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-d358a0a29d073ec92c52a435068302d65e1ffc772fa87120c34b69aa3c0f4464 | What is the fewest number of moves to go from 7 to 6965, if possible moves include adding 7 and multiplying by 7? | [
"13"
] | [
"What is the fewest number of moves to go from 7 to 6965, if possible moves include adding 7 and multiplying by 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 7,
"tir": 14
} | 25 | 0 | 0 |
omr-ws-9d3df0d942eab18364e1b4850347f92676ca58c4289dfe722fb5b5fe77aeaf59 | What is the fewest number of pieces that two squares, each with side length 1, need to be cut into so that the pieces will all fit, without overlapping, into a circle of radius 1? | [
"3"
] | [
"What is the fewest number of pieces that two squares, each with side length 1, need to be cut into so that the pieces will all fit, without overlapping, into a circle of radius 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-b011725fe4876ff44e3d888ba89fb4810a70b79dd3903faf779865a7a63c25f9 | What is the fewest number of seats that must be occupied in a row of 120 seats so that the next person to be seated must sit next to someone? | [
"40"
] | [
"What is the fewest number of seats that must be occupied in a row of 120 seats so that the next person to be seated must sit next to someone?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9883d26ae6a04262e7d9eb2339536756c5011377cc2e68d524375261131bdbbc | What is the fifth largest divisor of $2,014,000,000$? | [
"251,750,000"
] | [
"What is the fifth largest divisor of $2,014,000,000$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-a50eaea7e64f25dae5392d9dada5b3edd52ead75080ab0ad68262823f4129f99 | What is the final nonzero digit of $16!$? | [
"8"
] | [
"What is the final nonzero digit of $16!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-ca8e5ac5274e7c229f53a6ae4d4ac1b6c1a12a9d1e85d93f4dd274322add22de | What is the first (leftmost) digit of the smallest positive integer whose digits add up to 2019? | [
"3"
] | [
"What is the first (leftmost) digit of the smallest positive integer whose digits add up to 2019?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7dd271d3b419dc718a60f9b4f7126f8a0a798353efe71ded44d93778ebbe8bc8 | What is the first Fibonacci number that is divisible by 1,000,000? | [
"750000"
] | [
"What is the first Fibonacci number that is divisible by 1,000,000?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-a5a75cc37662f1b182a4683a91a7f89668c845842903db22f8d6d9911fbe4180 | What is the first digit (from left to right) of the base $8$ representation of $473_{10}$? | [
"7"
] | [
"What is the first digit (from left to right) of the base $8$ representation of $473_{10}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-490c0c594b9035ec3160d4d561fac4589643aae35faf4aeeaa853f0c3836d511 | What is the first digit after the decimal point in the expansion of $(\sqrt{2} + \sqrt{3})^{2008}$? | [
"9"
] | [
"What is the first digit after the decimal point in the expansion of $(\\sqrt{2} + \\sqrt{3})^{2008}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-55e2ab368284af31a315de73a26dff26ecbdb588566aa5748e57e1bdcb4edee2 | What is the first digit after the decimal point of the number \( m = \sqrt{n^2 + n + 1} \), where \( n \) is a natural number? | [
"5"
] | [
"What is the first digit after the decimal point of the number \\( m = \\sqrt{n^2 + n + 1} \\), where \\( n \\) is a natural number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ca4a1493821a13aa22c6f959e4c150e2e6d34820db93f4ed2cc95dc4aadec16 | What is the first digit after the decimal point of the number \( m = \sqrt{n^2 + n + 1} \)? | [
"5 when \\( n > 3 \\), 6 when \\( n = 2, 3 \\), and 7 when \\( n = 1 \\)."
] | [
"What is the first digit after the decimal point of the number \\( m = \\sqrt{n^2 + n + 1} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-2bed7a8eac9782265cdb433905fc2c8b2465ff70e0e42b699963cb0247a861ff | What is the first digit from the left when the decimal number \(2^{125}\) is expressed in the hexadecimal system? | [
"2"
] | [
"What is the first digit from the left when the decimal number \\(2^{125}\\) is expressed in the hexadecimal system?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f4970db94349b958fb4230d54ee3a77bf0611110790b4c3a2ba3f1919e24000 | What is the first digit of \(4^{11213}\)? | [
"7"
] | [
"What is the first digit of \\(4^{11213}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-dfe26c924731c6f4aefd1e53d31b413df93367928f2f36c19b861720e1354d25 | What is the first digit of the number $12^{144}$? | [
"2"
] | [
"What is the first digit of the number $12^{144}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a837a2c2a8772ddd9e24aa4d341a89b5d54a1cb31e739f30c57ba68fabdec031 | What is the first moment between 4:00 and 4:30 when the minute hand and the hour hand of a clock form a perfect right angle? Provide the answer to the fraction of the minute. | [
"\\( 4: \\frac{24}{11} \\)"
] | [
"What is the first moment between 4:00 and 4:30 when the minute hand and the hour hand of a clock form a perfect right angle? Provide the answer to the fraction of the minute."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 20
} | 29 | 0 | 0 |
omr-ws-bb42269c3457b12dd178c0dbf12b33edfc9f154a6954f5cbddaeebf193947251 | What is the first non-zero digit in $80!$? | [
"8"
] | [
"What is the first non-zero digit in $80!$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-5573ff80f1e336e0ef40e699b15ddf6d188ca09c75c3d2d252f7400b723dccd2 | What is the first nonzero digit in the decimal expansion of $\sqrt{1667^2+1}-1667$? | [
