divided by $100$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-a6d04257308915b68e30f9449a2c5a04693829c3c97a8ecb6710a8f27806e13e | What is the remainder when $(2183)^{2156}$ is divided by $100$? | [
"81"
] | [
"What is the remainder when $(2183)^{2156}$ is divided by $100$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-750c2936ce68319e42dc352d3e32ac97463dd0902197236b831e1f751f80fd46 | What is the remainder when $(257^{33} + 46)^{26}$ is divided by 50? | [
"29"
] | [
"What is the remainder when $(257^{33} + 46)^{26}$ is divided by 50?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-005410cb7b56029db7bcc03c468bee42b3d66a662be63e6567ee16f8dd9ec693 | What is the remainder when $(4(7^{23}) + 3^{11})$ is divided by $101$? | [
"0"
] | [
"What is the remainder when $(4(7^{23}) + 3^{11})$ is divided by $101$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ddb0f5167553ad28e0a04522d3a024d59a9ebc284a842151ddfe31b655bebd6 | What is the remainder when $(x + 1)^{2010}$ is divided by $x^2 + x + 1$? | [
"1"
] | [
"What is the remainder when $(x + 1)^{2010}$ is divided by $x^2 + x + 1$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e17a123dea1eddcffb0493be2e99d3254f3179781ec5358bdb739f3c0c4f6d8 | What is the remainder when $(x+1)^n$ is divided by $(x-1)^3$? | [
"\\( n(n-1) \\cdot 2^{n-3} (x-1)^2 + n \\cdot 2^{n-1} (x-1) + 2^n \\)"
] | [
"What is the remainder when $(x+1)^n$ is divided by $(x-1)^3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e66b3f3275ab5f239382e1bff31709cc03d041db006c3f058adc7fb0da144ffb | What is the remainder when $(x+1)^n$ is divided by $(x-1)^4$? | [
"\\(2^n + n2^{n-1}(x-1) + \\binom{n}{2}2^{n-2}(x-1)^2 + \\binom{n}{3}2^{n-3}(x-1)^3\\)"
] | [
"What is the remainder when $(x+1)^n$ is divided by $(x-1)^4$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-8e575b9aef0066eff88b0e38da46bc6d95c96e1196089994a169d89a72a7a6a3 | What is the remainder when $(x+1)^{2016}$ is divided by $x^2 + x + 1$? | [
"1"
] | [
"What is the remainder when $(x+1)^{2016}$ is divided by $x^2 + x + 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c8787254c85b7964728491a4b44c1872fa970e7a3cdb6fbda2709b302c7138a9 | What is the remainder when $1 + 2 + 3 + 4 + \dots + 9 + 10$ is divided by 8? | [
"7"
] | [
"What is the remainder when $1 + 2 + 3 + 4 + \\dots + 9 + 10$ is divided by 8?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-afbc84d3d5ed0f88c4ef5a7e922efbe9e7f6d11f12ab6d621284ece4fb4bd02c | What is the remainder when $1! + 2! + 3! + 4! + \cdots + 45!$ is divided by 47? | [
"18"
] | [
"What is the remainder when $1! + 2! + 3! + 4! + \\cdots + 45!$ is divided by 47?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0237964f64279717f618848c992ba465cc287dc142867ded648d449e550bc81c | What is the remainder when $1! + 2! + 3! + \ldots + 2012!$ is divided by 7? | [
"5"
] | [
"What is the remainder when $1! + 2! + 3! + \\ldots + 2012!$ is divided by 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-57993504025232aa0e4e8f54560174d698109c6d5d69852b93d967cacb3a8eb2 | What is the remainder when $1! + 2! + \cdots + 100!$ is divided by 18? | [
"9"
] | [
"What is the remainder when $1! + 2! + \\cdots + 100!$ is divided by 18?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82613b3665a31dd8a6dbd674da021eee21cef0d8624722a1381cc98e7fc2ec71 | 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": 47,
"tir": 20
} | 67 | 0 | 0 |
omr-ws-e37926e3d1df3139b2eb85cb3dc5e1ada34039743c21bfd38d0435a66063d75b | 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": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-ef97d9351c116811e7e33dcce78c59cb444c35d6d319abee6095607db623fc58 | What is the remainder when $1000!$ is divided by $2003$? | [
"2"
] | [
"What is the remainder when $1000!$ is divided by $2003$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-f96e52bdd3d4c93af0225c77dad6e37222a0ff0dc39c06265c850bf4a207670c | What is the remainder when $10^{24}$ is divided by 9997? | [
