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-2ce6f64b156028af4b4100c5a6eb0c1c053d9b97b7af806c616e7c3486b4440c | What is the probability that three randomly selected distinct squares from an $8 \times 8$ chessboard form an L-shaped pattern when viewed from one fixed side? | [
"\\(\\frac{1}{434}\\)"
] | [
"What is the probability that three randomly selected distinct squares from an $8 \\times 8$ chessboard form an L-shaped pattern when viewed from one fixed side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d5c0d4fcd38c43018f283174f85464328de800d49ca6d9be18dd33796430a89 | What is the probability that three randomly selected numbers between 0 and 1 can form a triangle? | [
"\\(\\frac{1}{2}\\)\n\nIt seems there was a slight misinterpretation in the forum discussion. The correct probability that three randomly selected numbers between 0 and 1 can form a triangle is \\(\\frac{1}{2}\\), not \\(\\frac{1}{3}\\). This is a well-known result in geometric probability."
] | [
"What is the probability that three randomly selected numbers between 0 and 1 can form a triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-bc0f180aac10d258dae50602b1ac2c429f273af270b6f4f17e6a0e6fe94a4b3e | What is the probability that three randomly selected numbers from the set $\{1, 2, \dots, (2n + 1)\}$ are in arithmetic progression? | [
"\\(\\frac{3n}{(2n-1)(2n+1)}\\)"
] | [
"What is the probability that three randomly selected numbers from the set $\\{1, 2, \\dots, (2n + 1)\\}$ are in arithmetic progression?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-3d59a18b2db6a7f48600482f4fbf1a49b5a7c741ca3af06bc2c58c68ed3ef622 | What is the probability that three sides, with lengths randomly chosen such that one side is between 0 and 1 (exclusive), one side is between 1 and 2, and the third side is between 2 and 3 units, can form a triangle? | [
"\\dfrac{1}{6}"
] | [
"What is the probability that three sides, with lengths randomly chosen such that one side is between 0 and 1 (exclusive), one side is between 1 and 2, and the third side is between 2 and 3 units, can form a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6605904da89115eb25d000c23bd5ec355a98b27ca0df4274347cf72d2520650 | What is the probability that three specific students - Linda, Martin, and Nancy - are in the same group when 15 students are randomly assigned into 5 lab groups of 3 students each? | [
"\\(\\frac{1}{91}\\)"
] | [
"What is the probability that three specific students - Linda, Martin, and Nancy - are in the same group when 15 students are randomly assigned into 5 lab groups of 3 students each?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-fe3c092b02bc484f88f9e3c28ad9dd234580461a95024770637bf9193e8caa85 | What is the probability that two 7s appear before both a 6 and an 8 appear when a pair of dice are repeatedly rolled and the sum of the two values are computed each time? | [
"\\(\\frac{9}{49}\\)"
] | [
"What is the probability that two 7s appear before both a 6 and an 8 appear when a pair of dice are repeatedly rolled and the sum of the two values are computed each time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-81418bf575fa7bb230b5c47278299a7003c5a71ca57a1b29e49ed5b703a2d896 | What is the probability that two cards drawn at random from a well-shuffled deck of 52 cards are both spades? | [
"\\(\\frac{1}{17}\\)"
] | [
"What is the probability that two cards drawn at random from a well-shuffled deck of 52 cards are both spades?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-6e191c6175b1c20a142cf285f1e29fe3c6f8f82c48b91449e1baee31373f51b7 | What is the probability that two cards drawn from a standard 52-card deck are from different suits? | [
"\\(\\frac{13}{17}\\)"
] | [
"What is the probability that two cards drawn from a standard 52-card deck are from different suits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1b02319b1206bc7b01d03f3aa87b0d6377e4d3daab08669ca091769b885d0b58 | What is the probability that two dice sum up to 7 twice in a row before they sum up to 12? | [
"\\(\\frac{6}{13}\\)"
] | [
"What is the probability that two dice sum up to 7 twice in a row before they sum up to 12?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-feb3df5a8ca7d118460673a8e62d6aa47eb67f83e887965ce6b5dae4bbc24e42 | What is the probability that two different cards (of any suit) will be next to each other in a randomly shuffled deck of 52 playing cards? | [
"1"
] | [
"What is the probability that two different cards (of any suit) will be next to each other in a randomly shuffled deck of 52 playing cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-461fe46ad77a1344acd5c83b4c9007d94c0b66e3e27a69fdaa648f5c0aecfceb | What is the probability that two face cards will NOT be found one after another in a shuffled deck of cards? (Note: An ace is not considered a face card.) | [
"\\dfrac{\\dbinom{41}{12}}{\\dbinom{52}{12}}"
] | [
