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-21845fb9107c56cceabc9295c4137ccf33f8e684bb0bdcc14770d9bc57cd5e46 | Bessie accidentally broke her analog clock, making it go $2015$ times faster. How many times during the day is her clock correct? | [
"4"
] | [
"Bessie accidentally broke her analog clock, making it go $2015$ times faster. How many times during the day is her clock correct?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9bb697f4a904d0ec0510035909875eca1a3677f381d0ccc8fbdeda7539110f7d | Bessie is playing a game with a square labeled \( A, B, C, D \) in clockwise order. She can perform 7 possible moves: rotate the square \( 90^\circ \), \( 180^\circ \), or \( 270^\circ \) about the center; flip across diagonal \( AC \), flip across diagonal \( BD \), flip the square horizontally, or flip the square ver... | [
"42"
] | [
"Bessie is playing a game with a square labeled \\( A, B, C, D \\) in clockwise order. She can perform 7 possible moves: rotate the square \\( 90^\\circ \\), \\( 180^\\circ \\), or \\( 270^\\circ \\) about the center; flip across diagonal \\( AC \\), flip across diagonal \\( BD \\), flip the square horizontally, or... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-53bae1ae54e2b7ab58cf17df6d0ce4b5ae259e7a93bcf9eac90320028098a350 | Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these points. He then chooses any $1011$ of these segments and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet. Let $S$ be the set of all possible segments that Vikram c... | [
"2020"
] | [
"Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these points. He then chooses any $1011$ of these segments and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet. Let $S$ be the set of all possible segments that Vik... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-274fd87f6b0446754058e7ed3dd398fd21069c171c6df62b4acbd6e705fefe71 | Bethan is playing a game on an $n \times n$ grid consisting of $n^2$ cells. A move consists of placing a counter in an unoccupied cell $C$ where the $2n-2$ other cells in the same row or column as $C$ contain an even number of counters. After making $M$ moves, Bethan realizes she cannot make any more moves. Determine t... | [
"\\( M = \\frac{n^2 + 1}{2} \\) if \\( n \\) is odd, or \\( M = \\frac{n^2}{2} + n \\) if \\( n \\) is even."
] | [
"Bethan is playing a game on an $n \\times n$ grid consisting of $n^2$ cells. A move consists of placing a counter in an unoccupied cell $C$ where the $2n-2$ other cells in the same row or column as $C$ contain an even number of counters. After making $M$ moves, Bethan realizes she cannot make any more moves. Deter... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2f3fbeb8fdc9a1986bcd75a1747635c14df8ace682f6e4f2a9fc2797c6b3c1a | Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$4$ per package, and $D$ batteries for $\$7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\$2021$, and no two of them purchased the same quantities of both types of battery. Bethany a... | [
"28548"
] | [
"Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\\$4$ per package, and $D$ batteries for $\\$7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\\$2021$, and no two of them purchased the same quantities of both types of battery. B... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-b1fe4f128d1c36ef2fd9e51155299dcc78f635e789ba516f90ea887f58d377d3 | Beto has $207$ cards on the table with arbitrary positive integers written on them. He aims to split these cards into $7$ piles such that the product of the differences between the sums of the numbers in any two piles is a multiple of $7^{100}$. Determine the minimum number of piles Beto must use to guarantee he can al... | [
"15"
] | [
"Beto has $207$ cards on the table with arbitrary positive integers written on them. He aims to split these cards into $7$ piles such that the product of the differences between the sums of the numbers in any two piles is a multiple of $7^{100}$. Determine the minimum number of piles Beto must use to guarantee he c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-815053d00108761db837c687b24fc7f034a43c110f7bf974b55cbe248072bda1 | Beto is playing a game on a $50 \times 50$ board where Ana has already placed some chips on some unit squares. Beto's goal is to place chips on at most 99 empty unit squares so that each row and each column of the board contains an even number of chips. Determine the maximum number of chips Beto can place on the board ... | [
"98"
] | [
"Beto is playing a game on a $50 \\times 50$ board where Ana has already placed some chips on some unit squares. Beto's goal is to place chips on at most 99 empty unit squares so that each row and each column of the board contains an even number of chips. Determine the maximum number of chips Beto can place on the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-58b305ff81face555ea00492bf30224733d4ff083ece63ac0e0923073d588cb7 | Beto is trying to guess a positive integer \( n \) that has exactly 250 positive divisors. The divisors are ordered as \( 1 = d_1 < d_2 < d_3 < \ldots < d_{249} < d_{250} = n \). Beto can pick an index \( j \) (where \( 2 \le j \le 249 \)) and receive the divisor \( d_j \). The index \( (251-j) \) cannot be picked if \... | [
"125"
] | [
"Beto is trying to guess a positive integer \\( n \\) that has exactly 250 positive divisors. The divisors are ordered as \\( 1 = d_1 < d_2 < d_3 < \\ldots < d_{249} < d_{250} = n \\). Beto can pick an index \\( j \\) (where \\( 2 \\le j \\le 249 \\)) and receive the divisor \\( d_j \\). The index \\( (251-j) \\) c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a488dec6c71b18037a01ca93f9101e25ed2c756ccf0d6eb97206be596d3aa4e | Beto plays a solitaire game where a machine chooses at random 26 positive integers between 1 and 2016 and writes them on a blackboard (numbers may be repeated). At each step, Beto chooses some of the numbers on the blackboard and subtracts the same non-negative integer $k$ from each of them, with the condition that the... | [
"11"
] | [
"Beto plays a solitaire game where a machine chooses at random 26 positive integers between 1 and 2016 and writes them on a blackboard (numbers may be repeated). At each step, Beto chooses some of the numbers on the blackboard and subtracts the same non-negative integer $k$ from each of them, with the condition tha... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-daffe481451d63bd48e474224abedbfd0ce5d9c9667443b7defdd8436c9c7364 | Bette visits her friend Keith and returns home by the same route. She always walks 2 km/h when going uphill, 6 km/h when going downhill, and 3 km/h when on level ground. If her total walking time is 6 hours, then the total distance she walks, in km, is? | [
"18"
] | [
"Bette visits her friend Keith and returns home by the same route. She always walks 2 km/h when going uphill, 6 km/h when going downhill, and 3 km/h when on level ground. If her total walking time is 6 hours, then the total distance she walks, in km, is?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 1
} | 32 | 0 | 0 |
omr-ws-086cbcf05db3ce811b742819509ba65f846aad38a46468d1249243eb1c4da2c8 | Bette visits her friend Keith and then returns home by the same route. She always walks 2 km/h when going uphill, 6 km/h when going downhill, and 3 km/h when on level ground. If her total walking time is 6 hours, then what is the total distance she walks, in km? | [
"18 km"
] | [
"Bette visits her friend Keith and then returns home by the same route. She always walks 2 km/h when going uphill, 6 km/h when going downhill, and 3 km/h when on level ground. If her total walking time is 6 hours, then what is the total distance she walks, in km?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-688084fee8f7deb7b397b94a793fcce8fcb5401705a40365ce5fc97ea8ef4e41 | Betty goes to the store to get flour and sugar. The amount of flour she buys, in pounds, is at least 6 pounds more than half the amount of sugar, and is no more than twice the amount of sugar. Find the least number of pounds of sugar that Betty could buy. | [
"4"
] | [
"Betty goes to the store to get flour and sugar. The amount of flour she buys, in pounds, is at least 6 pounds more than half the amount of sugar, and is no more than twice the amount of sugar. Find the least number of pounds of sugar that Betty could buy."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-369632381408144d057e3c1a155bfaa8c12e44cd460fa1cb95723c2b0e7734e2 | Betty is given a 200-page document to type. The number of errors Betty makes can be modeled by a Poisson distribution with a mean of 0.07 errors per page. Using a suitable approximation, find the probability that she makes more than 21 errors in the document. | [
"0.0225"
] | [
"Betty is given a 200-page document to type. The number of errors Betty makes can be modeled by a Poisson distribution with a mean of 0.07 errors per page. Using a suitable approximation, find the probability that she makes more than 21 errors in the document."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-8640ef6fe392af223148d6a07a58e57fd9b0316807431c883476ec923867d10b | Betty is given a 200-page document to type. The number of errors Betty makes can be modeled by a Poisson distribution with a mean of 0.07 per page. Using a suitable approximation, find the probability that she makes more than 21 errors in the document. | [
