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-73f5f9b1fa2fb1fdb690c5f3e8b215f40c80a1508377c6ee10af26b3329bd376 | What is the probability that a coin flip results in heads and the sum of the numbers on a 6-sided die and a 10-sided die is 8? | [
"\\(\\frac{1}{20}\\)"
] | [
"What is the probability that a coin flip results in heads and the sum of the numbers on a 6-sided die and a 10-sided die is 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-08826cd7eec0c9e818fab0fc2e52cfdcb0b89985fcf89bf31c77671f1593674f | What is the probability that a coin is facing the same way three times in a row (either three heads or three tails) exactly once in 21 tosses? | [
"\\dfrac{59155}{1048576}"
] | [
"What is the probability that a coin is facing the same way three times in a row (either three heads or three tails) exactly once in 21 tosses?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-01b89e012ca631b322984ebde950ab158ae351538d885cfd7eea71ba06baec82 | What is the probability that a coin lands heads up exactly 50 times out of 100 tosses? | [
"\\(\\frac{\\binom{100}{50}}{2^{100}}\\) or approximately \\(0.0795892374\\)"
] | [
"What is the probability that a coin lands heads up exactly 50 times out of 100 tosses?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-87025f00e5057fccc883b85920eb1567c34b53e4cf7ca3757dc380767ae9fdb0 | What is the probability that a coin will be flipped exactly 12 times to get a sequence of a head followed by two tails? | [
"\\(\\frac{143}{4096}\\)"
] | [
"What is the probability that a coin will be flipped exactly 12 times to get a sequence of a head followed by two tails?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 17
} | 36 | 0 | 0 |
omr-ws-2d97da9ee49fa3c1a316bcc5942111e71f70b2eccb74a4fb52c586c6cec365c3 | What is the probability that a computer will store three $1$s before two $0$s in a document, given that it stores a $1$ with probability $\frac23$ and a $0$ with probability $\frac13$, and each time it stores a $1$ it removes all $0$s, and each time it stores a $0$ it removes all $1$s? | [
"\\dfrac{32}{51}"
] | [
"What is the probability that a computer will store three $1$s before two $0$s in a document, given that it stores a $1$ with probability $\\frac23$ and a $0$ with probability $\\frac13$, and each time it stores a $1$ it removes all $0$s, and each time it stores a $0$ it removes all $1$s?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94abcf51117f22d6b9def32c59a4396815e21a852e0c031c12e8060f563934dd | What is the probability that a dart hitting a dart board lands inside the circle but outside the inscribed square? | [
"\\( 1 - \\frac{2}{\\pi} \\)"
] | [
"What is the probability that a dart hitting a dart board lands inside the circle but outside the inscribed square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-db01e052e46868d86cec9cd12fc9f36776488f4e8fc988ba5203a0b59f2242bd | What is the probability that a dart hitting a square board at random lands nearer the centre than the edge? | [
"\\(\\frac{4\\sqrt{2}-5}{3}\\)"
] | [
"What is the probability that a dart hitting a square board at random lands nearer the centre than the edge?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 16
} | 28 | 0 | 0 |
omr-ws-7aad16764151abb6b5152366b36b77b2bcba638606f0711200da8409197f1ba2 | What is the probability that a digital number (a number consisting of all the digits from 1 to 9, each used exactly once) will be divisible by 7? | [
"\\dfrac{1}{7}"
] | [
"What is the probability that a digital number (a number consisting of all the digits from 1 to 9, each used exactly once) will be divisible by 7?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7bba43a6b47557eb722411671654bd9cd8c4b09208dd468b0b01d4d7262f4ce8 | What is the probability that a digital number (a number consisting of all the digits from 1 to 9, each used exactly once) will be of the form $9p$, where $p$ is a prime? | [
"0"
] | [
"What is the probability that a digital number (a number consisting of all the digits from 1 to 9, each used exactly once) will be of the form $9p$, where $p$ is a prime?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-090086f94e3fca0861d402e8dde5e83b02aeb0f42aa3f3dca0af07f70c4436a5 | What is the probability that a fair coin flipped 10 times results in at least 4 consecutive heads? | [
"\\(\\frac{251}{1024}\\)"
] | [
"What is the probability that a fair coin flipped 10 times results in at least 4 consecutive heads?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1a1c24680ca73d6cf0578dbf7e8cd2b1e050842297c42e1500b6d1c613d5bde6 | What is the probability that a fair coin will land on heads exactly 51 times out of 100 flips? | [
"\\(\\frac{\\binom{100}{51}}{4^{50}}\\)"
] | [
"What is the probability that a fair coin will land on heads exactly 51 times out of 100 flips?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-cf29a623b2a7b3adeae2d74c0a26db3b7a91aae65f315e66239cb2e1e259ec8d | What is the probability that a fair coin will land on heads exactly \(N+1\) times out of \(2N\) flips? | [
"\\(\\frac{C_{2N}^{N+1}}{4^N}\\)"
] | [
"What is the probability that a fair coin will land on heads exactly \\(N+1\\) times out of \\(2N\\) flips?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-ee2ae4b86658fb3c19b72f39e5492cb9d35649a5b7082e207ea9f5d9f3303042 | What is the probability that a five-card hand has at least one of each of the four values: ace, king, queen, and jack? | [
"\\dfrac{32}{7735}"
] | [
"What is the probability that a five-card hand has at least one of each of the four values: ace, king, queen, and jack?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c2b38a20ce56f4933c1e64e7ce6938f42f23f3c70fffdc09085153c99eb588b | What is the probability that a five-card hand has the same number of hearts and spades? | [