"2"
] | [
"What is the first nonzero digit in the decimal expansion of $\\sqrt{1667^2+1}-1667$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-549b183f8846486a5832559021a1236c2a46c21c28a0240658e3a11ae64c867b | What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{129}$? | [
"7"
] | [
"What is the first nonzero digit to the right of the decimal point of the fraction $\\frac{1}{129}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a89b1893d719deac7b3f9e9fcb5d40d2653c1fea86786e54dd7ba6c09b0da45 | What is the first perfect cube? Should it be 0, 1, or another value, considering all integers including negatives? | [
"0"
] | [
"What is the first perfect cube? Should it be 0, 1, or another value, considering all integers including negatives?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-d97b49720b1717526086cfefefe65be45df298949dd308b34301ebcf02a0174f | What is the first perfect square? Is it 0 or 1? | [
"0"
] | [
"What is the first perfect square? Is it 0 or 1?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa1ecb09a57f094f9b372c0f6505af6c7680c21508e061988b62badaed22a68a | What is the first term in the Fibonacci sequence to contain 1000 digits? | [
"4787"
] | [
"What is the first term in the Fibonacci sequence to contain 1000 digits?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e9e036dd5d5c0fca605ec8a37658d37129c6405b3907698ddde6d39d12928e84 | What is the first time after $4:56$ (a.m.) when the 24-hour expression for the time has three consecutive digits that form an increasing arithmetic sequence with a difference of $1$? | [
"10:12"
] | [
"What is the first time after $4:56$ (a.m.) when the 24-hour expression for the time has three consecutive digits that form an increasing arithmetic sequence with a difference of $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-93a4fa199c94337752355088e0a4285fac75079d51d7a4f8939dcc250227d8dc | What is the first time after 1 PM at which the minute hand and hour hand of a clock point in exactly opposite directions? | [
"1:38 2/11",
"1:38\\frac2{11}\\text{ PM}"
] | [
"What is the first time after 1 PM at which the minute hand and hour hand of a clock point in exactly opposite directions?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 40
} | 72 | 0 | 0 |
omr-ws-25ae077619109493b2efb9b5fa346b5eae9bd4a32882dbe881bf98ca8c25ca26 | What is the first time after 12 o'clock at which the hour and minute hands of a clock meet? | [
"1:\\!05\\!\\frac{5}{11}",
"\\( 1: \\frac{60}{11} \\)"
] | [
"What is the first time after 12 o'clock at which the hour and minute hands of a clock meet?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 30
} | 62 | 0 | 0 |
omr-ws-022580e6f9e0a87122637dcb32231c36b8498b12a80ed58aac424e397f8ada9d | What is the first time after 12 o'clock that the hour and minute hands of a clock meet? (Avoid algebra!) | [
"1:05:27"
] | [
"What is the first time after 12 o'clock that the hour and minute hands of a clock meet? (Avoid algebra!)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-00ed55cd4c618002095b65be83ea00512044c310a99af4a2461e92b48ae7af79 | What is the first time after 8:00 when the minute and hour hands form equal angles with the vertical? | [
"8:18 \\(\\frac{6}{13}\\)"
] | [
"What is the first time after 8:00 when the minute and hour hands form equal angles with the vertical?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-ea54fa758141b6342bc09c299bd3f7ddda9f1cc58f96bfc896be72775e1862af | What is the first time between $4:00$ and $5:00$ that the hour and minute hand are exactly $10^\circ$ apart? | [
"4:20"
] | [
"What is the first time between $4:00$ and $5:00$ that the hour and minute hand are exactly $10^\\circ$ apart?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-257eca8e83736f46fadd86e95d8f1b07b533781807f81e2ecacb105a9fb145ca | What is the first year after 2007 where each digit is prime and the year itself is also prime? | [
"2357"
] | [
"What is the first year after 2007 where each digit is prime and the year itself is also prime?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e1a92050c50709df53ac720c2c1f7972acc2cf9c3f87ea8480561895a21c2b0 | What is the first year after 2007 where each digit is prime, distinct, and the year itself is also prime? | [
"2357"
] | [
"What is the first year after 2007 where each digit is prime, distinct, and the year itself is also prime?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 17
} | 40 | 0 | 0 |
omr-ws-4629eb8d2627fcbdb51b2bfccfe5e1c0f593c062cb066aee87f6f6c2760dc718 | What is the flaw in the following process of solving the equation $( -2)^k = 2$? Given: \[ ( -2)^{2k} = 2^2 \implies (( -2)^2)^k = 4 \implies 4^k \implies k = 1. \] | [
"The flaw is that squaring both sides introduces extraneous solutions."
] | [
"What is the flaw in the following process of solving the equation $( -2)^k = 2$?\n\nGiven:\n\\[\n( -2)^{2k} = 2^2 \\implies (( -2)^2)^k = 4 \\implies 4^k \\implies k = 1.\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-85aaee026a7dbfc2167f508cc6e90e5010cbddb19fcfaed7663e05493863b549 | What is the flaw in the following solution for forming 3 teams of 11 players each from 33 boys? We select $\binom{33}{11}$ players for the first team, then $\binom{22}{11}$ players for the second team, and the remaining players form the third team. The total number of ways is $\binom{33}{11} \times \binom{22}{11}$. | [
"The flaw is in treating the teams as distinguishable, leading to an overcount."