"0.0228"
] | [
"Betty is given a 200-page document to type. The number of errors Betty makes can be modeled by a Poisson distribution with a mean of 0.07 per page. Using a suitable approximation, find the probability that she makes more than 21 errors in the document."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-acc7bd811231f361976fa2893ebbebe76a3893eced3dcf883362ef79c42100c3 | Betty is given a 200-page document to type. Using a suitable approximation, find the probability that she makes more than 21 errors in the document. | [
"0.369"
] | [
"Betty is given a 200-page document to type. Using a suitable approximation, find the probability that she makes more than 21 errors in the document."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-446f5eaa3cff06bc82dd5e26e07d228b1c412621ded64dcadcffd8a7bd9cd79c | Betty is playing a game where she has an equal chance of winning or losing each game. Winning a game gives her one trophy, and losing a game removes one trophy. Starting with 108 trophies, what is the probability that she will reach 9 trophies before reaching 1008 trophies? Express the probability in the form $\frac{m}... | [
"211"
] | [
"Betty is playing a game where she has an equal chance of winning or losing each game. Winning a game gives her one trophy, and losing a game removes one trophy. Starting with 108 trophies, what is the probability that she will reach 9 trophies before reaching 1008 trophies? Express the probability in the form $\\f... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-a735db565b524a6ea636bff42555709c07ed76c4006ab1588d64a5619f6a19b4 | Between $12$ p.m. today and $12$ p.m. tomorrow, how many times do the hour hand and the minute hand on a clock form an angle of $120^{\circ}$? | [
"44"
] | [
"Between $12$ p.m. today and $12$ p.m. tomorrow, how many times do the hour hand and the minute hand on a clock form an angle of $120^{\\circ}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-35bb452e77ac083fb1b72f07abc3786063a94f6ed705d5ff0e33c88d9d188fa3 | Between $12:00$ midnight and $6:00$ a.m., how many times do the hour hand and the minute hand of a clock form a right angle? | [
"11"
] | [
"Between $12:00$ midnight and $6:00$ a.m., how many times do the hour hand and the minute hand of a clock form a right angle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa2f7547035b32762621228b0509b2bb64a0ac30c24c699320085cd08dbd0182 | Between $5~\text{pm}$ and $6~\text{pm}$, I looked at my watch. Mistaking the hour hand for the minute hand and the minute hand for the hour hand, I mistook the time to be $57$ minutes earlier than the actual time. Find the number of minutes past $5$ when I looked at my watch. | [
"\\dfrac{243}{11}"
] | [
"Between $5~\\text{pm}$ and $6~\\text{pm}$, I looked at my watch. Mistaking the hour hand for the minute hand and the minute hand for the hour hand, I mistook the time to be $57$ minutes earlier than the actual time. Find the number of minutes past $5$ when I looked at my watch."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-96bce69d3a62ff14ed8f0ebe19bc82af6e4f1be76a3280f6cb443c79950b3d8f | Between 5 and 6 o'clock, a lady mistook the hour hand for the minute hand and vice versa. As a result, she thought the time was approximately 55 minutes earlier. Exactly how many minutes earlier was the mistaken time? | [
"55"
] | [
"Between 5 and 6 o'clock, a lady mistook the hour hand for the minute hand and vice versa. As a result, she thought the time was approximately 55 minutes earlier. Exactly how many minutes earlier was the mistaken time?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-cbe2b366e838150b0aeec0b252a749f7e3603d6ec2b435325ec46a6b610e42a0 | Between 5 and 6, a lady looked at her watch and mistook the hour hand for the minute hand, thinking it was 57 minutes earlier than the correct time. What was the correct time? | [
"5:\\dfrac{2616}{133}"
] | [
"Between 5 and 6, a lady looked at her watch and mistook the hour hand for the minute hand, thinking it was 57 minutes earlier than the correct time. What was the correct time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-5cf85f39f7f82784950f465a37865400116c12f250f8a3874d89049fac525c7d | Between 5pm and 6pm, I looked at my watch and mistook the hour hand for the minute hand and the minute hand for the hour hand. As a result, I thought the time was 57 minutes earlier than the actual time. Find the number of minutes past 5 when I looked at my watch. | [
"24"
] | [
"Between 5pm and 6pm, I looked at my watch and mistook the hour hand for the minute hand and the minute hand for the hour hand. As a result, I thought the time was 57 minutes earlier than the actual time. Find the number of minutes past 5 when I looked at my watch."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-43a230f1c11602349b6097a75ee4fba22648b07303878b91826fc94a744bb3f6 | Between the roots of the quadratic equation \(3px^2 - 10px + 5^q = 0\), an odd number of arithmetic means (A.M.S) are inserted, and their sum exceeds their number by 10. Find the number of A.M.S inserted. | [
"15"
] | [
"Between the roots of the quadratic equation \\(3px^2 - 10px + 5^q = 0\\), an odd number of arithmetic means (A.M.S) are inserted, and their sum exceeds their number by 10. Find the number of A.M.S inserted."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ebe78b79a050e0008b67ae3345daa7778d52393c6e8bf3d486296e13b7f7a8f6 | Between the roots of the quadratic equation \(3px^2 - 10px + 5q = 0\) (where \(p > 0\) and \(\frac{q}{p} < \frac{2}{3}\)), an odd number of arithmetic means (A.M.S) are inserted, and their sum exceeds their number by 10. Find the number of A.M.S inserted. | [
"15"
] | [
"Between the roots of the quadratic equation \\(3px^2 - 10px + 5q = 0\\) (where \\(p > 0\\) and \\(\\frac{q}{p} < \\frac{2}{3}\\)), an odd number of arithmetic means (A.M.S) are inserted, and their sum exceeds their number by 10. Find the number of A.M.S inserted."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d30f83ad39dcaeba11436fb28f2cda7817b563c2ab91b2e31d4dbe8531e252b3 | Between town $A$ and city $E$ is a gasoline (petrol) station $O$ and a water supply station $B$, which divide the distance between $A$ and $E$ into three equal parts $(AO=OB=BE)$. A motorist and cyclist start out simultaneously from $A$ in the direction of $E$ and, at the same time, a truck starts out from $E$ in the d... | [
"The rate of the cyclist is 20 km/h, that of the truck, 40 km/h, of the Volga car 80 km/h. The distance from $A$ to $D$ is 60 km"
] | [
"Between town $A$ and city $E$ is a gasoline (petrol) station $O$ and a water supply station $B$, which divide the distance between $A$ and $E$ into three equal parts $(AO=OB=BE)$. A motorist and cyclist start out simultaneously from $A$ in the direction of $E$ and, at the same time, a truck starts out from $E$ in ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b68d8fdf1bdf23aecfa47e153615996e26a6f3c95f088d0f8f5c900cef567c5 | Between which two consecutive integers does \( S \) lie, where \[ S = \sqrt{18 + \sqrt{18^2 + \sqrt{18^3 + \cdots}}} \]? | [
"6 and 7"
] | [
"Between which two consecutive integers does \\( S \\) lie, where \\[ S = \\sqrt{18 + \\sqrt{18^2 + \\sqrt{18^3 + \\cdots}}} \\]?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7312d36dd17e51d0b7fcaf20274b65752f2b5206aa54786237bc4ae31ee9bc32 | Between which two consecutive proper irreducible fractions with two-digit numerators is the number $\frac{5}{8}$ located? | [
"\\(\\frac{13}{21} \\text{ and } \\frac{13}{20}\\)"
] | [
"Between which two consecutive proper irreducible fractions with two-digit numerators is the number $\\frac{5}{8}$ located?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d0c5d8f5223210393d4a909c16471b241f5f62aa33d868fd0d66797a83dcb69 | Bhairav runs a 15-mile race at 28 miles per hour, while Daniel runs at 15 miles per hour and Tristan runs at 10 miles per hour. What is the greatest length of time, in minutes, between consecutive runners' finishes? | [
"30"
] | [
"Bhairav runs a 15-mile race at 28 miles per hour, while Daniel runs at 15 miles per hour and Tristan runs at 10 miles per hour. What is the greatest length of time, in minutes, between consecutive runners' finishes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-8110d75f554bbd7921c3d187605279745770e501de13325e8d230368f340fbe1 | Big Corporation sells on average 1.5 units of ventilators each week. The lead time for this product is 2 weeks. They order weekly and use a base stock level of 5. What is their average on-hand inventory in units? | [
"2"
] | [
"Big Corporation sells on average 1.5 units of ventilators each week. The lead time for this product is 2 weeks. They order weekly and use a base stock level of 5. What is their average on-hand inventory in units?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-30dbf848ae05d0326eb08cb253dff8d28c8ac0b5aca5a5434c8c8e140f67f176 | Bilbo chooses real numbers $x, y$ and tells $x^n + y^n$ and $x^k + y^k$ to Gollum. Find all positive integers $n, k$ for which Gollum can determine $xy$. | [
"\\( n \\) and \\( k \\) are consecutive integers."