"What is the probability that two face cards will NOT be found one after another in a shuffled deck of cards? (Note: An ace is not considered a face card.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-cdda2e4ed32a64edc444ff2502b8a7f5fc7954ff1223bbe7f38eadbbf5c34fe0 | What is the probability that two heads will turn up in succession somewhere in a sequence of \( n \) coin tosses? | [
"1 - \\dfrac{F_{n+2}}{2^n}"
] | [
"What is the probability that two heads will turn up in succession somewhere in a sequence of \\( n \\) coin tosses?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da4f4c4164701715ab842e31ae8d65c5f92aa6bb4433e1d6a71519a6ea518ee5 | What is the probability that two people in a room have the same birthday month and day? | [
"\\dfrac{1}{365}"
] | [
"What is the probability that two people in a room have the same birthday month and day?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-75ab0250e24411416a5bd80a17a49ac614c47503356516d8bc494fc33676c40b | What is the probability that two people, each tossing a pair of dice, get the same total? | [
"\\(\\frac{73}{648}\\)"
] | [
"What is the probability that two people, each tossing a pair of dice, get the same total?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-1f3b3b563f9fca3dfd3281bb5bc59c5d339d0f87fc1ed00dff1d3246421c4193 | What is the probability that two persons receive the same number of balls when 7 identical balls are distributed among 10 persons, where each person may receive zero or more balls? | [
"1"
] | [
"What is the probability that two persons receive the same number of balls when 7 identical balls are distributed among 10 persons, where each person may receive zero or more balls?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-a3b7d41aa433a6c408e8393575e0c0e367298b15fddddc59575f3c00579d19fe | What is the probability that two queens placed randomly on a chessboard do not attack each other? | [
"\\(\\frac{23}{36}\\)"
] | [
"What is the probability that two queens placed randomly on a chessboard do not attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-63672890d4a890fa352b6853472ce4f558bd11cc42c1ce5cb6fa693481cbc025 | What is the probability that two randomly chosen $n$-domino tiles match, where an $n$-domino tile is defined by a pair $(k, m)$ with $k$ and $m$ taking values from $0$ to $n$ and pairs $(k, m)$ and $(m, k)$ are considered the same tile? | [
"\\(\\frac{4(n+1)}{(n+2)(n+3)}\\)"
] | [
"What is the probability that two randomly chosen $n$-domino tiles match, where an $n$-domino tile is defined by a pair $(k, m)$ with $k$ and $m$ taking values from $0$ to $n$ and pairs $(k, m)$ and $(m, k)$ are considered the same tile?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 5,
"tir": 7
} | 49 | 0 | 0 |
omr-ws-91765c7078fe8c017c2cee0f71881e814ba9aec3f352663669321ea5e3065846 | What is the probability that two randomly chosen $n$-element permutations commute? The problem can be solved by considering the group $S_n$ acting on itself by conjugation and applying Burnside's Lemma. | [
"\\dfrac{p(n)}{n!}"
] | [
"What is the probability that two randomly chosen $n$-element permutations commute? The problem can be solved by considering the group $S_n$ acting on itself by conjugation and applying Burnside's Lemma."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b47c6340226fab89248274cb9352e34f8b36c0444f6416d2c1f9ab705b4db92e | What is the probability that two randomly chosen chairs from a grid of 10 rows and 10 columns (total of 100 chairs) are side by side? | [
"\\(\\frac{1}{55}\\)"
] | [
"What is the probability that two randomly chosen chairs from a grid of 10 rows and 10 columns (total of 100 chairs) are side by side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-14071de0ff562b0a14d5f92011e09a74925eedab0262c8e2ec29f2b95ecd6b7c | What is the probability that two randomly chosen chords on a circle intersect? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that two randomly chosen chords on a circle intersect?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8c7b4d85ea95e2b88efbaeb04d64e057a2d3461d453c4c379ea2e9499f3619c2 | What is the probability that two randomly chosen integers are coprime (i.e., their greatest common divisor is 1)? | [
"\\dfrac{6}{\\pi^2}"
] | [
"What is the probability that two randomly chosen integers are coprime (i.e., their greatest common divisor is 1)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c4aaab47907bffc8b5a06ce45fced146ecde43fbd16319ca611cf2c0d4151b3 | What is the probability that two randomly chosen integers are coprime? | [
"\\(\\frac{6}{\\pi^2}\\)"
] | [
"What is the probability that two randomly chosen integers are coprime?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c36b3510f73b71963b3148a568a45262074079b94839ec4e81b03ecba4db54fe | What is the probability that two randomly chosen integers are relatively prime? | [
"\\(\\frac{6}{\\pi^2}\\)"
] | [