"\\( \\frac{12757}{49980} \\)"
] | [
"What is the probability that a five-card hand has the same number of hearts and spades?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-6e92a10052e69842cba50988ddd75729df653c69c5699803054c6084ded622c7 | What is the probability that a five-card poker hand has a straight (a set of five consecutive values with the initial card value between Ace and 9)? | [
"$\\frac{5}{1274}$"
] | [
"What is the probability that a five-card poker hand has a straight (a set of five consecutive values with the initial card value between Ace and 9)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-cc541f6b19e912b060676640180bba078f10ff38265bdf7d20a6a55052206079 | What is the probability that a five-digit number formed using the digits $0, 1, 2, 3, 4, 5$ without repetition is divisible by $11$? | [
"\\(\\frac{1}{10}\\)"
] | [
"What is the probability that a five-digit number formed using the digits $0, 1, 2, 3, 4, 5$ without repetition is divisible by $11$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-570abb07df00c9a58ab3b517d2d6f2b92db799e38315e4742cdc2cb6e18f69ad | What is the probability that a four-digit number formed by rolling a fair six-sided die four times is divisible by 7? | [
"\\(\\frac{1}{7} + \\frac{1}{7 \\cdot 6^3}\\)"
] | [
"What is the probability that a four-digit number formed by rolling a fair six-sided die four times is divisible by 7?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2f9b96137dcf3139e67880df537d43accbda679f940a5ee934be95707325822 | What is the probability that a four-digit number formed using the digits $1, 2, 3, 4, 5, 6,$ and $7$ (with all digits being different) is divisible by $5$? | [
"\\dfrac{1}{7}"
] | [
"What is the probability that a four-digit number formed using the digits $1, 2, 3, 4, 5, 6,$ and $7$ (with all digits being different) is divisible by $5$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca0e49d32a06ad27c9dd300639d465cae5271a4f69b89aabb03ec0c71e88151e | What is the probability that a four-digit number, formed at random from the digits $1, 2, 3, 4, 5$ with all digits being different, is greater than $2345$? | [
"\\(\\frac{17}{24}\\)"
] | [
"What is the probability that a four-digit number, formed at random from the digits $1, 2, 3, 4, 5$ with all digits being different, is greater than $2345$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eeaaa86cecad3f2f074982a30fb5d214efc96e593c2745be61237271a03a7a63 | What is the probability that a four-number lock combination forms a geometric sequence in order, where each number is between 0 and 39? | [
"\\dfrac{13}{640000}"
] | [
"What is the probability that a four-number lock combination forms a geometric sequence in order, where each number is between 0 and 39?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-50a42753031b7db5bd941b999d694f7255aad19b6bc79d70418d20025e6432fd | What is the probability that a four-number lock combination forms an arithmetic sequence in order, where each number is between 0 and 39? | [
"\\dfrac{267}{1280000}"
] | [
"What is the probability that a four-number lock combination forms an arithmetic sequence in order, where each number is between 0 and 39?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5cb1c5254c863302ab882706355655155ca44122993961d2667e9b133fd4c0fa | What is the probability that a geyser explodes in any given 23-minute interval, given that there is a 3/4 chance of it exploding in any given 10-minute interval? | [
"\\( 1 - \\left(\\frac{1}{4}\\right)^{23/10} \\)"
] | [
"What is the probability that a geyser explodes in any given 23-minute interval, given that there is a 3/4 chance of it exploding in any given 10-minute interval?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ee41ca84989740269687c2bf1b3cba2a75b5c6d7c1c5d3bb9baa6f74ca576122 | What is the probability that a given permutation \(a_1, a_2, \ldots, a_n\) is a derangement and also has no cycles of length 2? | [
"\\sum_{m=0}^{\\left\\lfloor \\frac{n}{2} \\right\\rfloor} \\frac{(-1)^m}{2^m m!} \\sum_{k=0}^{n - 2m} \\frac{(-1)^k}{k!}"
] | [
"What is the probability that a given permutation \\(a_1, a_2, \\ldots, a_n\\) is a derangement and also has no cycles of length 2?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-e9cd029150aff318416b71ced7a77c36665d0c402a125febe7552875806275cb | What is the probability that a given permutation of \(a_1, a_2, \ldots, a_n\) has no cycles of length 2? | [
"\\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\frac{(-1)^k}{2^k k!}"
] | [
"What is the probability that a given permutation of \\(a_1, a_2, \\ldots, a_n\\) has no cycles of length 2?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-24cb843677fb8b326780068dfec779b8078218a8b0db272b6b32c4b7d346a7b3 | What is the probability that a group of 5 cards dealt at random from a 52-card deck will contain exactly 2 pairs? | [
"\\(\\frac{198}{4165}\\)"
] | [
"What is the probability that a group of 5 cards dealt at random from a 52-card deck will contain exactly 2 pairs?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-9783fbd13da025122d29ee2e9a0c07995cf404d9e83c213bfe4b55141425d122 | What is the probability that a hand of 13 cards randomly dealt from a pack of 52 contains exactly 2 aces? | [
"0.6588"
] | [
"What is the probability that a hand of 13 cards randomly dealt from a pack of 52 contains exactly 2 aces?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16d16786d7cbf41dfab893da1dcac345a8f55c518ba791a672f6c1d6133a1630 | What is the probability that a hand of 5 cards dealt from a deck of 52 cards contains exactly one ace? | [