] | [
"Bilbo chooses real numbers $x, y$ and tells $x^n + y^n$ and $x^k + y^k$ to Gollum. Find all positive integers $n, k$ for which Gollum can determine $xy$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 6
} | 8 | 0 | 0 |
omr-ws-b0d6227992be34caec58c21c9126ea983fc4038e7350d24a9ac9a95723d36f22 | Bilbo composes a number triangle of zeroes and ones as follows: he fills the topmost row with any $n$ digits, and in other rows he writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. For $n=100$, in how many ways can Bilbo fill the topmost row so that each of the $n$ rows of the ... | [
"\\( 2^{99} \\)"
] | [
"Bilbo composes a number triangle of zeroes and ones as follows: he fills the topmost row with any $n$ digits, and in other rows he writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. For $n=100$, in how many ways can Bilbo fill the topmost row so that each of the $n$ rows of... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-ce990674dc1e1a8ce4c7203902a66139d859888a965583c304db0c4c2c5ea83c | Bilbo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any $n$ digits, and in other rows he always writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. In how many ways can Bilbo fill the topmost row for $n=100$ so that each of $n$ rows of... | [
"\\( 2^{98} \\)"
] | [
"Bilbo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any $n$ digits, and in other rows he always writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. In how many ways can Bilbo fill the topmost row for $n=100$ so that each of $n$ ro... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-f90a8e844b4c578880a9c111e4a76cc3d30fbb5418bb818f07ef3aec8d4ffde4 | Bilbo, Gandalf, and Nitzan play a game where Nitzan picks a whole number between $1$ and $2^{2022}$ inclusive and reveals it to Bilbo. Bilbo creates a string of length $4044$ using the letters $a$, $b$, and $c$. Nitzan then chooses one of the letters and removes all instances of that letter from the string. The modifie... | [
"3^{2022}"
] | [
"Bilbo, Gandalf, and Nitzan play a game where Nitzan picks a whole number between $1$ and $2^{2022}$ inclusive and reveals it to Bilbo. Bilbo creates a string of length $4044$ using the letters $a$, $b$, and $c$. Nitzan then chooses one of the letters and removes all instances of that letter from the string. The mo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-4645ea849698586bfa4a87ec85de16ac1ca965c356bc6b05a6e65d6e1691ed65 | Bill and Ben each have 2 fair coins. Each minute, both Bill and Ben flip all their coins at the same time, if they have any. If a coin lands heads, then the other person gets that coin. If a coin lands tails, then that coin stays with the same person. What is the probability that after exactly 3 minutes, they each end ... | [
"\\dfrac{3}{8}"
] | [
"Bill and Ben each have 2 fair coins. Each minute, both Bill and Ben flip all their coins at the same time, if they have any. If a coin lands heads, then the other person gets that coin. If a coin lands tails, then that coin stays with the same person. What is the probability that after exactly 3 minutes, they each... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06cfdb9eddfef1b7f61345ddfc36036e65028e59952b2ee174330b6d25bc4218 | Bill and Joe run a race. Bill starts at a rate of $0.2$ miles per minute and at the end of each minute reduces his rate by $0.01$ miles per minute. Joe starts at a rate of $0.1$ mile per minute and at the end of each minute increases his rate by $0.01$ mile per minute. (a) How far do Bill and Joe go in the first $5$ mi... | [
"\\(\\frac{2(r-s)}{d+e} - 1 < m < \\frac{2(r-s)}{d+e}\\)"
] | [
"Bill and Joe run a race. Bill starts at a rate of $0.2$ miles per minute and at the end of each minute reduces his rate by $0.01$ miles per minute. Joe starts at a rate of $0.1$ mile per minute and at the end of each minute increases his rate by $0.01$ mile per minute.\n(a) How far do Bill and Joe go in the first ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 10
} | 12 | 0 | 0 |
omr-ws-8199c196b9511cd950c3f31bac76e241deb3512be670ae699a3ac95b7f4b53c0 | Bill and John play in a background tournament. A player is the winner if he wins three games in a row or four games altogether. In what percent of all possible cases does the tournament end because John wins four games without winning three in a row? | [
"26%"
] | [
"Bill and John play in a background tournament. A player is the winner if he wins three games in a row or four games altogether. In what percent of all possible cases does the tournament end because John wins four games without winning three in a row?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d9fc9c953a7d9c90c0c258973c294b1ed879474cdbabdbe411f2f55e4fab465 | Bill and Peter play a game with the expression $1\oplus2\oplus3\oplus4\oplus5\oplus6\oplus7\oplus8\oplus9$. They replace each $\oplus$ with either $+$ or $\cdot$, taking turns. Bill can only use $+$ and Peter can only use $\cdot$. Bill aims to make the result an even number, while Peter aims to make it an odd number. A... | [
"141"
] | [
"Bill and Peter play a game with the expression $1\\oplus2\\oplus3\\oplus4\\oplus5\\oplus6\\oplus7\\oplus8\\oplus9$. They replace each $\\oplus$ with either $+$ or $\\cdot$, taking turns. Bill can only use $+$ and Peter can only use $\\cdot$. Bill aims to make the result an even number, while Peter aims to make it ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-00dbbb219c03f27ff163598a018ac97f12a5b37663d7d6fc7f08ce4ba1955439 | Bill and Ted are playing a game with 39 sticks. Each player, on their turn, can remove between 1 and 5 sticks. The player who is forced to take the last stick loses. If Bill goes first, how many sticks should he take to ensure he wins? | [
"2"
] | [
"Bill and Ted are playing a game with 39 sticks. Each player, on their turn, can remove between 1 and 5 sticks. The player who is forced to take the last stick loses. If Bill goes first, how many sticks should he take to ensure he wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-afab3c011ff2fded23b40a89894badacc70c5d29a6bd981bfbbeeb0ca494150f | Bill and Ted are playing a game with 59 sticks. Each player, on their turn, removes at least one but no more than 5 sticks. The player who is forced to take the last stick loses. If Bill goes first, how many sticks should he take to ensure he wins? | [
"4"
] | [
"Bill and Ted are playing a game with 59 sticks. Each player, on their turn, removes at least one but no more than 5 sticks. The player who is forced to take the last stick loses. If Bill goes first, how many sticks should he take to ensure he wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f59d5626199f7a9cf7bc20fb261060de8f480b3d4cd8b2a0c15588c656c169eb | Bill and his dog walk home from the shopping center. It takes Bill 36 minutes, and his dog walks twice as fast. They start together, but the dog reaches home before Bill and returns to meet Bill. After meeting Bill, the dog walks home again at double speed and then turns back to meet Bill again. Bill starts at noon to ... | [
"24"
] | [
"Bill and his dog walk home from the shopping center. It takes Bill 36 minutes, and his dog walks twice as fast. They start together, but the dog reaches home before Bill and returns to meet Bill. After meeting Bill, the dog walks home again at double speed and then turns back to meet Bill again. Bill starts at noo... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c53b585d43a1064c2ca2fe75c18befa26d4f800cf1c2287a61adf1366909ffb3 | Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers? | [
"$88"
] | [
"Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-18a403a2f9302a30e833107ce4bac87e9cbfe1a5690ea6d99f04357cbd3cf119 | Bill buys a stock that decreases by $20\%$ on the first day, and then on the second day the stock increases by $30\%$ of its value at the end of the first day. What was the overall percent increase in Bill's stock over the two days? | [