"What is the probability that two randomly chosen integers are relatively prime?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-ab39c3b9cd44ac4af4358e8120f17314bdf393f41592fd31658c1801c1850d53 | What is the probability that two randomly chosen numbers are relatively prime (i.e., their greatest common divisor is 1)? | [
"\\dfrac{6}{\\pi^2}"
] | [
"What is the probability that two randomly chosen numbers are relatively prime (i.e., their greatest common divisor is 1)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-090eb71994b63008a6a80f0e7c483802a1785d817591fcc34a9925c9127cd6ac | What is the probability that two randomly chosen permutations from the set {1, 2, ..., n} commute (i.e., fg = gf)? | [
"\\dfrac{p(n)}{n!}"
] | [
"What is the probability that two randomly chosen permutations from the set {1, 2, ..., n} commute (i.e., fg = gf)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-01d9b27384fb93064a43bfa2fbab10497ebe24a6a573d046e2087aa88d1e0e59 | What is the probability that two randomly chosen persons, A and B, have the same birthday? | [
"\\(\\frac{1}{365}\\)"
] | [
"What is the probability that two randomly chosen persons, A and B, have the same birthday?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-840512977866528b63904cc48f1d0d401b17e408c2a08dec7015d4f69b39e25d | What is the probability that two randomly chosen vertices of a regular dodecahedron are the endpoints of the same edge? | [
"\\(\\frac{2}{19}\\)"
] | [
"What is the probability that two randomly chosen vertices of a regular dodecahedron are the endpoints of the same edge?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0961974e36857961860bb1058a74bd5b37dd6a3080ccd72d03754f58b5493c6d | What is the probability that two randomly selected intervals from the interval [0,1] intersect? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that two randomly selected intervals from the interval [0,1] intersect?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-eff2f5428f22e2867095931613f581524f6a62df7fd853472521a842e087d665 | What is the probability that two specific players, X and Y, will be paired against each other in a bowling tournament with 8 players, where the players are randomly paired in the quarterfinals? | [
"\\dfrac{1}{7}"
] | [
"What is the probability that two specific players, X and Y, will be paired against each other in a bowling tournament with 8 players, where the players are randomly paired in the quarterfinals?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b5d120e02a200aa5cb6c60b98fbe1f50d5c716ea5ac36bdc639905089936bd0 | What is the probability that two students share a Spanish, Biology, English, and History class, given there are seven periods (A-G), with five periods being school subjects, and the other two periods not necessary? There are five Biology teachers, two Math teachers, four English teachers, three History teachers, and fi... | [
"\\(\\frac{2}{9}\\)"
] | [
"What is the probability that two students share a Spanish, Biology, English, and History class, given there are seven periods (A-G), with five periods being school subjects, and the other two periods not necessary? There are five Biology teachers, two Math teachers, four English teachers, three History teachers, a... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80d5de983e875ae6f0bec57ec30a234c8a676e837095fcda835163b024fe64d1 | What is the probability that we stop throwing a fair die after an odd number of throws, given that we stop when the last three numbers thrown are 1, 3, and 5 in that order? | [
"\\dfrac{216}{431}"
] | [
"What is the probability that we stop throwing a fair die after an odd number of throws, given that we stop when the last three numbers thrown are 1, 3, and 5 in that order?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6101185f3c53c14e07bb4769104631b656f2bddd136c358bd97319c81903874f | What is the probability that when 2 lots of shares are purchased, they are both purchased by the same person, given that both Al and Bob buy shares with probability \( p \) on each day, and Al buys first, followed by Bob, and so on? | [
"\\( 2p^2 (1 - p) \\)"
] | [
"What is the probability that when 2 lots of shares are purchased, they are both purchased by the same person, given that both Al and Bob buy shares with probability \\( p \\) on each day, and Al buys first, followed by Bob, and so on?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ccaae8d98b1eef68b716ad3a86b92091ec96f599eec798697f677a7ef2f87d0a | What is the probability that when 5 cards are drawn from a 52-card deck, at least 3 of them are spades? | [
"\\(\\frac{\\binom{13}{3}\\binom{39}{2} + \\binom{13}{4}\\binom{39}{1} + \\binom{13}{5}}{\\binom{52}{5}}\\)"
] | [
"What is the probability that when 5 cards are drawn from a 52-card deck, at least 3 of them are spades?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-340b76e0d351eb1cdd56597dbd08a04a168f5d0544eff356971e13473df19063 | What is the probability that when 8 fair coins are flipped, there is no sequence with two consecutive heads? | [
"55/256"
] | [