"\\(\\frac{3243}{10829}\\)"
] | [
"What is the probability that a hand of 5 cards dealt from a deck of 52 cards contains exactly one ace?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-6a268d7aa579e8f30497561a8fd11789da168166200af43d3a09807d37c69924 | What is the probability that a knight, moving randomly on a chessboard, is in one of the 4 corners at any given time? | [
"\\(\\frac{1}{42}\\)"
] | [
"What is the probability that a knight, moving randomly on a chessboard, is in one of the 4 corners at any given time?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-75f5c8d0de8f25f83c33a88e510bb1dc8108c46a93c3def98cf20c0eb745cc6b | What is the probability that a knight, moving randomly on a chessboard, is in one of the squares in the center 4x4 of the chessboard? | [
"\\(\\frac{8}{21}\\)"
] | [
"What is the probability that a knight, moving randomly on a chessboard, is in one of the squares in the center 4x4 of the chessboard?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-043258a2607933fda8790a36df39e74db1cb9b1f1e98a018d5d15db5963a5c55 | What is the probability that a license plate, which consists of 3 letters followed by 3 digits, has either a letter palindrome or a digit palindrome (or both)? | [
"7/52"
] | [
"What is the probability that a license plate, which consists of 3 letters followed by 3 digits, has either a letter palindrome or a digit palindrome (or both)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-9024d8d974f9b0195eb9adcc6a27eed347e30706136b200a77d294c99e8f03c1 | What is the probability that a line connecting two points randomly chosen from the surface of an outer cube (with side length 4) does not intersect an inner cube (with side length 2), given that each face of the outer cube is parallel to a face of the inner cube and 1 unit away from it? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that a line connecting two points randomly chosen from the surface of an outer cube (with side length 4) does not intersect an inner cube (with side length 2), given that each face of the outer cube is parallel to a face of the inner cube and 1 unit away from it?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-4eda039e233988427303644cc6737ca8f195c4b1ad67e0b601e6b6423ce1314c | What is the probability that a line through the origin, which makes a uniformly random angle with the $x$-axis, intersects the graph of $|x| + |y - \sqrt{3}| = 1$? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that a line through the origin, which makes a uniformly random angle with the $x$-axis, intersects the graph of $|x| + |y - \\sqrt{3}| = 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 11
} | 24 | 0 | 0 |
omr-ws-ebbc30883cb228632452df5c1ef6ce183668b9978926dfa4dab423813ca325a6 | What is the probability that a lottery ticket, which chooses five distinct numbers between 1 and 50 (inclusive), matches precisely $i$ of the five numbers that are randomly drawn ($i \in \{0,1,...,5\}$)? | [
"\\(\\frac{\\binom{5}{i} \\binom{45}{5-i}}{\\binom{50}{5}}\\)"
] | [
"What is the probability that a lottery ticket, which chooses five distinct numbers between 1 and 50 (inclusive), matches precisely $i$ of the five numbers that are randomly drawn ($i \\in \\{0,1,...,5\\}$)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-b12ec78bc609a7f24b66565fbed1bb023476da5be9ea266d87a4c706e64a9835 | What is the probability that a man flipping a coin 10 times gets at least 3 heads or at least 3 tails in a row? | [
"\\(\\frac{423}{512}\\)"
] | [
"What is the probability that a man flipping a coin 10 times gets at least 3 heads or at least 3 tails in a row?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-6c79f2fc561c02532de00d1018988b991aefb8a71c1b0ead0c77eb96f387b7ee | What is the probability that a multiple of 864 is divisible by 1994? | [
"\\(\\frac{1}{997}\\)"
] | [
"What is the probability that a multiple of 864 is divisible by 1994?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9e957723802689ce14c787176064d86a35f3073c0a5dc479af47bf30b2eb422e | What is the probability that a needle of length \( a \) thrown onto a grid of squares (each square having a side length of \( 2a \)) does not touch or cross any grid lines? | [
"\\(\\frac{4}{\\pi} - 1\\)"
] | [
"What is the probability that a needle of length \\( a \\) thrown onto a grid of squares (each square having a side length of \\( 2a \\)) does not touch or cross any grid lines?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17f5523073156e89de96120671ddde20d58f558d0de98a047768175a891ff612 | What is the probability that a non-increasing function \( f: A \to B \), where \( A = \{1, 2, \ldots, 6\} \) and \( B = \{1, 2, \ldots, 20\} \), satisfies \( f(x_1) = f(x_2) \) for all \( x_1 \ne x_2 \)? | [
"\\dfrac{1}{8855}"
] | [
"What is the probability that a non-increasing function \\( f: A \\to B \\), where \\( A = \\{1, 2, \\ldots, 6\\} \\) and \\( B = \\{1, 2, \\ldots, 20\\} \\), satisfies \\( f(x_1) = f(x_2) \\) for all \\( x_1 \\ne x_2 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28572ee5510c69bb4a770fc8ffabaa6a46658dace3e416f70fd5b9a1f0af0432 | What is the probability that a number \( j \) selected at random from the set \(\{1, 2, \ldots, 2024\}\) is divisible by both 9 and 15? | [
"\\(\\frac{1}{46}\\)"
] | [
"What is the probability that a number \\( j \\) selected at random from the set \\(\\{1, 2, \\ldots, 2024\\}\\) is divisible by both 9 and 15?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-d213e43b05dea63a95f7443069c3749da765f208eff8b77a3eb9ac81d1b35502 | What is the probability that a number chosen at random between 0 and 60 is less than a number chosen at random between 10 and 70? | [
"\\(\\frac{47}{72}\\)"
] | [