] | [
"What is the flaw in the following solution for forming 3 teams of 11 players each from 33 boys? We select $\\binom{33}{11}$ players for the first team, then $\\binom{22}{11}$ players for the second team, and the remaining players form the third team. The total number of ways is $\\binom{33}{11} \\times \\binom{22}... | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c1446f99f2b9b62fb51ff7ab3a7f15761853e6985cb96f5f7acc0a2ca0960c10 | What is the following limit? $$\lim_{x\to\infty}\left(x+|\csc(\pi x)|\right)$$ Is the limit infinite, or does the limit not exist? | [
"\\(\\infty\\)"
] | [
"What is the following limit?\n$$\\lim_{x\\to\\infty}\\left(x+|\\csc(\\pi x)|\\right)$$\nIs the limit infinite, or does the limit not exist?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ca3125d94eec158dff2f159f5682288708570bc440692ce3c65cad4db001de3c | What is the following value when expressed as a common fraction: $$\frac{1}{3^{1}}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}?$$ | [
"\\frac{364}{729}"
] | [
"What is the following value when expressed as a common fraction: $$\\frac{1}{3^{1}}+\\frac{1}{3^{2}}+\\frac{1}{3^{3}}+\\frac{1}{3^{4}}+\\frac{1}{3^{5}}+\\frac{1}{3^{6}}?$$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f93a0274ce84b340e18e7a16553d93c8c1eca2665d24fa065f09c08b3e9c277e | What is the form of the equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ after rotating it $\theta$ radians counterclockwise around a point $(p,q)$? | [
"\\frac{\\left((x - p)\\cos\\theta + (y - q)\\sin\\theta\\right)^2}{a^2} - \\frac{\\left(-(x - p)\\sin\\theta + (y - q)\\cos\\theta\\right)^2}{b^2} = 1"
] | [
"What is the form of the equation $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ after rotating it $\\theta$ radians counterclockwise around a point $(p,q)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-7b09d8a1c46e46a1f662e270eb1a7e85f78ec2531244b52ae7bdef7fc04f4dca | What is the form of the equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ after rotating it $\theta$ radians counterclockwise around the origin? | [
"\\(\\frac{x'^2 (\\cos^2 \\theta / a^2 - \\sin^2 \\theta / b^2)} + \\frac{y'^2 (\\sin^2 \\theta / a^2 - \\cos^2 \\theta / b^2)} - \\frac{2x'y' (\\cos \\theta \\sin \\theta / a^2 + \\cos \\theta \\sin \\theta / b^2)} = 1\\)"
] | [
"What is the form of the equation $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ after rotating it $\\theta$ radians counterclockwise around the origin?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-4943d5d3477507427a216f64d5f716a196ae6d1de16ae08dcd886e319e56fed3 | What is the formula for $\sum_{a=1}^n a^k$, where $k \in \mathbb{N}_{+}$? | [
"\\sum_{a=1}^n a^k = \\frac{1}{k+1} \\sum_{j=0}^k \\binom{k+1}{j} B_j n^{k+1 - j}"
] | [
"What is the formula for $\\sum_{a=1}^n a^k$, where $k \\in \\mathbb{N}_{+}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-b4376fd33a05826a03a2089f2c8ab16220ab7daf13c452ea2d54add8579d6797 | What is the formula for $\sum_{i=0}^n i^x$ where $x$ is an arbitrary integer? | [
"\\frac{B_{x+1}(n+1) - B_{x+1}(0)}{x+1}"
] | [
"What is the formula for $\\sum_{i=0}^n i^x$ where $x$ is an arbitrary integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-40acb11287997611dd4598a032a0e397ad66c21595bef51cab700027d4943540 | What is the formula for calculating combinations \(\binom{n}{k}\) for \(k > 2\)? Is there a shortcut to avoid going through all the calculations? | [
"\\(\\binom{n}{k} = \\frac{n!}{k!(n-k)!}\\)"
] | [
"What is the formula for calculating combinations \\(\\binom{n}{k}\\) for \\(k > 2\\)? Is there a shortcut to avoid going through all the calculations?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-882a0cc14d1434f18ab1ec342a1338f662a10e21c191375e0c5a7748fea74ea8 | What is the formula for generalized integration by parts for a triple integral? | [
"\\int_{V} u \\, \\nabla v \\, dV = \\int_{\\partial V} u v \\, \\mathbf{n} \\, dS - \\int_{V} v \\, \\nabla u \\, dV"
] | [
"What is the formula for generalized integration by parts for a triple integral?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-0c963bc48bc5527afd69e7e69f5331d3086ff904c8f8a39c0cfcb37a26c01868 | What is the formula for rotating the point $(x, y)$ about the point $(a, b)$ by $k$ degrees counterclockwise? Is there a closed form or a matrix form for this rotation? | [
"\\(((x-a)\\cos k - (y-b)\\sin k + a, (x-a)\\sin k + (y-b)\\cos k + b)\\)"
] | [
"What is the formula for rotating the point $(x, y)$ about the point $(a, b)$ by $k$ degrees counterclockwise? Is there a closed form or a matrix form for this rotation?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 25
} | 42 | 0 | 0 |
omr-ws-8b7027efe0f410d72d0db59bc697879773ff1bd3459a7b075c5280bcac37c0f1 | What is the formula for the Gamma function $\Gamma(z)$ when the real part of $z$ is less than or equal to 0, i.e., ${\cal R}(z) \leq 0$? | [
"$\\Gamma(z + 1) = z \\Gamma(z)$"
] | [
"What is the formula for the Gamma function $\\Gamma(z)$ when the real part of $z$ is less than or equal to 0, i.e., ${\\cal R}(z) \\leq 0$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a8ee26735b674d6724f7eca0402ea9ca80a80e762228ba094e3053b0008d92d | What is the formula for the angle subtended by a chord of length $l$ in a circle of radius $R$? | [
"\\(\\alpha = \\arccos \\left( 1 - \\frac{l^2}{2R^2} \\right)\\)"
] | [