"What is the probability that when 8 fair coins are flipped, there is no sequence with two consecutive heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-47e1d83ab94428a69b48bf85df95ef380ad12121cf80f2101a394ce1e921a625 | What is the probability that when Kayla rolls four fair 6-sided dice, at least one of the numbers is greater than 4 and at least two of the numbers are greater than 2? | [
"\\(\\frac{61}{81}\\)"
] | [
"What is the probability that when Kayla rolls four fair 6-sided dice, at least one of the numbers is greater than 4 and at least two of the numbers are greater than 2?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-df8eeab37831545519826a154dfb4f2a4e9dff1314f7b4fd6b2f3583675ec807 | What is the probability that when a coin is flipped six times, the 3rd flip is heads and there are more tails than heads? | [
"\\(\\frac{3}{32}\\)"
] | [
"What is the probability that when a coin is flipped six times, the 3rd flip is heads and there are more tails than heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-0d618b3b3634249aa330599dba443378257b272b42c70dca9b91ef78317c0742 | What is the probability that when a five-card hand is dealt from a standard 52-card deck, at least two of the cards will have the same denomination? Round the answer to the nearest hundredth. | [
"0.49"
] | [
"What is the probability that when a five-card hand is dealt from a standard 52-card deck, at least two of the cards will have the same denomination? Round the answer to the nearest hundredth."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-5be8bed257701594f8e1b3783848123ec3c8b74a9d4b8fc98eb14ac141afb554 | What is the probability that when drawing 5 cards from a deck of 40 cards (which contains exactly 2 Kuribohs and 38 other unique cards), at least 1 Kuriboh is drawn? | [
"\\(\\dfrac{37}{156}\\)"
] | [
"What is the probability that when drawing 5 cards from a deck of 40 cards (which contains exactly 2 Kuribohs and 38 other unique cards), at least 1 Kuriboh is drawn?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-36cf153df52aa8be180d8fc9601ac58f308fb4c60e71d9e64b519415287ea0b3 | What is the probability that when successively flipping a coin, one gets a chain of three consecutive heads before two consecutive tails? | [
"\\dfrac{3}{10}"
] | [
"What is the probability that when successively flipping a coin, one gets a chain of three consecutive heads before two consecutive tails?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-07bcf50d918c8a859ccdce92df6b553234d79d0b2bec6678a433e98140723e45 | What is the probability that when we roll four fair 6-sided dice, they won't all show the same number? | [
"\\frac{215}{216}"
] | [
"What is the probability that when we roll four fair 6-sided dice, they won't all show the same number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-603f16a83255e38af532f0da15faef3e0ac1abd1118c882c269164a93ec5ffe1 | What is the probability that when you combine Peter’s and Kyle’s dice into one set and roll all of them at once, you roll at least one $1$? Express your answer as a common fraction. | [
"\\(\\frac{12709}{16384}\\)"
] | [
"What is the probability that when you combine Peter’s and Kyle’s dice into one set and roll all of them at once, you roll at least one $1$? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a65352572c81dd6e76a1a276ab65881d7d8af427affe54d914039872d2d01c7 | What is the probability that you get more tails than your friend when you flip a coin 8 times and your friend flips a coin 7 times? | [
"\\(\\dfrac{1}{2}\\)"
] | [
"What is the probability that you get more tails than your friend when you flip a coin 8 times and your friend flips a coin 7 times?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 18 | 0 | 0 |
omr-ws-9181dd8c3f02265d522cbc1b6e5fcf60627c14d8f1ff4a86b141f8dab46a6550 | What is the probability that you will roll a 3 before you roll both a 1 and a 2 if you roll a fair 6-sided die repeatedly? | [
"\\(\\frac{2}{3}\\)"
] | [
"What is the probability that you will roll a 3 before you roll both a 1 and a 2 if you roll a fair 6-sided die repeatedly?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-d049afab15afdc1cd9037044c20615baa89f15d4c95c9cd338c6d7fbd5b81ffe | What is the probability that, after $n$ drawings with replacement from a bag containing three balls of distinct colors, all three colors occur? | [
"\\(\\frac{3^{n-1} - 2^n + 1}{3^{n-1}}\\)"
] | [
"What is the probability that, after $n$ drawings with replacement from a bag containing three balls of distinct colors, all three colors occur?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 18
} | 35 | 0 | 0 |
omr-ws-2ef726fea3a80592d78e6cdd0b777935f01199484904bb977ea8bbed590d3258 | What is the probability that, after $n$ drawings with replacement from a bag containing three balls of distinct colors, exactly two colors occur? | [
"\\(\\frac{2^n - 2}{3^{n-1}}\\)"
] | [