"What is the probability that a number chosen at random between 0 and 60 is less than a number chosen at random between 10 and 70?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-68f6dc4afb214c7246ae35169d4ca49b0b64080327dc4e85619fba198696701a | What is the probability that a number chosen at random from the set $\{1, 2, 3, \ldots, 2012\}$ has no prime factor in common with $10!$? | [
"\\(\\frac{115}{303}\\)"
] | [
"What is the probability that a number chosen at random from the set $\\{1, 2, 3, \\ldots, 2012\\}$ has no prime factor in common with $10!$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88c4ce851dd4f91db53a85611db9fea27388156acb7e2d62c846bec6e37fae58 | What is the probability that a number picked from the set $\{ 1,2,3,4,...,2010 \}$ cannot be expressed as the sum of two or more consecutive natural numbers? | [
"\\dfrac{11}{2010}"
] | [
"What is the probability that a number picked from the set $\\{ 1,2,3,4,...,2010 \\}$ cannot be expressed as the sum of two or more consecutive natural numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6525c473df62c3fcdecce959030857776a0ea059c132af52bd2b3827e4183c61 | What is the probability that a particular song will be played on exactly 5 days out of 7 days, given that the probability of the song being played on any single day is 0.38? | [
"\\( 21 \\cdot (0.38)^5 \\cdot (0.62)^2 \\)"
] | [
"What is the probability that a particular song will be played on exactly 5 days out of 7 days, given that the probability of the song being played on any single day is 0.38?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
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] | {
"cot": 3,
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} | 19 | 0 | 0 |
omr-ws-d2a8a2dfb0396a7fdf6b8192c5765a2f909928de174ce243c3e1fbbd049f1269 | What is the probability that a person flipping a coin repeatedly will get a string of at least four heads before getting a string of at least two tails? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the probability that a person flipping a coin repeatedly will get a string of at least four heads before getting a string of at least two tails?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-f3f7169906c5bb6192f361da54e8edcdea11a6af72a9a82b6fdb2d7bff65e3cb | What is the probability that a person has two boys, given that they have two children and one of them is a boy born on a Tuesday? Assume an equal chance of giving birth to either sex and an equal chance of giving birth on any day. | [
"\\(\\frac{13}{27}\\)"
] | [
"What is the probability that a person has two boys, given that they have two children and one of them is a boy born on a Tuesday? Assume an equal chance of giving birth to either sex and an equal chance of giving birth on any day."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
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"cot": 32,
"tir": 59
} | 91 | 0 | 0 |
omr-ws-46d40ff3091336a2f965c08517dbf97bf635b065152dc14ca1ed88a1deaa83df | What is the probability that a person will qualify for the AIME (120 points) by randomly guessing each question on the test and leaving none blank? | [
"\\dfrac{57795621}{298023223876953125}"
] | [
"What is the probability that a person will qualify for the AIME (120 points) by randomly guessing each question on the test and leaving none blank?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-d51b1c99b2231a5930142c9bdac4977790df040427a0a807033990c2183336c9 | What is the probability that a person will qualify for the AIME after first answering 15 questions correctly and then randomly guessing the rest of the questions? | [
"\\(\\binom{10}{5} \\left(\\frac{1}{5} \\right)^5 \\left(\\frac{4}{5} \\right)^5\\)"
] | [
"What is the probability that a person will qualify for the AIME after first answering 15 questions correctly and then randomly guessing the rest of the questions?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
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"cot": 1,
"genselect": 1,
"tir": 6
} | 8 | 0 | 0 |
omr-ws-a536287bed4f164c98cbd45b439a524bbf4445ececd01a7ba33b0d9eeadc3cfb | What is the probability that a person, looking through a shuffled deck of cards and sequentially searching for an Ace, then a 2, and so on up to a King, succeeds in finding all cards from Ace to King in this manner? | [
"\\dfrac{1}{13!}"
] | [
"What is the probability that a person, looking through a shuffled deck of cards and sequentially searching for an Ace, then a 2, and so on up to a King, succeeds in finding all cards from Ace to King in this manner?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-696fdedebb045cf2954ba76630e1c2510598d4be8063d571535c9afcc035813c | What is the probability that a phone number consisting of eight digits, starting with the digit 8, contains at most five different digits? | [
"0.41032"
] | [
"What is the probability that a phone number consisting of eight digits, starting with the digit 8, contains at most five different digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 23
} | 69 | 0 | 0 |
omr-ws-e01c3af0ae3a35d635974c63309e2f169526a9d4169a9a03833036c5b2622a6f | What is the probability that a player will get a yahtzee (all five dice showing the same number) in three or fewer rolls in the game of Yahtzee? | [
"0.0465675"
] | [
"What is the probability that a player will get a yahtzee (all five dice showing the same number) in three or fewer rolls in the game of Yahtzee?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2b0ef6f5111d607395a4048623636f7940774b4d48cee491a10cf244eccb4c6 | What is the probability that a point \((x, y)\), chosen at random in the rectangle \([-1, 1] \times [0, 1]\), satisfies \(y > x^2\)? | [