"4"
] | [
"Bill buys a stock that decreases by $20\\%$ on the first day, and then on the second day the stock increases by $30\\%$ of its value at the end of the first day. What was the overall percent increase in Bill's stock over the two days?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9ddf5d680c9e4bc9be09703e60469539fb06f95127810173b24233538f9b837 | Bill groups his ties by color: red, green, and blue. He then takes his entire red pile and splits it into 3 identical piles. Each group has 2 “size” piles, and all the “size” piles now contain 2 ties each. What is the probability that, if Bill selects 3 ties at random from all the piles and trashes them, he will trash ... | [
"\\dfrac{67}{91}"
] | [
"Bill groups his ties by color: red, green, and blue. He then takes his entire red pile and splits it into 3 identical piles. Each group has 2 “size” piles, and all the “size” piles now contain 2 ties each. What is the probability that, if Bill selects 3 ties at random from all the piles and trashes them, he will t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-6d825dd1d015e45fa09589cef15a28b1b4d4bbfdd61535aef61cbd3ae1c4b8e5 | Bill has 5 hens, 5 rats, 5 pigeons, 5 rabbits, and 5 hamsters, each with their own distinct cage. Each type of animal does not want to stay in the cage designated for its own type. In how many ways can the animals be placed in cages such that no animal is in a cage designated for its own type? | [
"164916224"
] | [
"Bill has 5 hens, 5 rats, 5 pigeons, 5 rabbits, and 5 hamsters, each with their own distinct cage. Each type of animal does not want to stay in the cage designated for its own type. In how many ways can the animals be placed in cages such that no animal is in a cage designated for its own type?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-bad383e4da5892beb017d98ed770a063f67069dfe36b91802466164cd7be892e | Bill has a $2 \times 2 \times 2$, a $3 \times 3 \times 3$, a $4 \times 4 \times 4$, and a $5 \times 5 \times 5$ Rubik's cube. Sticky Fingers Sue steals one of them with equal probability, and then Pick-Pocket Pearl takes either the largest one or the smallest one still there, again with equal probability. What is the p... | [
"\\(\\frac{5}{8}\\)"
] | [
"Bill has a $2 \\times 2 \\times 2$, a $3 \\times 3 \\times 3$, a $4 \\times 4 \\times 4$, and a $5 \\times 5 \\times 5$ Rubik's cube. Sticky Fingers Sue steals one of them with equal probability, and then Pick-Pocket Pearl takes either the largest one or the smallest one still there, again with equal probability. ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-4481a2978671ec08d502d3a9a3c3ed0cde05632e387ef39551b5b3cc538b7292 | Bill has a set of 6 black cards and a set of 6 red cards, each numbered from 1 to 6. He shuffles all 12 cards, turns over 4 cards, and looks for pairs of cards that have the same value. What is the probability that Bill finds at least one pair of cards that have the same value? | [
"\\(\\frac{17}{33}\\)"
] | [
"Bill has a set of 6 black cards and a set of 6 red cards, each numbered from 1 to 6. He shuffles all 12 cards, turns over 4 cards, and looks for pairs of cards that have the same value. What is the probability that Bill finds at least one pair of cards that have the same value?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-ee9b33b04ed19b3bdb4cf0ed60762ea57b94685fa8e4a8da7bb3042435c3314f | Bill has a set of 6 black cards and a set of 6 red cards. Each card has a number from 1 through 6, such that each of the numbers 1 through 6 appears on 1 black card and 1 red card. Bill shuffles all 12 cards, turns over 4 cards, and looks for pairs of cards that have the same value. What is the probability that Bill fi... | [
"\\dfrac{17}{33}"
] | [
"Bill has a set of 6 black cards and a set of 6 red cards. Each card has a number from 1 through 6, such that each of the numbers 1 through 6 appears on 1 black card and 1 red card. Bill shuffles all 12 cards, turns over 4 cards, and looks for pairs of cards that have the same value. What is the probability that Bi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f87286553ac8798ce35d5f43c9d58b0888c9867495ae33b61bb8da2fc1bd96ec | Bill has an ample supply of 10 cent and 15 cent stamps. He will buy an ample supply of stamps for one more denomination, which is a whole number. He needs to combine the stamps to make any amount ≥ 30 cents. In cents, what is the greatest denomination of stamp he can buy that will allow him to do so? | [
"6"
] | [
"Bill has an ample supply of 10 cent and 15 cent stamps. He will buy an ample supply of stamps for one more denomination, which is a whole number. He needs to combine the stamps to make any amount ≥ 30 cents. In cents, what is the greatest denomination of stamp he can buy that will allow him to do so?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-5e8dbd951e651f4b70661d6e4886eba0e8e686f2bdf786e9e08942d82cac628f | Bill has rods with lengths $1, 2, 3, \cdots, 7$. He repeatedly takes two rods of lengths $a$ and $b$, forms a right triangle with them as legs, and replaces them with a new rod of the length of the hypotenuse. After repeating this process until only two rods remain, he forms a right triangle with those two rods as legs... | [
"35"
] | [
"Bill has rods with lengths $1, 2, 3, \\cdots, 7$. He repeatedly takes two rods of lengths $a$ and $b$, forms a right triangle with them as legs, and replaces them with a new rod of the length of the hypotenuse. After repeating this process until only two rods remain, he forms a right triangle with those two rods a... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-ff875494b15204755338f47f98442a48b35b45decd3ded09e73460f4b9774f1d | Bill is sent to a donut shop to purchase exactly six donuts. If the shop has four kinds of donuts and Bill is to get at least one of each kind, how many combinations will satisfy Bill's order requirements? | [
"10"
] | [
"Bill is sent to a donut shop to purchase exactly six donuts. If the shop has four kinds of donuts and Bill is to get at least one of each kind, how many combinations will satisfy Bill's order requirements?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4093c83246bb781cab81796800dfe88e008e646d0142d79d3869777e1fc55c8e | Bill plays a game in which he rolls two fair standard six-sided dice. He wins if the number on one of the dice is three times the number on the other die. If Bill plays this game three times, compute the probability that he wins at least once. | [
"\\(\\frac{217}{729}\\)"
] | [
"Bill plays a game in which he rolls two fair standard six-sided dice. He wins if the number on one of the dice is three times the number on the other die. If Bill plays this game three times, compute the probability that he wins at least once."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5b18a3fb9031d491015646fe1ddc1ef9b7c0c239d6628633eb984a4d6fb15afa | Bill plays roulette and bets \$5 on his first bet; thereafter (i) If he won the previous bet then he bets \$5 on the next bet. (ii) If he lost \$N on the previous bet then he bets \$(2N - 4) on the next bet. Assuming the payoff is the next bet: (a) If Bill bets 6 times and wins 2 of the times, what are the possible net... | [
"-6"
] | [
"Bill plays roulette and bets \\$5 on his first bet; thereafter\n(i) If he won the previous bet then he bets \\$5 on the next bet.\n(ii) If he lost \\$N on the previous bet then he bets \\$(2N - 4) on the next bet.\nAssuming the payoff is the next bet:\n(a) If Bill bets 6 times and wins 2 of the times, what are the... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-26ef18666a1c7cf8ee24fbad40765f90b20070dd8a43a1a27a41c0594444bf67 | Bill purchased 1 ticket for a draw to win a backstage pass. There are 900 people at the concert, and 2/3 of these people purchased tickets for the draw. Each person is equally likely to purchase 1, 2, or 3 tickets. What is the probability that Bill will win the backstage pass? | [