"What is the probability that, after $n$ drawings with replacement from a bag containing three balls of distinct colors, exactly two colors occur?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-2a5ffe13ef195edbfbfd0cc617765de3080cdb23aab2a81c37892fffe86c3472 | What is the probability that, after rolling Jim's 3-sided die (with faces 0, 1, and 2) three times and Barbara's 6-sided die (with faces 1, 2, 3, 4, 5, and 6) once, the sum of Jim's rolls will be greater than Barbara's roll? | [
"\\dfrac{55}{162}"
] | [
"What is the probability that, after rolling Jim's 3-sided die (with faces 0, 1, and 2) three times and Barbara's 6-sided die (with faces 1, 2, 3, 4, 5, and 6) once, the sum of Jim's rolls will be greater than Barbara's roll?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-492a2f813e497d4cf1a5e11b14223b61451bc4afd380a1e704a2ac39b8489fcb | What is the probability that, from 6 randomly picked balls (each with an equal probability of 0.2 of being one of 5 different colors: white, black, blue, red, green), there are both white and black balls? | [
"\\(\\frac{8162}{15625}\\)"
] | [
"What is the probability that, from 6 randomly picked balls (each with an equal probability of 0.2 of being one of 5 different colors: white, black, blue, red, green), there are both white and black balls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-8ad51fd399802bc4c284a6a9668aca8c1633585860149090837a91c96771cdc4 | What is the probability that, given a class of 30 students, 2 or more of them have the same birthday? | [
"0.7063"
] | [
"What is the probability that, given a class of 30 students, 2 or more of them have the same birthday?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 27
} | 45 | 0 | 0 |
omr-ws-f00feb883e4cbb2de70ae8f0b7751cdee5e7a62674d8b2b28f3f062891bedc0d | What is the probability that, when 6 dice are rolled, we can choose four of them such that the sum of the numbers on their upper faces is 11? | [
"\\dfrac{56}{1443}"
] | [
"What is the probability that, when 6 dice are rolled, we can choose four of them such that the sum of the numbers on their upper faces is 11?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-5e0342a8904710a533adf239b56997071baefbdeec1380c9e834c285b6e31fdb | What is the probability that, when converting a random three-digit number to base 8, any three consecutive digits from the resulting number will form a Pythagorean triple? | [
"\\dfrac{1}{225}"
] | [
"What is the probability that, when converting a random three-digit number to base 8, any three consecutive digits from the resulting number will form a Pythagorean triple?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-00ab6e2336a4af154da0480c69fb50782319f6c17505013a94eda93686235205 | What is the probability that, when throwing a fair coin consecutively, you will get 10 heads in a row before you get 10 tails (not necessarily in a row)? | [
"\\(1 - \\left(\\frac{1023}{1024}\\right)^{10}\\)"
] | [
"What is the probability that, when throwing a fair coin consecutively, you will get 10 heads in a row before you get 10 tails (not necessarily in a row)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-4a817fb8e326804826a114284c819d87d5818722eb5653a5136868c84f7709a4 | What is the probability to arrange 5 men and 9 women around a circle such that no man is alone? (Each person adjacent to the man is another woman) | [
"\\(\\frac{14}{143}\\)"
] | [
"What is the probability to arrange 5 men and 9 women around a circle such that no man is alone? (Each person adjacent to the man is another woman)"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-eb7306a78aa6bb7b819fe5f6a45646aac4f4280bdaaccf25d325fb01768c6199 | What is the probability, expressed as a decimal, of drawing one marble which is either red or blue from a bag containing 3 red, 2 blue, and 5 yellow marbles? | [
"0.5"
] | [
"What is the probability, expressed as a decimal, of drawing one marble which is either red or blue from a bag containing 3 red, 2 blue, and 5 yellow marbles?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11d5514e2efaf8d5257aec2897d56dd17bb4463a19105e32a1338c666de6ea0e | What is the probability, in an infinite sequence of tosses of a fair coin, to obtain exactly one occurrence of the sequence HHH and never have more than three consecutive heads (H)? | [
"0"
] | [
"What is the probability, in an infinite sequence of tosses of a fair coin, to obtain exactly one occurrence of the sequence HHH and never have more than three consecutive heads (H)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 40
} | 40 | 0 | 0 |
omr-ws-2a48181b3ea992b5caa5cedc776fd57f5ef34837aca3a076a157025e2cf9a75d | What is the procedure for converting the matrix $ \begin{pmatrix} 1 & 1 \\ 2 & 4 \\ \end{pmatrix} $ to Jordan form? | [
"\\begin{pmatrix} \\frac{5 + \\sqrt{17}}{2} & 0 \\\\ 0 & \\frac{5 - \\sqrt{17}}{2} \\end{pmatrix}"
] | [
"What is the procedure for converting the matrix $ \\begin{pmatrix} 1 & 1 \\\\ 2 & 4 \\\\ \\end{pmatrix} $ to Jordan form?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ecd5ffb37dd29d52a08fb694f217911428d3515e3eabe4e49f0530117de818e | What is the procedure to find the continued fraction representation of $\sqrt{7}$? | [