"\\(\\frac{2}{3}\\)"
] | [
"What is the probability that a point \\((x, y)\\), chosen at random in the rectangle \\([-1, 1] \\times [0, 1]\\), satisfies \\(y > x^2\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-36abdc53772f8581f2e361e036174b37bded9b0922862b908a47334189276bdf | What is the probability that a point chosen at random within a square is closer to the centre than to the boundary? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that a point chosen at random within a square is closer to the centre than to the boundary?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c3e181f0113624f21b2976a1647a37cd7598ce8fbf71babfe58ccdde6627cb6 | What is the probability that a point chosen randomly from the interior of a cube is closer to the cube’s center than it is to any of the cube’s eight vertices? | [
"\\(\\frac{1}{2}\\)",
"\\dfrac{1}{2}"
] | [
"What is the probability that a point chosen randomly from the interior of a cube is closer to the cube’s center than it is to any of the cube’s eight vertices?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-2afef8220191f8f6bb59bfd81cd1b41d2fbbf88b2b12026f4db55e6685bee310 | What is the probability that a point randomly selected in a rectangular prism with dimensions $5$ by $12$ by $13$ is closer to a fixed space diagonal than to any vertex? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that a point randomly selected in a rectangular prism with dimensions $5$ by $12$ by $13$ is closer to a fixed space diagonal than to any vertex?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-1db8c0ff0e7de64d70aa842c5cc1f4f99c504eb680fb4d7063b860be884b6dcb | What is the probability that a point struck on a rectangular dashboard (with length twice the breadth) is closer to the center of the dashboard than to the edges? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the probability that a point struck on a rectangular dashboard (with length twice the breadth) is closer to the center of the dashboard than to the edges?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 19
} | 31 | 0 | 0 |
omr-ws-f823e6590ceca7bad13a71209e222c398de49b2f30af3cd658e93ef11f1123cf | What is the probability that a poker hand has no aces? | [
"\\(\\frac{1712304}{2598960}\\) or approximately \\(0.658\\)"
] | [
"What is the probability that a poker hand has no aces?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-8bf12ed75f8b24ef7f679818fe4f04140ca78f8242082bb5d2b1242a9d85de69 | What is the probability that a positive integer less than or equal to 24 is a factor of 24? Express your answer as a common fraction. | [
"\\frac{1}{3}"
] | [
"What is the probability that a positive integer less than or equal to 24 is a factor of 24? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0943e6a5f4997c433e1fd7035c40b81bed4d3dd0efa99bb52d0d9c0019efb5f2 | What is the probability that a positive integer randomly selected from the positive integers 1-1000 inclusive is divisible by 5 or 11? | [
"\\dfrac{34}{125}"
] | [
"What is the probability that a positive integer randomly selected from the positive integers 1-1000 inclusive is divisible by 5 or 11?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a1f114567ee13480ed0273a526e6d42c382e1d59d33c2fc76d5d7bb7807837ec | What is the probability that a positive two-digit number selected at random has its tens digit at least three more than its unit digit? | [
"\\(\\frac{14}{45}\\)"
] | [
"What is the probability that a positive two-digit number selected at random has its tens digit at least three more than its unit digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-658bf1b6c0bc44466d45ca6cc9dd27d957f96ccee9cfdfdb9c0a9b2b18d40617 | What is the probability that a quadrilateral formed by 4 randomly chosen points in a 1x1 square is convex? | [
"\\(\\frac{35}{12\\pi^2}\\)"
] | [
"What is the probability that a quadrilateral formed by 4 randomly chosen points in a 1x1 square is convex?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a0bac931f2404c16d7e000dfb4cb2d6db42aa38beb771b4da97defac6f420ec | What is the probability that a quadrilateral formed by choosing 4 random points inside a circle contains the center of the circle? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that a quadrilateral formed by choosing 4 random points inside a circle contains the center of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-518ba0b30c8bda361070e567ae745cabcad1f67eb7aabccbf8a1304c030a206c | What is the probability that a random 2-digit base \( b \) number will be relatively prime with its digit reversal? | [
"\\( \\frac{41}{90} \\) or \\( \\frac{40}{81} \\)"
] | [
"What is the probability that a random 2-digit base \\( b \\) number will be relatively prime with its digit reversal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a68915a65e0444c4e51e559205b72550a6a3c4aa04da9b5c47c9025d88099795 | What is the probability that a random arrangement of the acronym WYSIWYG has the Ws and Ys in consecutive spots? | [
"\\(\\frac{8}{21}\\)"
] | [
"What is the probability that a random arrangement of the acronym WYSIWYG has the Ws and Ys in consecutive spots?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca8310d60da8881f952e12098c6acb15f6fe2b83d11e09ba8ce95c7106280e2a | What is the probability that a random arrangement of the letters in the word "MATHEMATICAL" contains no adjacent A's? | [
"\\(\\frac{6}{11}\\)"
] | [
"What is the probability that a random arrangement of the letters in the word \"MATHEMATICAL\" contains no adjacent A's?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-a30f63fe0cf60751823180f0ed2cb579da09e1cf962a24c2cedb5c2fae649e6b | What is the probability that a random arrangement of the numbers $1, 2, 3, 4, 5$ forms a strongly non-monotonic sequence, where a sequence is strongly non-monotonic if no three consecutive terms form a monotone sequence? | [