"1/1200"
] | [
"Bill purchased 1 ticket for a draw to win a backstage pass. There are 900 people at the concert, and 2/3 of these people purchased tickets for the draw. Each person is equally likely to purchase 1, 2, or 3 tickets. What is the probability that Bill will win the backstage pass?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 15
} | 19 | 0 | 0 |
omr-ws-8e1b428f676350324803dffe0a047f0a0b6b1085e0474f2320ccbba0a8c6155a | Bill receives $20 per hour and Jenny makes $35 per hour. They agree to split the cost of 2 tickets to a concert, with each ticket costing $350. Given that neither of them has any money right now, how many more hours per day must Bill work than Jenny in order to buy a ticket? | [
"\\(\\frac{15}{14}\\)"
] | [
"Bill receives $20 per hour and Jenny makes $35 per hour. They agree to split the cost of 2 tickets to a concert, with each ticket costing $350. Given that neither of them has any money right now, how many more hours per day must Bill work than Jenny in order to buy a ticket?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2481cb37c30b057b3ffe67130a3da63e308f1797d687adca3ec4de65bc58ae67 | Bill the Belociraptor is at the top corner of the fourth iteration of Sierpiński's triangle with side length $16$ and trying to get to either of the two bottom corners. Every second, he uniformly randomly chooses a segment at the intersection that he is at (possibly even the segment which he just travelled on) and trav... | [
"\\(5^4\\) or 625."
] | [
"Bill the Belociraptor is at the top corner of the fourth iteration of Sierpiński's triangle with side length $16$ and trying to get to either of the two bottom corners. Every second, he uniformly randomly chooses a segment at the intersection that he is at (possibly even the segment which he just travelled on) and... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-7845cd59035e78dc0657552dc20ad35296007851a034761f5d96dba36d4998d7 | Bill the magician has cards \( A \), \( B \), and \( C \) as shown. For his act, he asks a volunteer to pick any number from 1 through 8 and tell him which cards among \( A \), \( B \), and \( C \) contain it. He then uses this information to guess the volunteer’s number (for example, if the volunteer told Bill “\( A \... | [
"\\(\\frac{8}{35}\\)"
] | [
"Bill the magician has cards \\( A \\), \\( B \\), and \\( C \\) as shown. For his act, he asks a volunteer to pick any number from 1 through 8 and tell him which cards among \\( A \\), \\( B \\), and \\( C \\) contain it. He then uses this information to guess the volunteer’s number (for example, if the volunteer ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-842bfd202a65045afbfb2de3345c5fc6d5efabe0392f639d42f5a5b89754787e | Bill wrote $n$ numbers $a_i$ on the blackboard such that $2a_j > a_{j+1} + a_{j-1}$ for all $2 \leq j \leq n-1$. Each $a_i$ is a positive integer less than or equal to 56. Find the maximum number of numbers Bill could have written on the blackboard. | [
"22"
] | [
"Bill wrote $n$ numbers $a_i$ on the blackboard such that $2a_j > a_{j+1} + a_{j-1}$ for all $2 \\leq j \\leq n-1$. Each $a_i$ is a positive integer less than or equal to 56. Find the maximum number of numbers Bill could have written on the blackboard."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-09a03f533348952852dc1ea3210d2342cf74e78462e434aae45b3af41aa053ef | Bill, Bob, and Ben write their favorite numbers on a sheet of paper. Bob points out that each number has its digits sum to $4$, Bill points out that they are all divisible by the same prime number greater than $10$, and Ben points out that none of them have $0$’s. Find the sum of Bill’s, Bob’s, and Ben’s favorite numbe... | [
"35"
] | [
"Bill, Bob, and Ben write their favorite numbers on a sheet of paper. Bob points out that each number has its digits sum to $4$, Bill points out that they are all divisible by the same prime number greater than $10$, and Ben points out that none of them have $0$’s. Find the sum of Bill’s, Bob’s, and Ben’s favorite ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76ccf392aa0f81598e13bdeb6db2e45c232d4ef407c9fb1309f9582ae3b59231 | Billy Goats invested some money in stocks and bonds. The total amount he invested was $\$165,\!000$. If he invested 4.5 times as much in stocks as he did in bonds, what was his total investment in stocks? | [
"135,\\!000"
] | [
"Billy Goats invested some money in stocks and bonds. The total amount he invested was $\\$165,\\!000$. If he invested 4.5 times as much in stocks as he did in bonds, what was his total investment in stocks?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb666c54c196f880e6a3bb55a3b8ce723bb227242531198451f646d1a025ee98 | Billy and Bill want to play a FTW game together. Billy can only stay for 15 minutes and Bill can only stay for 20 minutes. They decide to come at random times between 8 and 9 pm. Given that they have to start and finish the game between 8 and 9 pm, and that each game lasts 3 minutes, what is the expected number of game... | [
"\\dfrac{5}{3}"
] | [
"Billy and Bill want to play a FTW game together. Billy can only stay for 15 minutes and Bill can only stay for 20 minutes. They decide to come at random times between 8 and 9 pm. Given that they have to start and finish the game between 8 and 9 pm, and that each game lasts 3 minutes, what is the expected number of... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-848bb002cbec776ed547f0eaae8d94ef1a0bd321dc353d473a864badcc00ada8 | Billy and Bob decide to build a rectangular field on the side of the river, so they don't have to contribute any fencing to the side of the field against the river. Billy will only agree to contribute fencing to one side of the field that is perpendicular to the river, while Bob has agreed to contribute fencing to the ... | [
"1225 ft\\(^2\\)"
] | [
"Billy and Bob decide to build a rectangular field on the side of the river, so they don't have to contribute any fencing to the side of the field against the river. Billy will only agree to contribute fencing to one side of the field that is perpendicular to the river, while Bob has agreed to contribute fencing to... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-62563278a69a8061a581fb63612259b4434717ab19ec320a0b04f418a4567ccd | Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, and Bobbi's number is a multiple of 24. What is the probability that they selected the same number? Express your answer as a common fraction. | [
"\\frac{1}{44}"
] | [
"Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, and Bobbi's number is a multiple of 24. What is the probability that they selected the same number? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60f198fa5fb47f40f0c9379caecb24b8ef2c1e1169f756f7ca8924d7bb72ffb7 | Billy and Cindy want to meet at their favorite restaurant, and they have made plans to do so sometime between 1:00 and 2:00 this Sunday. Unfortunately, they didn’t decide on an exact time, so they both decide to arrive at a random time between 1:00 and 2:00. Silly Billy is impatient, though, and if he has to wait for C... | [
"\\dfrac{23}{32}"
] | [
"Billy and Cindy want to meet at their favorite restaurant, and they have made plans to do so sometime between 1:00 and 2:00 this Sunday. Unfortunately, they didn’t decide on an exact time, so they both decide to arrive at a random time between 1:00 and 2:00. Silly Billy is impatient, though, and if he has to wait ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d952d2d9d2c70bcf8896664f162aae5e2e80b1f47e3f443605540d59db68d482 | Billy and Joe are playing a game with these rules: - Billy and Joe decide on a positive integer $k$ and a positive integer $n$, with $n > k$. - Billy will always go first. - Let $k$ be the maximum number of consecutive positive integers each person can say on their turn. - This cycle repeats until someone reaches $n$; ... | [
"6"
] | [