"[2; \\overline{1, 1, 1, 4}]"
] | [
"What is the procedure to find the continued fraction representation of $\\sqrt{7}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-43658477509997b52d08962ccea75cde4dee6e477fbed7246c4c2e97f4ff23ba | What is the product of $-4\sqrt{2} + 4\sqrt{2}i$ and $3 + \sqrt{3}i$ in polar form (in radians)? | [
"$16e^{i\\frac{11\\pi}{12}}$"
] | [
"What is the product of $-4\\sqrt{2} + 4\\sqrt{2}i$ and $3 + \\sqrt{3}i$ in polar form (in radians)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-a357802780b8f50cdd63544523f26ef2e1a1333730e22085ecebdd5c939bfc2d | What is the product of $\cos\left(\frac{r\pi}{2n}\right)$ as $r$ varies from 1 to $n-1$? | [
"\\dfrac{\\sqrt{n}}{2^{n-1}}"
] | [
"What is the product of $\\cos\\left(\\frac{r\\pi}{2n}\\right)$ as $r$ varies from 1 to $n-1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-77e9e11e8581acc6ae7368aa3c0e8a3c90f96ae91ba5e92333df359f6d42235b | What is the product of $\frac{1}{5}$ and $\frac{3}{7}$ ? | [
"\\frac{3}{35}"
] | [
"What is the product of $\\frac{1}{5}$ and $\\frac{3}{7}$ ?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e8d84e888f65d4058f81fb7f0fbd92d1e5e1c4185305745a6841717e0d4f006 | What is the product of $\sqrt[3]{4}$ and $\sqrt[4]{8}$? | [
"\\(2 \\sqrt[12]{32}\\)"
] | [
"What is the product of $\\sqrt[3]{4}$ and $\\sqrt[4]{8}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-eb10de1eaade5eed1a0c83dc2d630b4a40a6e6d3e3415edd5f0ca84167901e3c | What is the product of 385 and 1001? | [
"385385"
] | [
"What is the product of 385 and 1001?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13b97b460318abd81841dbd1b283d1afac52251ed458c00d3a7ab94ea7c45a62 | What is the product of all odd integers less than $10000$? (Express in factorial form.) | [
"\\(\\frac{10000!}{2^{5000} \\times 5000!}\\)"
] | [
"What is the product of all odd integers less than $10000$? (Express in factorial form.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 8
} | 26 | 0 | 0 |
omr-ws-6b906758b99266a55ba1a8ed2fe83af9615d0a8642aef1b5ea11107769675278 | What is the product of all odd positive integers less than 10000? | [
"\\(\\frac{10000!}{2^{5000} \\cdot 5000!}\\)"
] | [
"What is the product of all odd positive integers less than 10000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-7b8d4c1b638ad04f9e83f380e65aa15dd7abd82e2f6eb6bededb6ea7dbebb953 | What is the product of all positive odd integers less than $1000$? (Using factorial notation where necessary.) | [
"\\(\\frac{1000!}{2^{500} \\cdot 500!}\\)"
] | [
"What is the product of all positive odd integers less than $1000$? (Using factorial notation where necessary.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 9
} | 29 | 0 | 0 |
omr-ws-20809c7370f569824f6238618d85bb4e98ec7c16bc84e7080dac73956cd1f58f | What is the product of all prime numbers less than or equal to 200 modulo 9? | [
"0"
] | [
"What is the product of all prime numbers less than or equal to 200 modulo 9?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34f4eefeef254c3a884a85c1c95fcbf5a3470810327b2f6a0efa3f2bc1645f59 | What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$? | [
"81"
] | [
"What is the product of all real numbers $x$ such that the distance on the number line between $\\log_6x$ and $\\log_69$ is twice the distance on the number line between $\\log_610$ and $1$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-8dbd8c961a468874f6963e930242e85a3f5faa97c02eb61b9c8c480e5720df0c | What is the product of all real numbers that are doubled when added to their reciprocals? | [
"-1"
] | [
"What is the product of all real numbers that are doubled when added to their reciprocals?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88629aa0f08b1edf4c60968443b1b419d7a418a7d77c9bfa156198e75a6ff0da | What is the product of all real solutions \( x \) of the equation \[\log_{7x}2023 \cdot \log_{289x}2023 = \log_{2023x}2023?\] | [
"1"
] | [
"What is the product of all real solutions \\( x \\) of the equation \\[\\log_{7x}2023 \\cdot \\log_{289x}2023 = \\log_{2023x}2023?\\]"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-3eccc4838f57ab78c4b5515c8ba2213d987ddb76c4b53ebfb02f5fc91fa2d7ca | What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$? | [
"10"
] | [
"What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-694d20c9543189b48e62e425e2b5592aa905d729e66488bb5b0b78e9fce7dfb9 | What is the product of all the positive integer divisors of 12? | [
"1728"
] | [
"What is the product of all the positive integer divisors of 12?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54edd6d68cc502bbb85d16e7de9b6f2f50c4edf7bad4c50d0bddfea980b839b3 | What is the product of all the roots of the equation \(x^{10} = 1\)? | [
"-1"
] | [