"\\(\\frac{4}{15}\\)"
] | [
"What is the probability that a random arrangement of the numbers $1, 2, 3, 4, 5$ forms a strongly non-monotonic sequence, where a sequence is strongly non-monotonic if no three consecutive terms form a monotone sequence?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 26
} | 39 | 0 | 0 |
omr-ws-9ebc9702ca12dcc9f983c5c7e51411e7839d54bb93f53a121945c7c53bf442eb | What is the probability that a random permutation of the digits {1, 2, 3, 4, 5} changes from increasing to decreasing, or decreasing to increasing at most once? Acceptable permutations include 12345, 54321, 12543, and 53214, but not 13245 or 53241. | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability that a random permutation of the digits {1, 2, 3, 4, 5} changes from increasing to decreasing, or decreasing to increasing at most once? Acceptable permutations include 12345, 54321, 12543, and 53214, but not 13245 or 53241."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-8049fee21cbb546696d6ca6efdec24d923f983b9e032f5fdfd7335407230bd1a | What is the probability that a random permutation of the letters of the word "SIPNAYAN" contains the word "PINAS" in that order? | [
"\\(\\frac{1}{420}\\)"
] | [
"What is the probability that a random permutation of the letters of the word \"SIPNAYAN\" contains the word \"PINAS\" in that order?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 23
} | 43 | 0 | 0 |
omr-ws-b7cfe21ac9f1051e9041f4abc8617ce5079180e96b6e2c912656f5c56b4c8213 | What is the probability that a random point \( P \) drawn within an equilateral triangle \( ABC \) is closer to one of the vertices than to the center of the triangle? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that a random point \\( P \\) drawn within an equilateral triangle \\( ABC \\) is closer to one of the vertices than to the center of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-6d89c16283ef17628bd3b7fe5df560b377999e27ccc06c401666ac1b3d801ba8 | What is the probability that a random power of 2 starts with the digit 1? | [
"\\(\\log_{10} 2\\)"
] | [
"What is the probability that a random power of 2 starts with the digit 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ca28614632f664ba243504b40e72903dea5ea432b4b06e9284eb3e053285e335 | What is the probability that a random triangle formed by connecting the three vertices of a dodecagon contains the center of the dodecagon? | [
"\\dfrac{5}{11}"
] | [
"What is the probability that a random triangle formed by connecting the three vertices of a dodecagon contains the center of the dodecagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-84da0d4110b8a0213effa4337817bad142c702e6282f4d006f1d27a960a99d7d | What is the probability that a random walk in $\mathbb{Z}^2$, starting at the origin and stopping when the origin is reached again, passes through the point $(0,1)$? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that a random walk in $\\mathbb{Z}^2$, starting at the origin and stopping when the origin is reached again, passes through the point $(0,1)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-0ad7ea539d83b4f06255d6cf6e3f381c328a66501a5b8870237ced47874c73c8 | What is the probability that a randomly chosen 3-digit number has exactly 3 factors? | [
"\\(\\frac{1}{150}\\)"
] | [
"What is the probability that a randomly chosen 3-digit number has exactly 3 factors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-048ede7b911e6a18125e86a19492f90412c9536e9d460cf4a1b53e5f0a02759b | What is the probability that a randomly chosen 4-digit number, composed from the digits $1, 1, 2, 2, 3, 3, 3,$ and $3$ (with the order of digits being important), contains exactly 3 different digits and the digit $1$ repeats exactly 2 times? | [
"\\dfrac{4}{21}"
] | [
"What is the probability that a randomly chosen 4-digit number, composed from the digits $1, 1, 2, 2, 3, 3, 3,$ and $3$ (with the order of digits being important), contains exactly 3 different digits and the digit $1$ repeats exactly 2 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9a81b8acc36376f58b5de5bf5403adac019402ac16daff95d2325bd28217195 | What is the probability that a randomly chosen 5-digit number (formed from 5 out of the 10 digits 0,1,2,3,4,5,6,7,8,9 without replacement) does not contain 0, is odd, and has odd and even digits alternating? | [
"\\(\\frac{1}{21}\\)"
] | [
"What is the probability that a randomly chosen 5-digit number (formed from 5 out of the 10 digits 0,1,2,3,4,5,6,7,8,9 without replacement) does not contain 0, is odd, and has odd and even digits alternating?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 17
} | 25 | 0 | 0 |
omr-ws-ce75f2967a3ea8d8ed6e36d4f1b6e38716c6861d30f3d11ec6826c24b0bc7f6c | What is the probability that a randomly chosen 5-digit number (formed from 5 out of the 10 digits 0,1,2,3,4,5,6,7,8,9 without replacement) is an odd number exceeding 10000? | [
"\\(\\frac{9}{20}\\)"
] | [
"What is the probability that a randomly chosen 5-digit number (formed from 5 out of the 10 digits 0,1,2,3,4,5,6,7,8,9 without replacement) is an odd number exceeding 10000?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3022d8d86b2eaffb74f09d641ca10eb252af7c586bc4dc5d50ce073ed1f6155a | What is the probability that a randomly chosen 5-digit number is made from exactly 2 different digits? | [
"\\(\\frac{1}{80}\\)"
] | [