"729"
] | [
"What is the remainder when $10^{24}$ is divided by 9997?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-474f8087db23a4c685d7450ac01365f11adce02fe2cc7ebf490c4762154fa31f | What is the remainder when $11065+11067+11069+11071+11073+11075+11077$ is divided by $14$? | [
"7"
] | [
"What is the remainder when $11065+11067+11069+11071+11073+11075+11077$ is divided by $14$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dbf37a8da91713443644340f806cadb1c1f4ba9c33d6a74f4bd8e602478fa8d6 | What is the remainder when $11^{11^{11111}}$ is divided by $25$? | [
"11"
] | [
"What is the remainder when $11^{11^{11111}}$ is divided by $25$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9be8eca1104e39868f8a02255535689a12acf5aaf237c8c74063546054470702 | What is the remainder when $11^{2019}+13^{2019}$ is divided by $144$? | [
"0"
] | [
"What is the remainder when $11^{2019}+13^{2019}$ is divided by $144$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27780d75775a2c2d44f26ee9e7cf1bbc4aaf11772d269667588daba01581e139 | What is the remainder when $12 \cdot 11 \cdot 10 \cdot \ldots \cdot 3 \cdot 2 \cdot 2 + 14$ is divided by $13$? | [
"12"
] | [
"What is the remainder when $12 \\cdot 11 \\cdot 10 \\cdot \\ldots \\cdot 3 \\cdot 2 \\cdot 2 + 14$ is divided by $13$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-86a5814a119a5c62009e816c8480c0c5c6a66ddc33cd8e32b405eb302cbf43be | What is the remainder when $121^{103}$ is divided by 101? | [
"21"
] | [
"What is the remainder when $121^{103}$ is divided by 101?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-147eb7a111f068eab36897f0b58c8a037102a3ca862142af676b8e71fce57283 | What is the remainder when $1234567890$ is divided by $99$? | [
"72"
] | [
"What is the remainder when $1234567890$ is divided by $99$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-644a702f56af5b98fd10cffb0618b94586d4f3ed14f586e955467ec9b1c1d60b | What is the remainder when $128^{2023}$ is divided by $126$? | [
"2"
] | [
"What is the remainder when $128^{2023}$ is divided by $126$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-302c9d3b926b9aabfd5cc3b59f6f68c68fbc38a670c6e3ca64c4e48ee66768cf | What is the remainder when $12^9$ is divided by 1000? | [
"352"
] | [
"What is the remainder when $12^9$ is divided by 1000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-cfbdf5bcb90702eeeaafb158015a3bcd635ae906d82e7a4e62fdd1a766960189 | What is the remainder when $13 \cdot 12 \cdot \ldots \cdot 2 \cdot 1$ is divided by $169$? | [
"156"
] | [
"What is the remainder when $13 \\cdot 12 \\cdot \\ldots \\cdot 2 \\cdot 1$ is divided by $169$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-00b9613eecb5e2a4b9122b874fc1cab66e71e95a946072344ec48cdccc0de79a | What is the remainder when $13^{51}$ is divided by 5? | [
"2"
] | [
"What is the remainder when $13^{51}$ is divided by 5?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4065f1553396aadddf2d484371019b783d7e8d72ed20dbd097bc70ef5abc5c9c | What is the remainder when $13^{99} + 19^{93}$ is divided by $81$? | [
"29"
] | [
"What is the remainder when $13^{99} + 19^{93}$ is divided by $81$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-6d6b513ceb5202ce91dcc61ae0f224870519abd382e6571507fcadb450d91300 | What is the remainder when $13^{99}$ is divided by $81$? | [
"55"
] | [
"What is the remainder when $13^{99}$ is divided by $81$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-a9f658a7838665eba8c6dd81fa904cf8b5bf1d962aa5e4c4074214af20a7da4d | What is the remainder when $1492\cdot 1999$ is divided by $500$? | [
"8"
] | [
"What is the remainder when $1492\\cdot 1999$ is divided by $500$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7629ef3009e1890421ba12ab6aa63026e5066edf9e98a5f79f72ef33e00c3396 | What is the remainder when $15^{17} + 17^{15}$ is divided by 32? | [
"0"
] | [