"What is the product of all the roots of the equation \\(x^{10} = 1\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e32770b920381c0dafe9606245f8a65d8792a79b8cdd99a52f7d1f5c880d03e | What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$ | [
"1"
] | [
"What is the product of all the solutions to the equation $$\\log_{7x}2023 \\cdot \\log_{289x} 2023 = \\log_{2023x} 2023?$$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ea241387b6695ca9152b8a8f7e169786f82a726e74960d22fc6c22428d6a13b5 | What is the product of all values of $x$ that satisfy the equation $x^2 + 47|x| = 1340$? | [
"-400"
] | [
"What is the product of all values of $x$ that satisfy the equation $x^2 + 47|x| = 1340$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-fe68671bef40f2cf8ce0d7ef456bee2ed0917776d5ae1f71a80c0dec9f19348c | What is the product of real numbers $a$ which make $x^2 + ax + 1$ a negative integer for only one real number $x$? | [
"-8"
] | [
"What is the product of real numbers $a$ which make $x^2 + ax + 1$ a negative integer for only one real number $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-5a4a42cb080182c10363b0c4ffafd438df4da28624b0b55b593e34f81c49d5c9 | What is the product of the coordinates of the midpoint of a line segment with endpoints at $(2,3)$ and $(-6,5)$? | [
"-8"
] | [
"What is the product of the coordinates of the midpoint of a line segment with endpoints at $(2,3)$ and $(-6,5)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60f17014f894ec90822a66766c990205f098c83871656b91f9788a3a56232700 | What is the product of the digits in the base 8 representation of $6543_{10}$? | [
"168"
] | [
"What is the product of the digits in the base 8 representation of $6543_{10}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f02ca86dd536b6c064af4416e57a06a698e6baf810e71a8c9967be1cc97b844d | What is the product of the digits of $9!$ plus the product of the digits of $4!$? | [
"8"
] | [
"What is the product of the digits of $9!$ plus the product of the digits of $4!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-663f513ca1d324e98e9f3021008229b35a00ab5214dec60878e0c28cab34d9d3 | What is the product of the least common multiple and the greatest common factor of $20$ and $90$? | [
"1800"
] | [
"What is the product of the least common multiple and the greatest common factor of $20$ and $90$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed20c9d64b42a8fd423eae3893331bada89df218114cba12a14edefafdc3be96 | What is the product of the lengths of all sides and diagonals emanating from a single vertex of a regular n-gon inscribed in the unit circle? | [
"n"
] | [
"What is the product of the lengths of all sides and diagonals emanating from a single vertex of a regular n-gon inscribed in the unit circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-08fa8684a70a7c3ed017ab896b696e4fa69e55e1ba3d7ea51197bb9b5ccb9460 | What is the product of the lengths of the diagonals and sides of a regular $n$-gon inscribed in a unit circle? | [
"\\( n^{\\frac{n}{2}} \\)"
] | [
"What is the product of the lengths of the diagonals and sides of a regular $n$-gon inscribed in a unit circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 8
} | 56 | 0 | 0 |
omr-ws-906ebb45c16b50864703ace2a03847d19a04d48dd43e2e696ba8bc68f1ce59a1 | What is the product of the lengths of the sides and diagonals of a regular $n$-gon inscribed in a unit circle? | [
"\\( n^{\\frac{n}{2}} \\)"
] | [
"What is the product of the lengths of the sides and diagonals of a regular $n$-gon inscribed in a unit circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-0e8e722a92ab7735c05c6af72d4bf524106f90667614cd9dd1cc5f2ea3af8247 | What is the product of the numerator and the denominator when $0.\overline{009}$ is expressed as a fraction in lowest terms? | [
"111"
] | [
"What is the product of the numerator and the denominator when $0.\\overline{009}$ is expressed as a fraction in lowest terms?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b072831710de8723f5d4e1945e2f758b5ad3023d816e608d1f4ddefdc96b0bc7 | What is the product of the positive divisors of 6480 that are multiples of 12? | [
"\\(540^{12} \\cdot 12^{24}\\)"
] | [
"What is the product of the positive divisors of 6480 that are multiples of 12?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 19
} | 36 | 0 | 0 |
omr-ws-f7a89e4ff0f8befa9646d9fefd4dfccf22423895e39833b89d31570c16eb4c9e | What is the product of the positive divisors of \( n \in \mathbb{Z}^+ \) where \( n = \prod_{i=1}^r p_i^{r_i} \) and \( p_i \)'s are primes with \( r_i \ge 0 \)? | [
"\\( n^{\\frac{k}{2}} \\)"
] | [
"What is the product of the positive divisors of \\( n \\in \\mathbb{Z}^+ \\) where \\( n = \\prod_{i=1}^r p_i^{r_i} \\) and \\( p_i \\)'s are primes with \\( r_i \\ge 0 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-fc1145dd2a372dd9c42beb9cd3b31aeee13bd14b974ff0b4ee0811b4753c1257 | What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$? | [
"020"
] | [
"What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d7af20d678741bf57abcb8036d7f48f9c368fbcf2e95698be8fe5759f17cd824 | What is the product of the real roots of the equation $x^2 + 18x + 30 = 2\sqrt{x^2 + 18x + 45}$? | [