"Billy and Joe are playing a game with these rules:\n- Billy and Joe decide on a positive integer $k$ and a positive integer $n$, with $n > k$.\n- Billy will always go first.\n- Let $k$ be the maximum number of consecutive positive integers each person can say on their turn.\n- This cycle repeats until someone reac... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0e61b59d1b14a4fc4a7ec6c0b2381380ad097564a60601bb8643078a8dfd39e | Billy draws the points \( (x, x^3) \) for \( x \in \{-10, -9, \ldots, 9, 10\} \). How many distinct lines can Jimmy draw that pass through at least two of Billy’s points? | [
"110"
] | [
"Billy draws the points \\( (x, x^3) \\) for \\( x \\in \\{-10, -9, \\ldots, 9, 10\\} \\). How many distinct lines can Jimmy draw that pass through at least two of Billy’s points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-419c2f127d756c0c4a7606f72148d5b914a1cbdfff156bd84fbc69706482af4d | Billy is attending "Moore Academy", an all-boys school on a river that is consistently 3 km wide and straight. Edgar Prep School for Girls is located 8 km downriver from Moore Academy, but on the other side of the river. Billy can row at 6 km/hr and run at 8 km/hr. How far down the river should he row in order to reach... | [
"3.4 km"
] | [
"Billy is attending \"Moore Academy\", an all-boys school on a river that is consistently 3 km wide and straight. Edgar Prep School for Girls is located 8 km downriver from Moore Academy, but on the other side of the river. Billy can row at 6 km/hr and run at 8 km/hr. How far down the river should he row in order t... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 13
} | 16 | 0 | 0 |
omr-ws-2670b5e24ee795b01b6fc382ffd45e6a2b6b52eb2c97a9ea1b62e5178b42092b | Billy is climbing a staircase with 15 stairs. He takes one second to maneuver past each stair and has a $2/3$ chance of overcoming each obstacle. If he fails, he falls back to the bottom and tries again. What is the probability that Billy reaches the top of the stairs within 20 seconds? | [
"\\dfrac{2^{18}}{3^{16}}"
] | [
"Billy is climbing a staircase with 15 stairs. He takes one second to maneuver past each stair and has a $2/3$ chance of overcoming each obstacle. If he fails, he falls back to the bottom and tries again. What is the probability that Billy reaches the top of the stairs within 20 seconds?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-cc9a613a04f106e3e482f9a330ce26b52e70e4657e8a77ee87193346e0db39b3 | Billy is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How far is he from his starting point? Express your answer in simplest radical form. | [
"2\\sqrt{19}"
] | [
"Billy is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How far is he from his starting point? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-0b2f44d39d6cc6ff8bb8a3313006c18ad7a6b05962b26b14d91b43c10f398b29 | Billy the 8-legged spider has 8 identical socks and 8 identical shoes. He wants to put on his socks and shoes so that he gets one sock and one shoe on each leg, with the condition that he must put a sock on a leg before putting a shoe on that same leg. How many different ways can he put on all of his socks and shoes? | [
"\\(\\frac{16!}{2^8}\\)"
] | [
"Billy the 8-legged spider has 8 identical socks and 8 identical shoes. He wants to put on his socks and shoes so that he gets one sock and one shoe on each leg, with the condition that he must put a sock on a leg before putting a shoe on that same leg. How many different ways can he put on all of his socks and sho... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-a552f57c19b54b805e0004acb6ad93bb5cd993c57c67860d5b56f40d42c90817 | Billy the Cyclops wants to paint a sphere hanging from his ceiling from a cord of length 6 meters. Standing directly under the center of the sphere with his eye a distance of 13 meters from the center of the sphere, he stares up and 100\(\pi\) square meters of the area of the ceiling is blocked from his vision. If it t... | [
"461"
] | [
"Billy the Cyclops wants to paint a sphere hanging from his ceiling from a cord of length 6 meters. Standing directly under the center of the sphere with his eye a distance of 13 meters from the center of the sphere, he stares up and 100\\(\\pi\\) square meters of the area of the ceiling is blocked from his vision.... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-574df7774f7f8274b9f7bf8a141ca3655b09b2829419d472f2fffff43d344fc5 | Billy the boy has an infinitely large piece of paper on which there are the 5 vertices of a regular pentagon. At each minute, he constructs regular pentagons that share at least one side with a pentagon drawn in the previous minute. No pentagons drawn at minute \( n \) may overlap entirely with a pentagon drawn at minu... | [
"5 \\times 4^{2017}"
] | [
"Billy the boy has an infinitely large piece of paper on which there are the 5 vertices of a regular pentagon. At each minute, he constructs regular pentagons that share at least one side with a pentagon drawn in the previous minute. No pentagons drawn at minute \\( n \\) may overlap entirely with a pentagon drawn ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-bf84fa928d2bde95e78ac0e5d126a04be9ffa1480b260d8952d5bde186d99177 | Billy tosses a fair six-sided die four times and records the outcomes. The product of the four outcomes is 120. How many possible combinations of 4 rolls could he have rolled? (Note: Rolling a 1,1,2,2 is considered the same combination as 1,2,2,1.) | [
"3"
] | [
"Billy tosses a fair six-sided die four times and records the outcomes. The product of the four outcomes is 120. How many possible combinations of 4 rolls could he have rolled? (Note: Rolling a 1,1,2,2 is considered the same combination as 1,2,2,1.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4073ffd9427b355a5ba7d73da224c91ac9ab2884e6837aace19e82c05bd30e39 | Billy wants to determine the average number of cookies in a package of cookies. He purchases 7 packages of cookies, opens them, and counts the number of cookies in each one. He finds that the packages have 8, 10, 12, 15, 16, 17, and 20 cookies. Based on these observations, what is the average number of cookies in a pac... | [
"14"
] | [
"Billy wants to determine the average number of cookies in a package of cookies. He purchases 7 packages of cookies, opens them, and counts the number of cookies in each one. He finds that the packages have 8, 10, 12, 15, 16, 17, and 20 cookies. Based on these observations, what is the average number of cookies in ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f27a64f79c3e7c19b2663619909663ed15ac8a7eae099fd0c61c270d52aef39 | Billy was given a polynomial $p(x)$ by his teacher. His teacher instructed him to write $p(x - 1)$ in standard form (i.e., of the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0$). Miraculously, the coefficients of $p(x)$ and $p(x - 1)$ were the same except for one term! Given that $p(0) = 0$ and $p(1) = 1$, find the sum ... | [
"1279"
] | [
"Billy was given a polynomial $p(x)$ by his teacher. His teacher instructed him to write $p(x - 1)$ in standard form (i.e., of the form $a_nx^n + a_{n-1}x^{n-1} + \\ldots + a_0$). Miraculously, the coefficients of $p(x)$ and $p(x - 1)$ were the same except for one term! Given that $p(0) = 0$ and $p(1) = 1$, find th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-4e1879d7905d5a4a9f642136261215152cf0fac97839e6141283ac97009d9032 | Billy's age is twice Joe's age and the sum of their ages is 45. How old is Billy? | [
"30"
] | [
"Billy's age is twice Joe's age and the sum of their ages is 45. How old is Billy?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c01dad8b9d3734987575f1183b68c05aed3ce55df1ea45cc3efea76becb5dcc2 | Bin Laden is playing hide and seek with the US government. He has 20 available hideouts numbered 1 through 20. On day 1, Bin Laden picks a random hideout, and on subsequent nights he switches from hideout \(i\) to hideout \(i-1\) or \(i+1\), staying at the hideout during the day. The US government can check one hideout... | [