"What is the formula for the angle subtended by a chord of length $l$ in a circle of radius $R$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-76236d491d565d5e70eb95039a7f6e36a7d7dfac9fdaa3502dbd21d918dd74c9 | What is the formula for the area of a triangle in terms of its side lengths $a$, $b$, and $c$, where $s$ is the semiperimeter? | [
"\\(\\sqrt{s(s-a)(s-b)(s-c)}\\)"
] | [
"What is the formula for the area of a triangle in terms of its side lengths $a$, $b$, and $c$, where $s$ is the semiperimeter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-dabc7215ea7c1867e5f9e41b9c5a4af318bf08252b71e6953dbcf98f54cf302b | What is the formula for the inradius of a right triangle in terms of its side lengths \(a\), \(b\), and \(c\)? | [
"\\( \\frac{ab}{a + b + c} \\)"
] | [
"What is the formula for the inradius of a right triangle in terms of its side lengths \\(a\\), \\(b\\), and \\(c\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72299757a777051499015e9974f36c6758cb55fb7fa9e05d79d951cd432821c2 | What is the formula for the intersection of lines $ab$ and $cd$, where $a, b, c, d$ are arbitrary complex numbers? | [
"a - \\dfrac{\\operatorname{Im}( \\overline{(d - c)}(c - a) )}{\\operatorname{Im}( \\overline{(b - a)}(d - c) )} (b - a)"
] | [
"What is the formula for the intersection of lines $ab$ and $cd$, where $a, b, c, d$ are arbitrary complex numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-00f863f0391bc7a1d96913c9e6886c0704dba0145dbdedba365e512fb1496dc0 | What is the formula for the mean value of many rolls of \( n \) dice, where you always take the greatest value of the dice? For example, if you roll two dice, the highest value from each roll is recorded (e.g., if you get 1 and 2, the number to add is 2; if you get 2 and 3, the number to add is 3; if you get 1 and 6, t... | [
"\\( 6 - \\left(\\frac{5}{6}\\right)^n - \\left(\\frac{4}{6}\\right)^n - \\left(\\frac{3}{6}\\right)^n - \\left(\\frac{2}{6}\\right)^n - \\left(\\frac{1}{6}\\right)^n \\)"
] | [
"What is the formula for the mean value of many rolls of \\( n \\) dice, where you always take the greatest value of the dice? For example, if you roll two dice, the highest value from each roll is recorded (e.g., if you get 1 and 2, the number to add is 2; if you get 2 and 3, the number to add is 3; if you get 1 a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-4cedefe14c3395266806b0b40bdb09ac3751d10872cb68ae8e14225b898cbd5c | What is the formula for the number of derangements of \( n \) objects, where a derangement is defined as an arrangement of \( n \) objects such that none of the objects are in their natural positions? | [
"D(n) = n! \\sum_{k=0}^n \\frac{(-1)^k}{k!}"
] | [
"What is the formula for the number of derangements of \\( n \\) objects, where a derangement is defined as an arrangement of \\( n \\) objects such that none of the objects are in their natural positions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b193db301ec52f1647fb960a6643e891015b0b17a26e5dc7ba1b0851b9acc133 | What is the formula for the number of ways that N distinguishable items can be placed in K indistinguishable boxes? | [
"\\sum_{i=1}^{K} S(N, i)"
] | [
"What is the formula for the number of ways that N distinguishable items can be placed in K indistinguishable boxes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-c5c0c0de64c33368a575faef0f114b5767fa3ee91d51d1c9e662e24973602146 | What is the formula for the number of ways to distribute \( n^2 \) objects into \( n \) groups of \( n \) objects each? | [
"\\(\\frac{(n^2)!}{(n!)^{n+1}}\\)"
] | [
"What is the formula for the number of ways to distribute \\( n^2 \\) objects into \\( n \\) groups of \\( n \\) objects each?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-ded62fe1db2533e86b452fd097b4bad5f114e4048c4f4185c4a490b0e19225f4 | What is the formula for the product of cosines $\prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right)$? | [
"\\[\n\\prod_{k=1}^{n-1} \\cos\\left(\\frac{k\\pi}{n}\\right) = \\begin{cases} \n0 & \\text{if } n \\text{ is even} \\\\\n( - 1)^{\\frac{n - 1}{2}} \\cdot 2^{1 - n} & \\text{if } n \\text{ is odd}\n\\end{cases}\n\\]"
] | [
"What is the formula for the product of cosines $\\prod_{k=1}^{n-1} \\cos\\left(\\frac{k\\pi}{n}\\right)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 26
} | 35 | 0 | 0 |
omr-ws-2c7640c8591e9b56b80ac40cf79b9573598a8a3490e06d5fa079456680659c9c | What is the formula for the solutions to the system of differential equations given by: \[ x' = x \] \[ y' = 2x + 3y \] | [
"\\( x = Ae^t \\) and \\( y = -Ae^t + CA^3e^{3t} \\)"
] | [
"What is the formula for the solutions to the system of differential equations given by:\n\\[ x' = x \\]\n\\[ y' = 2x + 3y \\]"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ce9b9889fca60b25b11847aa01b1e1d3104222185df4e3b31047161e438f1489 | What is the formula for the sum $1 + x + x^2 + x^3 + \ldots + x^n$? | [
"\\(\\frac{x^{n+1}-1}{x-1}\\)"
] | [
"What is the formula for the sum $1 + x + x^2 + x^3 + \\ldots + x^n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-4d17dc843e5858a082deeed08dda7615525f1a49f8c8ac37f2bcd33bc0985bb1 | What is the formula for the sum $1^1 + 2^2 + 3^3 + \cdots + n^n$? | [
"\\sum_{k=1}^n k^k"
] | [
"What is the formula for the sum $1^1 + 2^2 + 3^3 + \\cdots + n^n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90fe299843e1c463040655de8cc13f0c28a68452487036f203a6f5abe2585365 | What is the formula for the sum $1^8 + 2^8 + \cdots + n^8$? | [