"20"
] | [
"What is the product of the real roots of the equation $x^2 + 18x + 30 = 2\\sqrt{x^2 + 18x + 45}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-65ce7358f520d0558445a42c29cb61b50b61ab99aa2e2f10c5a2cb53a277779b | What is the product of the real roots of the equation \( 3x^2 + 10x + 61 = 2\sqrt{3x^2 + 10x + 76} \)? | [
"17"
] | [
"What is the product of the real roots of the equation \\( 3x^2 + 10x + 61 = 2\\sqrt{3x^2 + 10x + 76} \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-13d8725c2daf96e899d1d82e4781cfa88e84c263fcc7bad04d91fdc797d8199f | What is the product of the real roots of the equation \( x^2 + 18x + 30 = 2\sqrt{x^2 + 18x + 45} \)? | [
"20"
] | [
"What is the product of the real roots of the equation \\( x^2 + 18x + 30 = 2\\sqrt{x^2 + 18x + 45} \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-46290b4d76d990a20d0881dfabe1b60b78c4f75918b1a3e98637282be88c53fb | What is the product of the real roots of the equation \( x^{2} + 18x + 30 = 2\sqrt{x^{2} + 18x + 45} \)? | [
"20"
] | [
"What is the product of the real roots of the equation \\( x^{2} + 18x + 30 = 2\\sqrt{x^{2} + 18x + 45} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-43ba1a03c9eacd930a51357e0ea4caff91e5c611db7016efd9551b64816cdd28 | What is the product of the real roots of the equation \(x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\)? | [
"20"
] | [
"What is the product of the real roots of the equation \\(x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-e8c0022808d74962b7a02612fe02623b6162ddb2d3f658dbca2c91dd3a929bb1 | What is the product of the real roots of the equation \(x^2 + 18x + 30 = 2\sqrt{x^2 + 18x + 45}\)? | [
"20"
] | [
"What is the product of the real roots of the equation \\(x^2 + 18x + 30 = 2\\sqrt{x^2 + 18x + 45}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-109234eb6fe41382e5df502fe90ac11ba278b448dca186ac20c8d60dedd118e3 | What is the product of the real roots to the equation \(x^2 + 18x + 30 = \sqrt{x^2 + 18x + 45}\)? | [
"\\dfrac{59 - \\sqrt{61}}{2}"
] | [
"What is the product of the real roots to the equation \\(x^2 + 18x + 30 = \\sqrt{x^2 + 18x + 45}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73349f2475825c58d83e9ec19cd72c5aa837cd59a132415fd2d001a649af4d41 | What is the product of the real zeros of the polynomial \( p(x) = x^4 + 4x^3 + 6x^2 + 4x - 2011 \)? | [
"\\( 1 - \\sqrt{2012} \\)"
] | [
"What is the product of the real zeros of the polynomial \\( p(x) = x^4 + 4x^3 + 6x^2 + 4x - 2011 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 7
} | 16 | 0 | 0 |
omr-ws-8c7117b8a096c5e34ae781b8074a005cada8fb894d1c1d07c9df04ec30d5c395 | What is the product of the roots of the equation \(\sqrt[3]{x+8} + \sqrt[3]{x-8} = 1\)? | [
"\\(\\frac{1729}{8}\\)"
] | [
"What is the product of the roots of the equation \\(\\sqrt[3]{x+8} + \\sqrt[3]{x-8} = 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-6fdf26da6e5fe3a331e14dcb4fa8781ee1ac2c1410004726ded3c453a419aa34 | What is the product of the solutions of the equation $-35=-x^2-2x?$ | [
"-35"
] | [
"What is the product of the solutions of the equation $-35=-x^2-2x?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39732757e7241518ca26f2b215eb23ab6f232aaffeece3e508a23306ab3afc8d | What is the product of the two largest one-digit primes and the largest two-digit prime? | [
"3395"
] | [
"What is the product of the two largest one-digit primes and the largest two-digit prime?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-27aa5166ca3c7fb932946b17c94c289e5915f80707572f67d37f487ce6440f8a | What is the product of the two smallest prime factors of \( 2^{1024} - 1 \)? | [
"15"
] | [
"What is the product of the two smallest prime factors of \\( 2^{1024} - 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e86312efc620f9deb1e8c9e234b6da0329291202f3fdd4eb6abfedb78c2b1fb | What is the product of the uncommon real roots of the polynomials $x^4 + 2x^3 - 8x^2 - 6x + 15$ and $x^3 + 4x^2 - x - 10$? | [
"6"
] | [
"What is the product of the uncommon real roots of the polynomials $x^4 + 2x^3 - 8x^2 - 6x + 15$ and $x^3 + 4x^2 - x - 10$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eeb7d466e06092ff0987cf81118c46dc8e83d446a49450a9519098a0611c1faf | What is the projection of the point $(4,5)$ onto the line $x = y$? | [
"\\(\\left( \\frac{9}{2}, \\frac{9}{2} \\right)\\)"
] | [
"What is the projection of the point $(4,5)$ onto the line $x = y$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-9d2ea4d998079c06649d6f4af283eeac28a0e0746ca955196fa19a33b351803e | What is the proof for the minimum number of knights required to cover an 8x8 chessboard, which is known to be 32? | [
"32"
] | [
"What is the proof for the minimum number of knights required to cover an 8x8 chessboard, which is known to be 32?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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