"15"
] | [
"Bin Laden is playing hide and seek with the US government. He has 20 available hideouts numbered 1 through 20. On day 1, Bin Laden picks a random hideout, and on subsequent nights he switches from hideout \\(i\\) to hideout \\(i-1\\) or \\(i+1\\), staying at the hideout during the day. The US government can check ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-00dbde434ab102d4e466bd21a3674fd70df3e939cdd134920c1b790a8aa53adc | Birbal has 24 hours to mark an area on the ground using wooden sticks. He takes 60 seconds to mark each spot and takes a 15-minute break every hour. His average speed is 25 km/hr (on horseback). What is the maximum area of land Birbal can mark within these constraints? | [
"16000 km²"
] | [
"Birbal has 24 hours to mark an area on the ground using wooden sticks. He takes 60 seconds to mark each spot and takes a 15-minute break every hour. His average speed is 25 km/hr (on horseback). What is the maximum area of land Birbal can mark within these constraints?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-b991ad4b10d2a831da1dfbd19d501c0a597b54e2219792b23e66d668b555cfb1 | Bisharp has a $\frac{1}{2}$ chance every time to defeat another Bisharp. On average, how many Bisharp must it battle to evolve? | [
"6"
] | [
"Bisharp has a $\\frac{1}{2}$ chance every time to defeat another Bisharp. On average, how many Bisharp must it battle to evolve?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-77237c3c1cf826c355601734801b4b824a51f6152c8f9cfc22f7ade1b1d2e15e | Bisharp now learns from its previous wins and has a $\frac{k+1}{k+2}$ chance to defeat another Bisharp, given that it has already defeated $k$ Bisharp. On average, now how many Bisharp must it battle to evolve? | [
"\\dfrac{29}{6}"
] | [
"Bisharp now learns from its previous wins and has a $\\frac{k+1}{k+2}$ chance to defeat another Bisharp, given that it has already defeated $k$ Bisharp. On average, now how many Bisharp must it battle to evolve?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-5097a6e49184a16c218d22b718b657011d1edfd809ffab8f952f219eb9c66769 | Björk has $64$ sugar cubes, each of size $1 \times 1 \times 1$, and each cube is either white, demerara, or muscovado in flavor. She piles the sugar cubes into a neat $4 \times 4 \times 4$ cube. Find the maximum number of sugar cubes of the same flavor that can be put into $6$ disjoint pairs such that the distance betw... | [
"12"
] | [
"Björk has $64$ sugar cubes, each of size $1 \\times 1 \\times 1$, and each cube is either white, demerara, or muscovado in flavor. She piles the sugar cubes into a neat $4 \\times 4 \\times 4$ cube. Find the maximum number of sugar cubes of the same flavor that can be put into $6$ disjoint pairs such that the dist... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c26c6165ab38cd81dc91351340b0fbac2bf01cb496e0c09174b0f1926b0b00d | Black and white coins are placed on some of the squares of a $418 \times 418$ grid. All black coins that are in the same row as any white coin(s) are removed. After that, all white coins that are in the same column as any black coin(s) are removed. If $b$ is the number of black coins remaining and $w$ is the number of ... | [
"1534"
] | [
"Black and white coins are placed on some of the squares of a $418 \\times 418$ grid. All black coins that are in the same row as any white coin(s) are removed. After that, all white coins that are in the same column as any black coin(s) are removed. If $b$ is the number of black coins remaining and $w$ is the numb... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-97ef057e3dee8555536be62e4c0e101ddbcebe921e9125314095d862c6b2fdb4 | Blahaj has two rays with a common endpoint \( A_0 \) that form an angle of \( 1^\circ \). They construct a sequence of points \( A_0, \ldots, A_n \) such that for all \( 1 \le i \le n \), \( |A_{i-1}A_i| = 1 \), and \( |A_iA_0| > |A_{i-1}A_0| \). Find the largest possible value of \( n \). | [
"90"
] | [
"Blahaj has two rays with a common endpoint \\( A_0 \\) that form an angle of \\( 1^\\circ \\). They construct a sequence of points \\( A_0, \\ldots, A_n \\) such that for all \\( 1 \\le i \\le n \\), \\( |A_{i-1}A_i| = 1 \\), and \\( |A_iA_0| > |A_{i-1}A_0| \\). Find the largest possible value of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2b08bbccb0dc6584729ae46b2ff428265a429add03ac0280c8b46a1c7e9c3564 | Blaise constructed a convex hexagon with three non-adjacent right angles, where the two sides adjacent to each of these right angles are of the same length. He then constructs a second hexagon by replacing each right angle vertex with the symmetrical vertex with respect to the vertex opposite in the original hexagon, w... | [
"2"
] | [
"Blaise constructed a convex hexagon with three non-adjacent right angles, where the two sides adjacent to each of these right angles are of the same length. He then constructs a second hexagon by replacing each right angle vertex with the symmetrical vertex with respect to the vertex opposite in the original hexag... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-59edecff01852972ddd0ce2b180d4c12685a613d6cbb219b7f3c4fa65d21f100 | Blaise creates a circle of 100 unlit candles and randomly lights ten of them. Every second, each unlit candle has a \( \frac{k}{2} \) chance of becoming lit, where \( k \) is the number of lit candles adjacent to the unlit candle. What is the average number of seconds that pass before all 100 candles are lit? | [
"50"
] | [
"Blaise creates a circle of 100 unlit candles and randomly lights ten of them. Every second, each unlit candle has a \\( \\frac{k}{2} \\) chance of becoming lit, where \\( k \\) is the number of lit candles adjacent to the unlit candle. What is the average number of seconds that pass before all 100 candles are lit?... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-cbcb9da21be2166b527a2f882c65099f6fff2669fe5e87262d5b7bdc007f0478 | Bloggy has $1200$ treats. He arranges them in $n$ rows, where $n$ is a factor of $1200$. For the $x$th row, there is a $\frac{1}{x}$ chance that Bloggy will eat all the treats in that row. Given that the number $n$ is randomly chosen, find the expected value of the number of eaten treats after Bloggy is finished eating... | [
"257"
] | [
"Bloggy has $1200$ treats. He arranges them in $n$ rows, where $n$ is a factor of $1200$. For the $x$th row, there is a $\\frac{1}{x}$ chance that Bloggy will eat all the treats in that row. Given that the number $n$ is randomly chosen, find the expected value of the number of eaten treats after Bloggy is finished ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-0c6efae74e121345b6bacb182c383c15ff4151d6b8e9793d38dccb0f83f1bc06 | Bo has 10 distinct images and 6 distinct social media platforms. He must post each image on at least one platform. How many ways can Bo post the images so that he keeps his job? | [
"\\(6^{10}\\)"
] | [
"Bo has 10 distinct images and 6 distinct social media platforms. He must post each image on at least one platform. How many ways can Bo post the images so that he keeps his job?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a85bbbc7709cb78764189ac22a64696e63bd9641a66d6625ca598925dc4ef821 | Bo is in charge of the Penn Knitting Club and wants to admit members randomly over several hours. The probability of admission decreases each hour. If 60 people applied and Bo wants the club to have an expected number of 48 people, how many hours should he hold the admission process open for? | [
"4"
] | [
"Bo is in charge of the Penn Knitting Club and wants to admit members randomly over several hours. The probability of admission decreases each hour. If 60 people applied and Bo wants the club to have an expected number of 48 people, how many hours should he hold the admission process open for?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-ee7766fc79407e7bfcf12140f1b431a15ee92d8038f2612bf4e71e0416121253 | Bo, Darren, Annie, and Anna raced and earned first, second, third, and fourth place, respectively. If each person can move up or down at most one place in the next race, how many possibilities are there for the outcome of the second race? | [