"\\(\\frac{1}{90}(10n^9 + 45n^8 + 60n^7 - 42n^5 + 20n^3 - 3n)\\)"
] | [
"What is the formula for the sum $1^8 + 2^8 + \\cdots + n^8$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 23
} | 35 | 0 | 0 |
omr-ws-a33755ed44aebf8769482f18ab5518d5fc0a55bbb18035adff8af0e30bf82f75 | What is the formula for the sum \( \frac{1}{1 \cdot 3} + \frac{1}{2 \cdot 4} + \cdots + \frac{1}{n(n+2)} \)? | [
"\\(\\frac{1}{2}\\left(1 - \\frac{1}{n + 1} - \\frac{1}{n + 2}\\right)\\)"
] | [
"What is the formula for the sum \\( \\frac{1}{1 \\cdot 3} + \\frac{1}{2 \\cdot 4} + \\cdots + \\frac{1}{n(n+2)} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d10093bba883545881c6b5d534a09df88daf96ad4013aa7ef85fb20d83115f9e | What is the formula for the sum \(1^1 + 2^2 + 3^3 + \cdots + n^n\)? | [
"\\sum_{k=1}^n k^k"
] | [
"What is the formula for the sum \\(1^1 + 2^2 + 3^3 + \\cdots + n^n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-71af7ec9efc2f014cf8ba028649c41dfd772a79c949aba09d7689e5093b3e788 | What is the formula for the sum of the area of a dodecagon with side length \( a \)? | [
"\\( 3a^2 \\)"
] | [
"What is the formula for the sum of the area of a dodecagon with side length \\( a \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98465719477806ee88e2dd953836c04b0d8c855fb54f40e320b247f515e107e9 | What is the formula for the sum of the first N Fibonacci numbers? | [
"\\( F_{N+2} - 1 \\)"
] | [
"What is the formula for the sum of the first N Fibonacci numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-626d031335a9994e9dfcf87a5ce39cb67c606ce46863194cd01217c18d10cab1 | What is the formula for the sum of the fourth powers of the first \( n \) natural numbers, i.e., \( 1^4 + 2^4 + 3^4 + \cdots + n^4 \)? | [
"\\(\\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}\\)"
] | [
"What is the formula for the sum of the fourth powers of the first \\( n \\) natural numbers, i.e., \\( 1^4 + 2^4 + 3^4 + \\cdots + n^4 \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-56a077f10e70d91e5b207d0377075fcf7adf2ea863de7e90321956dcfce18e5c | What is the formula for the sum of the series \(1^n + 2^n + 3^n + \cdots + k^n\) for a given positive integer \(n\)? | [
"\\sum_{i=1}^k i^n = \\frac{1}{n+1} \\sum_{m=0}^n \\binom{n+1}{m} B_m k^{n+1 - m}"
] | [
"What is the formula for the sum of the series \\(1^n + 2^n + 3^n + \\cdots + k^n\\) for a given positive integer \\(n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-1ebe5b6a3f894dfa149ad6d92ffc530acb1cc41e533408ae40fbd9812f88d9cf | What is the formula for the surface area of the ellipsoid given by $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$? | [
"4\\pi \\int_{0}^{\\pi} \\int_{0}^{2\\pi} \\sqrt{a^2 b^2 \\sin^4\\theta \\cos^2\\phi + a^2 c^2 \\sin^4\\theta \\sin^2\\phi + b^2 c^2 \\sin^2\\theta \\cos^2\\theta} \\, d\\phi \\, d\\theta"
] | [
"What is the formula for the surface area of the ellipsoid given by $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}+\\frac{z^{2}}{c^{2}}=1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-d5540ced2f623d9d3e10af5896deceedc80f8eb353efce408f42f38302426a63 | What is the formula for the symmedian point using complex numbers when the vertices of the triangle lie on the unit circle? | [
"\\dfrac{a^3 + b^3 + c^3}{a^2 + b^2 + c^2}"
] | [
"What is the formula for the symmedian point using complex numbers when the vertices of the triangle lie on the unit circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-00833017e413dddea86ea5b8e9305fd1528ecb1ca0afd33cf6af57f14a5a7938 | What is the formula for the volume of any tetrahedron given its edge lengths using Heron's formula? Provide an example if possible. | [
"\\( K^2 = \\frac{-1}{16} \\det A \\) where \\( A \\) is the matrix defined above."
] | [
"What is the formula for the volume of any tetrahedron given its edge lengths using Heron's formula? Provide an example if possible."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-656df01a9c05da69c206088d0763b734c78ae3e5b15d541177a6d50eaedd1ace | What is the formula related to quartic residues, and how can it be used to solve the congruence \(x^2 \equiv y^2 \pmod{2004}\)? | [
"x \\equiv \\pm y \\pmod{2004}"
] | [
"What is the formula related to quartic residues, and how can it be used to solve the congruence \\(x^2 \\equiv y^2 \\pmod{2004}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-c000692a2f5a7ecbbdb4f85ae84f39faa75f20dcb3f78e704da1a7dfa7a8b1c8 | What is the formula that gives the coefficients of a polynomial of degree \( n \) from its roots? | [
"\\( a_{n-k} = (-1)^k a_n \\sum_{1 \\leq i_1 < i_2 < \\cdots < i_k \\leq n} r_{i_1} r_{i_2} \\cdots r_{i_k} \\)"
] | [
"What is the formula that gives the coefficients of a polynomial of degree \\( n \\) from its roots?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 17
} | 19 | 0 | 0 |
omr-ws-5c63f060275b9dfa80aea287822f70ca67866ff63d89fc1c86edd126ebf5f4c2 | What is the formula to calculate the area of the Morley triangle of triangle $ABC$ in terms of the area of triangle $ABC$? | [
"\\frac{8 \\sqrt{3} \\sin^2\\left(\\frac{A}{3}\\right) \\sin^2\\left(\\frac{B}{3}\\right) \\sin^2\\left(\\frac{C}{3}\\right)}{\\sin A \\sin B \\sin C} \\cdot \\text{Area of } ABC"
] | [