"What is the probability that a randomly chosen 5-digit number is made from exactly 2 different digits?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e898cd8bf275fc6b99314261642bb00fefafe85c9ced02d5d69de17ca6bd514 | What is the probability that a randomly chosen chord of a circle with diameter $d$ has a length between $\frac{2}{3}d$ and $\frac{5}{6}d$? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability that a randomly chosen chord of a circle with diameter $d$ has a length between $\\frac{2}{3}d$ and $\\frac{5}{6}d$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 3,
"tir": 11
} | 42 | 0 | 0 |
omr-ws-13b29277e011147ed736cd63cf342109e96ef0d282ee545bec4ba075448b67db | What is the probability that a randomly chosen chord of a circle with diameter $d$ has a length between $\frac{d}{2}$ and $\frac{3d}{4}$? | [
"\\(\\frac{2}{3} - \\frac{1}{\\pi}\\cos^{-1}\\left(\\frac{1}{8}\\right)\\)"
] | [
"What is the probability that a randomly chosen chord of a circle with diameter $d$ has a length between $\\frac{d}{2}$ and $\\frac{3d}{4}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77e863e69d2381b79ab24df181629ad64edf469e736702f29549d777a0d7b6a1 | What is the probability that a randomly chosen integer between 1000 and 9999, inclusive, is an odd integer whose digits are all distinct? | [
"\\(\\frac{56}{225}\\)"
] | [
"What is the probability that a randomly chosen integer between 1000 and 9999, inclusive, is an odd integer whose digits are all distinct?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 62
} | 94 | 0 | 0 |
omr-ws-62f6d37a112f3377874cc8b6c23f0bd3738e5483dbbbef1c58efdf3a8dd0a227 | What is the probability that a randomly chosen integer from 100 to 999 (inclusive) will have at least one digit as 0? | [
"\\(\\frac{19}{100}\\)"
] | [
"What is the probability that a randomly chosen integer from 100 to 999 (inclusive) will have at least one digit as 0?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-ae6356167a0d2d572f2934650aafd7014420a9be161303c5c95b9b4fb3e4186d | What is the probability that a randomly chosen number from one to twelve is a divisor of 12? (Unity is assumed to be a divisor of any number.) | [
"5/12"
] | [
"What is the probability that a randomly chosen number from one to twelve is a divisor of 12? (Unity is assumed to be a divisor of any number.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5676122435d0e975b58f7523c211ef7c1ab1ac8f0b22762ab0055189b3701e3 | What is the probability that a randomly chosen pair of diagonals in a convex n-gon intersect? | [
"\\(\\frac{(n-1)(n-2)}{3(n^2 - 3n - 1)}\\)"
] | [
"What is the probability that a randomly chosen pair of diagonals in a convex n-gon intersect?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f597ad2ab693b01025e0826b970c57498994c5fbf9b52b6d60a75aa014c068cd | What is the probability that a randomly chosen partition of the number $50$ contains a prime number? | [
"1"
] | [
"What is the probability that a randomly chosen partition of the number $50$ contains a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-f99c40b181dfce9450dbfba0e5e369d9d66411c6db6128b20066027c657d03e7 | What is the probability that a randomly chosen real number between 0 and 1 is rational? | [
"\\(0\\)"
] | [
"What is the probability that a randomly chosen real number between 0 and 1 is rational?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce7c828ec66cce989d29e83eb0f4e8b03d795bee87e538325f068d622883935e | What is the probability that a randomly chosen real number from the interval $[0,1]$ has a $3$ in its decimal expansion? | [
"1"
] | [
"What is the probability that a randomly chosen real number from the interval $[0,1]$ has a $3$ in its decimal expansion?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0716ea8ef9a54a3e6ccaee931d906d761898200e3647832cb504a4f3e25213e0 | What is the probability that a randomly chosen relation from a set \( A = \{1, 2, 3, \ldots, n\} \) to itself is a symmetric relation? | [
"\\( 2^{-\\frac{n(n-1)}{2}} \\)"
] | [
"What is the probability that a randomly chosen relation from a set \\( A = \\{1, 2, 3, \\ldots, n\\} \\) to itself is a symmetric relation?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-e7d07b552a32d795b4fb5dec9941925985257def7c8ea8f4d71118a50abd6900 | What is the probability that a randomly chosen relation with 10 elements on the set \( A = \{a, b, c, d\} \) is reflexive? | [
"\\(\\frac{\\left( \\begin{matrix}\n 12 \\\\\n 6 \\\\\n\\end{matrix} \\right)}{\\left( \\begin{matrix}\n 16 \\\\\n 10 \\\\\n\\end{matrix} \\right)}=\\frac{\\frac{12!}{6!6!}}{\\frac{16!}{10!6!}}=\\frac{12!\\times 10!}{16!\\times 6!}\\)"
] | [
"What is the probability that a randomly chosen relation with 10 elements on the set \\( A = \\{a, b, c, d\\} \\) is reflexive?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-780bfd1e3827367aaeb0d99946e0fa2a5f0999da2ec5f84bae9b7d364084619a | What is the probability that a randomly chosen subset $A$ is a proper subset of another randomly chosen subset $B$ from a set containing $n$ elements? | [
"\\(\\frac{3^n - 2^n}{4^n}\\)"
] | [
"What is the probability that a randomly chosen subset $A$ is a proper subset of another randomly chosen subset $B$ from a set containing $n$ elements?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-16f59b1a5c64e1ff353fe5c94d8ed4da504642e22c3e0ddc94dcb34e891268a6 | What is the probability that a randomly chosen three-element subset of the set $\{-10, -9, -8, \ldots, 8, 9, 10\}$ sums to zero? | [
"$\\frac{5}{133}$"
] | [