"5"
] | [
"Bo, Darren, Annie, and Anna raced and earned first, second, third, and fourth place, respectively. If each person can move up or down at most one place in the next race, how many possibilities are there for the outcome of the second race?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-448f7d79a986562859b398b4291af0e28ece9534f2cc6c0375e2c04f103c32b8 | Boards that are 8 feet long, 2 inches thick, and 6 inches wide are used to make a raised garden bed. If the boards are placed on level ground, how many cubic feet of soil are needed to completely fill the bed? Express your answer to the nearest whole number. | [
"29"
] | [
"Boards that are 8 feet long, 2 inches thick, and 6 inches wide are used to make a raised garden bed. If the boards are placed on level ground, how many cubic feet of soil are needed to completely fill the bed? Express your answer to the nearest whole number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-19ca4a747cf62a30ea0f2fa083c68bd33889f9c98026abb814088fb2e02575b0 | Boat A and Boat B leave the dock at the same time and each sails at a constant speed. Boat A sails due north and Boat B due east. After 3 hours, they are 65 miles apart. Boat B sails 7 miles per hour faster than Boat A. Find the speed of each boat to the nearest tenth of a mile per hour. | [
"Boat A: 11.4 mph, Boat B: 18.4 mph"
] | [
"Boat A and Boat B leave the dock at the same time and each sails at a constant speed. Boat A sails due north and Boat B due east. After 3 hours, they are 65 miles apart. Boat B sails 7 miles per hour faster than Boat A. Find the speed of each boat to the nearest tenth of a mile per hour."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 29
} | 33 | 0 | 0 |
omr-ws-e1b7a45dba3d9ce39d552ff9a180fa5ee60efb67ce6b3eebf73394d228d40438 | BoatWorks built 3 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatWorks by the end of March of this year? | [
"21"
] | [
"BoatWorks built 3 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatWorks by the end of March of this year?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39836d521d88563764cc540d2950148dbb8b22177484a4f117414acb82bf6d63 | Boats A and B leave the same place at the same time. Boat A heads due north at 12 km/hr, and Boat B heads due east at 18 km/hr. After 2.5 hours, how fast is the distance between the boats increasing (in km/hr)? | [
"21.63"
] | [
"Boats A and B leave the same place at the same time. Boat A heads due north at 12 km/hr, and Boat B heads due east at 18 km/hr. After 2.5 hours, how fast is the distance between the boats increasing (in km/hr)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-b2eb3e09acecb08c43a6a78b13584f729f1ff934d3c615c94766517ad28ac53e | BoatsRUs built 7 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatsRUs by the end of May of this year? | [
"217"
] | [
"BoatsRUs built 7 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatsRUs by the end of May of this year?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-23d3a87d3f84b2817df9d3ca5c6586141e571a421eb1e8e90c2321c29a032f24 | Bob and Alice are playing a game with an infinite number of rounds. In the nth round, Bob chooses a family $F_n$ of rational open intervals (i.e., intervals of the form $\{x \in \mathbb{Q} \mid p < x < q\}$ for $p, q \in \mathbb{Q}$) such that $\mathbb{Q} = \bigcup_{I \in F_n} I$. Alice then chooses an interval $I_n \i... | [
"\\text{Alice wins by choosing an interval containing the } n\\text{th rational in each round } n."
] | [
"Bob and Alice are playing a game with an infinite number of rounds. In the nth round, Bob chooses a family $F_n$ of rational open intervals (i.e., intervals of the form $\\{x \\in \\mathbb{Q} \\mid p < x < q\\}$ for $p, q \\in \\mathbb{Q}$) such that $\\mathbb{Q} = \\bigcup_{I \\in F_n} I$. Alice then chooses an i... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5bc1e56bc3974437e4cbe1f08a6fa16ad4cd05a6d7cc88525624fe6125713c12 | Bob and Alice each have 50 friends in a town of 1000 people (excluding themselves). What is the probability that they have $j$ mutual friends? | [
"\\(\\frac{\\binom{50}{j}\\binom{950}{50-j}}{\\binom{1000}{50}}\\)"
] | [
"Bob and Alice each have 50 friends in a town of 1000 people (excluding themselves). What is the probability that they have $j$ mutual friends?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-2113b46e4de184f88796554b637a42f46101587a00a28a43c504e9acec9d840a | Bob and Alice each have a bag containing one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process, the conten... | [
"\\(\\frac{1}{3}\\)"
] | [
"Bob and Alice each have a bag containing one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process, the c... | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-bde7172117abe3dc3f337664861e228d5413c48b3943478013f18c8ebeab8f94 | Bob and Chad have been given a task to clean 24 cars in 15 days. If Bob fell ill after 12 days, Chad needs an additional 2 days to complete the task. If Chad fell ill after 12 days, Bob needs an additional \( n \) days to complete the task. Find the value of \( n \). | [
"4.5"
] | [
"Bob and Chad have been given a task to clean 24 cars in 15 days. If Bob fell ill after 12 days, Chad needs an additional 2 days to complete the task. If Chad fell ill after 12 days, Bob needs an additional \\( n \\) days to complete the task. Find the value of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-9510e7c44a1522f74e8bdff54f6c4f769aeac857d9bc7741b7e5e7ad1bf4bc41 | Bob and Jane each have a deck of twelve cards, with three cards of each color: red, green, yellow, and blue. They shuffle their decks and take turns dealing one card at a time onto a pile, with Jane going first. Find the probability that Jane deals all her red cards before Bob deals any of his red cards. | [
"\\(\\frac{1}{20}\\)"
] | [
"Bob and Jane each have a deck of twelve cards, with three cards of each color: red, green, yellow, and blue. They shuffle their decks and take turns dealing one card at a time onto a pile, with Jane going first. Find the probability that Jane deals all her red cards before Bob deals any of his red cards."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ef8716854fa5a51cd10295558430ed962cd016210cb03f1042a01c438ed0da0 | Bob and Jeff are on a quest to find magical numbers. Jeff writes all positive integers on a board. Bob chooses two numbers $a$ and $b$, erases them, and writes $\frac{ab}{\sqrt{a^2+b^2}}$ on the board. He repeats this process until one number is left, which is $\frac{\sqrt{n}}{\pi}$ where $n$ is a positive integer. Fin... | [
"6"
] | [
"Bob and Jeff are on a quest to find magical numbers. Jeff writes all positive integers on a board. Bob chooses two numbers $a$ and $b$, erases them, and writes $\\frac{ab}{\\sqrt{a^2+b^2}}$ on the board. He repeats this process until one number is left, which is $\\frac{\\sqrt{n}}{\\pi}$ where $n$ is a positive in... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a4811fc4ed1c351b3dc078cf6dcce84ebcd4e28a2f67f3ad3b7bbb5ca75ccd45 | Bob and Jill are trying to find a day that they both have off so they can go to the zoo. Jill works for 6 days and gets off on the 7th day. Bob works for 12 days and then gets the 13th and 14th days off. What is the first day they both have off? | [
"14"
] | [
"Bob and Jill are trying to find a day that they both have off so they can go to the zoo. Jill works for 6 days and gets off on the 7th day. Bob works for 12 days and then gets the 13th and 14th days off. What is the first day they both have off?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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