"What is the formula to calculate the area of the Morley triangle of triangle $ABC$ in terms of the area of triangle $ABC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-7be1d7914b688642a866143520f2ae9358caead662a6c051e49c2851d732056f | What is the formula to compute a fixed percentage of profit over time (20 years), given a beginning number? | [
"\\( P = I\\left(1 + \\frac{p}{n}\\right)^{20n} \\)"
] | [
"What is the formula to compute a fixed percentage of profit over time (20 years), given a beginning number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f956672c0211ddd26aa8904db1ae83f8c69fc11c8da7c85c9670aeb177006cf | What is the formula to find the radius of a sphere inscribed in an irregular tetrahedron when all edges are given? Specifically, given the edges WX, WY, WZ, XY, XZ, and YZ, determine the radius \( r \) of the inscribed sphere. | [
"\\( r = \\frac{3V}{SA} \\)"
] | [
"What is the formula to find the radius of a sphere inscribed in an irregular tetrahedron when all edges are given? Specifically, given the edges WX, WY, WZ, XY, XZ, and YZ, determine the radius \\( r \\) of the inscribed sphere."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-ee73a14eb14cfe3fa4ed905ec3b2033039dcb75fb4208fa13899368273e524e1 | What is the fourth Wilson prime? A Wilson prime is a prime number \( p \) such that \( (p-1)! + 1 \) is divisible by \( p^2 \). The first three Wilson primes are 5, 13, and 563. | [
"\\text{Unknown}"
] | [
"What is the fourth Wilson prime? A Wilson prime is a prime number \\( p \\) such that \\( (p-1)! + 1 \\) is divisible by \\( p^2 \\). The first three Wilson primes are 5, 13, and 563."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-8e0615b1c16a44cb926034cd6da6c8cac8c5652bc7644937197f274122aa6854 | What is the fourth digit of the greatest possible 7-digit passcode where all digits are different, no consecutive numbers are next to each other, no even digit is next to an even digit, and no odd digit is next to an odd digit? | [
"8"
] | [
"What is the fourth digit of the greatest possible 7-digit passcode where all digits are different, no consecutive numbers are next to each other, no even digit is next to an even digit, and no odd digit is next to an odd digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-508773d183077ab9af830dbae39e022b1978668e2ab63f269a7ab03f76a9882a | What is the fourth largest real number \( a \) such that \( 3 \) is a [i]gingado[/i] of \( a \)? Here, a [i]gingado[/i] of \( a \) is defined as \( G = |a_n| + |a_{n-1}| + \cdots + |a_1| + |a_0| \), where \( a \) is a root of the polynomial \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) with integer coef... | [
"\\( 0 \\)"
] | [
"What is the fourth largest real number \\( a \\) such that \\( 3 \\) is a [i]gingado[/i] of \\( a \\)? \n\nHere, a [i]gingado[/i] of \\( a \\) is defined as \\( G = |a_n| + |a_{n-1}| + \\cdots + |a_1| + |a_0| \\), where \\( a \\) is a root of the polynomial \\( P(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b121350e1dd68fd77a38c81c6c966275c129defdb3517155e0db3cfde10eb816 | What is the fourth power of $\sqrt{1+\sqrt{1+\sqrt{1}}}$? | [
"\\(3+2\\sqrt{2}\\)"
] | [
"What is the fourth power of $\\sqrt{1+\\sqrt{1+\\sqrt{1}}}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 28
} | 76 | 0 | 0 |
omr-ws-8c0e5a7a0e494baf76f1c8eb8a8a4db684edda87f2209ea5d743f1be2fbd2943 | What is the fourth-degree coefficient in the polynomial $p(x)=(x-1)(x+1)(x-2)(x+2)(x-3)(x+3)$? | [
"-14"
] | [
"What is the fourth-degree coefficient in the polynomial $p(x)=(x-1)(x+1)(x-2)(x+2)(x-3)(x+3)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-0f46f134053dc0bc0364c639669121ec1ba6014cce1b64ede2a7256dfabed165 | What is the fundamental group of a Möbius band minus a point when the point is anywhere else on the band? | [
"\\(\\mathbb{Z} * \\mathbb{Z}\\)"
] | [
"What is the fundamental group of a Möbius band minus a point when the point is anywhere else on the band?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 5
} | 22 | 0 | 0 |
omr-ws-f75ceb8db99fd739ae83c2f5d6c797d554daea6f42cf8473b667b7477cc0315e | What is the fundamental group of a Möbius band minus a point when the point is on the boundary of the strip? | [
"\\mathbb{Z} * \\mathbb{Z}"
] | [
"What is the fundamental group of a Möbius band minus a point when the point is on the boundary of the strip?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-c5cf03828d653e553a1791c629f45c4d482d0fcd6f855fa4f8214e3dae843c0b | What is the fundamental group of a Möbius band minus a point when the point is on the central circle? | [
"\\mathbb{Z} * \\mathbb{Z}"
] | [
"What is the fundamental group of a Möbius band minus a point when the point is on the central circle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-e8cd499de1092c6a2b19d47ea575b2d7083dc6ba386320f08595b45079cc390e | What is the fundamental group of a projective plane with 2 points removed? | [
"$\\mathbb{Z} * \\mathbb{Z}$"
] | [
"What is the fundamental group of a projective plane with 2 points removed?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-517fdc13cf77c9d14284c74393c2d6c71b183b782a03be8fde6a2b8df379d7b2 | What is the fundamental group of the complement of the lines \( z = 0 \), \( z = 1 \), \( w = 0 \), \( w = 1 \), and \( z = w \) in \( \mathbb{C}^2 \)? | [
"\\mathbb{Z} \\times F_2 \\times F_2"
] | [