"What is the probability that a randomly chosen three-element subset of the set $\\{-10, -9, -8, \\ldots, 8, 9, 10\\}$ sums to zero?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-63c3088d345b25e5378dd830eeafffcbf1a2f4aa87ad30fd87129467ef1e6595 | What is the probability that a randomly drawn chord in a circle is greater than the side of an equilateral triangle inscribed in the circle? (Assume the distance of the chord from the center is chosen at random.) | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that a randomly drawn chord in a circle is greater than the side of an equilateral triangle inscribed in the circle? (Assume the distance of the chord from the center is chosen at random.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ff3671c07e5bd4cec9958c50630f222ac21af97807b7b39483de86c1a5a29443 | What is the probability that a randomly filled 3x3 array of circles forms a figure with mirror symmetry? | [
"\\dfrac{27}{64}"
] | [
"What is the probability that a randomly filled 3x3 array of circles forms a figure with mirror symmetry?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-18451f1864456240f8aea4f34d0d8c960582f1cf0b5f73aabb2d0aa304cf3559 | What is the probability that a randomly generated solution to the equation $x_1 + x_2 + x_3 + x_4 = 17$, where $|x_i| \le 5$ for all $i \in \{1, 2, 3, 4\}$, has all positive entries? | [
"1"
] | [
"What is the probability that a randomly generated solution to the equation $x_1 + x_2 + x_3 + x_4 = 17$, where $|x_i| \\le 5$ for all $i \\in \\{1, 2, 3, 4\\}$, has all positive entries?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f7ae78f2338d4f6fed440dbfbd22b17699f2e632cf470836666c9b606244f37b | What is the probability that a randomly generated string of six symbols using only the letters $A$, $B$, and $C$ will not contain three consecutive $A$'s? | [
"\\(\\frac{8}{9}\\)"
] | [
"What is the probability that a randomly generated string of six symbols using only the letters $A$, $B$, and $C$ will not contain three consecutive $A$'s?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-7f49d5a12ae76538ce6673769c7f18afb49bb9bdd340062103d9f43e65dd9d7e | What is the probability that a randomly generated string of six symbols using only the letters A, B, and C will not contain three consecutive A's? | [
"\\(\\frac{8}{9}\\)"
] | [
"What is the probability that a randomly generated string of six symbols using only the letters A, B, and C will not contain three consecutive A's?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 60
} | 91 | 0 | 0 |
omr-ws-6250ca74359f48e2c7b4941a0f9795b7bf27b5ca4c454d0631e8ecf708a0309b | What is the probability that a randomly ordered sequence of the symbols ".", "$\le$", "4", "5", and "6" forms a true inequality? | [
"\\(\\frac{1}{5}\\)"
] | [
"What is the probability that a randomly ordered sequence of the symbols \".\", \"$\\le$\", \"4\", \"5\", and \"6\" forms a true inequality?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 5
} | 10 | 0 | 0 |
omr-ws-d05e4504c3cf051945b75c5a5749474c8722356b2add073fc1eb902520e13965 | What is the probability that a randomly picked (side length $2$ cm) cube, when tossed like a die, has no blue paint on any of its 5 faces showing after a yellow cube with side length $8$ cm is painted blue and cut into smaller cubes of side length $2$ cm? | [
"$\\frac{3}{16}$"
] | [
"What is the probability that a randomly picked (side length $2$ cm) cube, when tossed like a die, has no blue paint on any of its 5 faces showing after a yellow cube with side length $8$ cm is painted blue and cut into smaller cubes of side length $2$ cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-3a6e864583dce863f519410de321bc373f29e25db0e5514808d6c10be62fadcf | What is the probability that a randomly selected 4-digit number is not divisible by any pair of primes between 5 and 23, inclusive? | [
"\\(\\frac{7403}{9000}\\)"
] | [
"What is the probability that a randomly selected 4-digit number is not divisible by any pair of primes between 5 and 23, inclusive?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8606cc0212993dd424b10ab2983e08710ade737505beee5d3fd07f3477daab50 | What is the probability that a randomly selected adult long-tongued fruit bat will weigh at most 15 grams, given that the weights are normally distributed with a mean of 20.22 grams and a standard deviation of 3.23 grams? | [
"\\(1 - \\Phi(1.616)\\)"
] | [
"What is the probability that a randomly selected adult long-tongued fruit bat will weigh at most 15 grams, given that the weights are normally distributed with a mean of 20.22 grams and a standard deviation of 3.23 grams?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec78fd2af179e99228acd6e10c0a9c3284bf369c4e3b533f9c231a21b59d6cbb | What is the probability that a randomly selected divisor of $20!$ is a multiple of $2000$? | [
"\\(\\frac{34}{105}\\)"
] | [
"What is the probability that a randomly selected divisor of $20!$ is a multiple of $2000$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4d6e7e77ae6bdfa4c81ca3fc42dd27fe36b8801e4119329a96807767a034035 | What is the probability that a randomly selected factor of \(66^{11}\) is a multiple of \(66^6\)? Express your answer as a common fraction. | [
"\\(\\frac{343}{1728}\\)"
] | [
"What is the probability that a randomly selected factor of \\(66^{11}\\) is a multiple of \\(66^6\\)? Express your answer as a common fraction."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98c06836ac485dd02fcfac32f16d6d19db4df8133841d073e76a422c273d700a | What is the probability that a randomly selected five-digit number has odd digits in the odd positions and even digits in the even positions, with no repetition of digits? | [
"\\(\\frac{2}{150}\\)"
] | [
"What is the probability that a randomly selected five-digit number has odd digits in the odd positions and even digits in the even positions, with no repetition of digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 22
} | 35 | 0 | 0 |
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