"What is the remainder when $15^{17} + 17^{15}$ is divided by 32?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-4a92996e557f61549f742a17d6319f527c084abaf10fcd6a6c57f4e357e75e55 | What is the remainder when $16^{101} + 8^{101} + 4^{101} + 2^{101} + 1$ is divided by $2^{100} + 1$? | [
"11"
] | [
"What is the remainder when $16^{101} + 8^{101} + 4^{101} + 2^{101} + 1$ is divided by $2^{100} + 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cec26a1a9082e25456314405c0b57442c53428ebbbae5966998aa0b6fd97fbf6 | 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": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-10fd917cc95975199b4444158f3b7e7dbddebcfe67c30787a047229a0a62349e | What is the remainder when $1776^{2011!}$ is divided by $2000$? | [
"1376"
] | [
"What is the remainder when $1776^{2011!}$ is divided by $2000$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-11997d98924382d8645c61ca886bb10fe94ac5734bc17940d40e14fb5a88ec9d | What is the remainder when $19^{92}$ is divided by 92? | [
"49"
] | [
"What is the remainder when $19^{92}$ is divided by 92?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-84042548c5081bba8997726b6d83d32917845bd6bf0a180b0b4cf630aa69be07 | What is the remainder when $1^1 + 2^2 + 3^3 + 4^4 + \cdots + (31!)^{(31!)}$ is divided by $31$? | [
"30"
] | [
"What is the remainder when $1^1 + 2^2 + 3^3 + 4^4 + \\cdots + (31!)^{(31!)}$ is divided by $31$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-fc4e48ea10a2873335f12c9b4d68470607731b284123514eca31f50111831f4d | What is the remainder when $1^1 + 2^2 + \cdots + 23^{23}$ is divided by 10? | [
"2"
] | [
"What is the remainder when $1^1 + 2^2 + \\cdots + 23^{23}$ is divided by 10?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-24935e657e249e8efb01331f71aaf641136feefcc1cc4eb78f6ea27dd5909f41 | What is the remainder when $1^2 \cdot 3^2 \cdot \ldots \cdot 55^2 \cdot 57^2$ is divided by $59$? | [
"1"
] | [
"What is the remainder when $1^2 \\cdot 3^2 \\cdot \\ldots \\cdot 55^2 \\cdot 57^2$ is divided by $59$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-26c7ac358e4bb847d375f943826f935a2a9d628ba11e67739bb375884ef3565d | What is the remainder when $1^2 \times 2! + 2^2 \times 3! + 3^2 \times 4! + \ldots + 2025^2 \times 2026!$ is divided by $2017$? | [
"2"
] | [
"What is the remainder when $1^2 \\times 2! + 2^2 \\times 3! + 3^2 \\times 4! + \\ldots + 2025^2 \\times 2026!$ is divided by $2017$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-10fb0541b40759f907e8a204161fcab1dd39d4bcfb8bca1653c0ff55d5d77b3b | What is the remainder when $1^6 + 2^6 + \cdots + 2012^6$ is divided by 41? | [
"15"
] | [
"What is the remainder when $1^6 + 2^6 + \\cdots + 2012^6$ is divided by 41?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 19
} | 42 | 0 | 0 |
omr-ws-fcea0b549aa950fc5581ab96a0c76e40a9417f1a5961071c7eb8d8dd914f7365 | What is the remainder when $2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005$ is divided by 19? | [
"11"
] | [
"What is the remainder when $2001 \\cdot 2002 \\cdot 2003 \\cdot 2004 \\cdot 2005$ is divided by 19?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa3b571f2cda72aa2076a5faa87e301909f7378c286894b3b458de608573f5d0 | 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": 7
} | 23 | 0 | 0 |
omr-ws-de1691261816f0a88e8a1e7c6a8801cd49437353b98cb8dd829883a5a58fd3f5 | What is the remainder when $2007$ is divided by $81$? | [
"63"
] | [
"What is the remainder when $2007$ is divided by $81$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5f50eb7025898a2a5c5eb78ac27e1779bb41ba04aad8c3eab0674fb3e751ebe | What is the remainder when $2008^{2008}$ is divided by $2009$? | [
"1"
] | [
"What is the remainder when $2008^{2008}$ is divided by $2009$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-86e72e633139f6eae967018cb7a6554ea62e10b1c9857b0441e1f6e5a26f69cf | What is the remainder when $2009^{2009}$ is divided by $11$? | [
"8"
] | [
"What is the remainder when $2009^{2009}$ is divided by $11$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.