"What is the fundamental group of the complement of the lines \\( z = 0 \\), \\( z = 1 \\), \\( w = 0 \\), \\( w = 1 \\), and \\( z = w \\) in \\( \\mathbb{C}^2 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-74fcd185e2932b2231fa8643b3ff2349cc973af9394a5e52db868b2c56adc8bf | What is the fundamental period of the function $\frac{\sin^5 4x}{\sin^2 3x}$? | [
"\\pi"
] | [
"What is the fundamental period of the function $\\frac{\\sin^5 4x}{\\sin^2 3x}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85f503c22b5dfbdd64e3af6147227287f0ec5e2864eafa4f57e0336f2e2a5b8f | What is the fundamental period of the function $\sin^5 4x + \sin^2 3x$? | [
"$\\pi$"
] | [
"What is the fundamental period of the function $\\sin^5 4x + \\sin^2 3x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-ddde2f9c0062a8f469458ea0cff8fc09936e60d754011a3feefd6e158082686e | What is the fundamental period of the function $\sin^5 4x \times \sin^2 3x$? | [
"\\pi"
] | [
"What is the fundamental period of the function $\\sin^5 4x \\times \\sin^2 3x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e3081b59e56cd69cada3f26608e4409191be55150a2a17b077470a0678f9c243 | What is the general Cartesian equation of a quadratic curve in 3D? | [
"\\( a \\cdot x^2 + b \\cdot x + c \\cdot y^2 + d \\cdot y + e \\cdot z^2 + f \\cdot z + g = 0 \\)"
] | [
"What is the general Cartesian equation of a quadratic curve in 3D?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5af9422732ca96bf2399a02123b51fce13176e9a7b81d088c1dcd95bdc3dce47 | What is the general approach for computing the inverse Laplace transform of functions of the form $\frac{P(s)}{(Q(s))^n}$, where $Q(s)$ is a polynomial of order 2 and $P(s)$ is a polynomial of order at most 1? | [
"\\text{Use partial fractions for real roots or complete the square for complex roots, then apply known Laplace transform pairs involving } t^{n-1}e^{\\lambda t}\\text{ and trigonometric functions.}"
] | [
"What is the general approach for computing the inverse Laplace transform of functions of the form $\\frac{P(s)}{(Q(s))^n}$, where $Q(s)$ is a polynomial of order 2 and $P(s)$ is a polynomial of order at most 1?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-eb339d5d6a4c580373f5131299e8006bcc47f23eaa40dc01761728db2cf01759 | What is the general equation for an ellipse whose foci do not lie on a straight line parallel to either the x-axis or y-axis? | [
"Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0"
] | [
"What is the general equation for an ellipse whose foci do not lie on a straight line parallel to either the x-axis or y-axis?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-4ce7fa39447afc1774fec1badea4b221d821da177e8c153664e5eb242a91439a | What is the general equation of a fourth-degree plane curve $\mathcal{C}^4$ such that the corresponding curve in the complexified projective plane $(\mathbb{R}P^2)^\mathbb{C}P$ has the isotropic points $[(0,1,i)]$ and $[(0,1,-i)]$ as double points? | [
"(y^2 + z^2)^2 + x Q(x, y, z) = 0"
] | [
"What is the general equation of a fourth-degree plane curve $\\mathcal{C}^4$ such that the corresponding curve in the complexified projective plane $(\\mathbb{R}P^2)^\\mathbb{C}P$ has the isotropic points $[(0,1,i)]$ and $[(0,1,-i)]$ as double points?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-af27008e13e92c0120a20ee6691fdb448558145fb12b46f66d7c80e086681e6a | What is the general expansion for \((a+bi)^{c+di}\)? | [
"(a^2 + b^2)^{c/2} e^{-d \\arctan\\left(\\frac{b}{a}\\right)} \\left[ \\cos\\left(c \\arctan\\left(\\frac{b}{a}\\right) + \\frac{d}{2} \\ln(a^2 + b^2)\\right) + i \\sin\\left(c \\arctan\\left(\\frac{b}{a}\\right) + \\frac{d}{2} \\ln(a^2 + b^2)\\right) \\right]"
] | [
"What is the general expansion for \\((a+bi)^{c+di}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-faedaba4ca6122c68cc472b44b0e6839b5d26a2c7d089db7bc720c58ed6d5a27 | What is the general factorization of $x^n - 1$? | [
"\\( x^n - 1 = \\prod_{d | n} \\Phi_d(x) \\)"
] | [
"What is the general factorization of $x^n - 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-d789eaecbb770de8b2aa0b98bee97064ac873aa9a2e88e61ba34bf4496000f7b | What is the general form for $s_n$, where $s_n$ is the sum of the products of the roots $r_i$ taken $n$ at a time for a polynomial with leading coefficient $a_n$? | [
"\\( s_n = (-1)^n \\frac{a_0}{a_n} \\)"
] | [
"What is the general form for $s_n$, where $s_n$ is the sum of the products of the roots $r_i$ taken $n$ at a time for a polynomial with leading coefficient $a_n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-02dbee288aa652d1ddf2599d9490bf0d7a7c6079afb33954fe8d8ecb856fb6b8 | What is the general form of a rational function which has absolute value $1$ on the circle $|z|=1$? In particular, how are the zeroes and poles related to each other? | [
"R(z) = e^{i\\theta} z^k \\prod_{j=1}^n \\frac{z - a_j}{1 - \\overline{a_j} z}"
] | [
"What is the general form of a rational function which has absolute value $1$ on the circle $|z|=1$? In particular, how are the zeroes and poles related to each other?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-4bff57a68990b50e65a75ee5c256a1dd0b00fbbb1cd63ae08ff9ed2d9af7d12b | What is the general form of a rational function with absolute value $1$ on the circle $\vert z\vert=1$? How are the zeroes and poles related to each other? | [
"\\( f(z)=\\alpha z^p\\prod_{j=1}^m\\frac{z-w_j}{1-\\overline{w_j}z}\\prod_{k=1}^n \\frac{1-\\overline{v_k}z}{z-v_k} \\) with \\( |\\alpha|=1 \\) and \\( p\\in\\mathbb{Z} \\)."
] | [
"What is the general form of a rational function with absolute value $1$ on the circle $\\vert z\\vert=1$? How are the zeroes and poles related to each other?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 8
} | 17 | 0 | 0 |
omr-ws-ddfe7841567e5266abc49e5a19dd75005f58e8abbdafbe454b9cddc1ffa89602 | What is the general form of the inverse of an upper triangular matrix \( A \) given by \[ A = \left[\begin{array}{cccc} A_{1,1} & A_{1,2} & \ldots & A_{1,n} \\ 0 & A_{2,2} & \ldots & A_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & A_{n,n} \end{array}\right]? \] | [
"The inverse of an upper triangular matrix is an upper triangular matrix, with rows \\( L_i \\) given by the recurrence relation:\n\\[ L_n = \\left( \\frac{1}{A_{n,n}} \\right) E_n \\]\n\\[ L_i = \\left( \\frac{1}{A_{i,i}} \\right) \\left( E_i - A_{i,i+1} L_{i+1} - \\cdots - A_{i,n} L_n \\right) \\text{ for } i < n... | [
"What is the general form of the inverse of an upper triangular matrix \\( A \\) given by\n\n\\[ A = \\left[\\begin{array}{cccc} \nA_{1,1} & A_{1,2} & \\ldots & A_{1,n} \\\\ \n0 & A_{2,2} & \\ldots & A_{2,n} \\\\ \n\\vdots & \\vdots & \\ddots & \\vdots \\\\ \n0 & 0 & \\ldots & A_{n,n} \n\\end{array}\\right]? \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-53b6a37b493166aefe16eb59eb8885bef5814b049174d5981f975dcfc6694202 | What is the general formula for a polar equation of a parabola? | [
"r = \\frac{d}{1 + \\cos \\theta}"
] | [
"What is the general formula for a polar equation of a parabola?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc19f94385ec244852efafebc93143f2751cb0aa580575c4ab4bbfe541e92e64 | What is the general formula for choosing \(i\) numbers \(a_1, a_2, \ldots, a_i\) such that \(a_1 + a_2 + \cdots + a_i = n\) for some natural number \(n\)? | [
"\\(\\binom{n+i-1}{i-1}\\)"
] | [
"What is the general formula for choosing \\(i\\) numbers \\(a_1, a_2, \\ldots, a_i\\) such that \\(a_1 + a_2 + \\cdots + a_i = n\\) for some natural number \\(n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 6
} | 11 | 0 | 0 |
omr-ws-4381675ddaf159700ee1c19722de65f30ec1c9bee3076cbb247d6f73a84655f3 | What is the general formula for the inradius of a triangle in terms of its side lengths \(a\), \(b\), and \(c\)? | [
"\\( r = \\frac{\\sqrt{s(s - a)(s - b)(s - c)}}{s} \\)"
] | [
"What is the general formula for the inradius of a triangle in terms of its side lengths \\(a\\), \\(b\\), and \\(c\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-7b5ff8dea79137df15d440ce3036cbcc6ce4d1300d095870012a4c3a5c795e10 | What is the general formula for the maximum number of space diagonals in a polyhedron with \( n \) vertices? | [
"\\dfrac{n^2 - 7n + 12}{2}"
] | [
"What is the general formula for the maximum number of space diagonals in a polyhedron with \\( n \\) vertices?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b71d2cb10012ecc01d3ca2522d8337a0a636060cbff530ede21bd171a654a91 | What is the general formula for the product of all positive factors of a positive integer \( n \)? | [
"n^{\\tau(n)/2}"
] | [
"What is the general formula for the product of all positive factors of a positive integer \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-978af9af29e3fa58213462fae7ad0a25987e85dd6378c9f3cc153959161ebac7 | What is the general formula for the recurrence relation $a_{n}^{2}=2+a_{n-1}$ where $a_{1}=\sqrt{2}$? | [
"\\( a_n = 2\\cos\\frac{\\pi}{2^{n+1}} \\)"
] | [
"What is the general formula for the recurrence relation $a_{n}^{2}=2+a_{n-1}$ where $a_{1}=\\sqrt{2}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-b91dcb41f4f1387c5a63063f7fd5f5e6bc08e015a5d1b6041e0b2c8cc611202e | What is the general formula for the solutions of $x^n = 1$ over the complex numbers? | [
"\\(\\cos \\frac{2\\pi k}{n} + i \\sin \\frac{2\\pi k}{n}\\), where \\(k \\in \\{0,1,2,...,n-1\\}\\) or equivalently \\(e^{2\\pi i k/n}\\)."
] | [
"What is the general formula for the solutions of $x^n = 1$ over the complex numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4c620dde3ff930bea42054bc464895705748a0a65613357f42dcfb2e929dccc4 | What is the general formula for the sum of the exponents of \(x\) and \(y\) in the expansion of \((x+y)^k\)? | [
"\\(k(k+1)\\)"
] | [
"What is the general formula for the sum of the exponents of \\(x\\) and \\(y\\) in the expansion of \\((x+y)^k\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 16
} | 30 | 0 | 0 |
omr-ws-91dc9036f2ac6f5a89a0b1afeb6eb78fd5fa8030bce009f4991fac8a98ca9fee | What is the general formula to find the coefficient of $x^r$ in the expansion of $\left(\sum_{i=1}^{n} x^i\right)^p$? | [
"\\sum_{k=0}^{\\left\\lfloor \\frac{r - p}{n} \\right\\rfloor} (-1)^k \\binom{p}{k} \\binom{r - kn - 1}{p - 1}"
] | [
"What is the general formula to find the coefficient of $x^r$ in the expansion of $\\left(\\sum_{i=1}^{n} x^i\\right)^p$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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