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-1b87ecd47593a08698907a8a85e5cd26185d821fb953b04ea209988e5320682d | At 12:00 noon, there are two pirate ships, A and B. Ship A is 50 km east of ship B. If ship A sails west at 10 km/h and ship B sails south at 20 km/h, when will these two ships be closest to each other? What is the distance between them at that time? | [
"\\( 20\\sqrt{5} \\)"
] | [
"At 12:00 noon, there are two pirate ships, A and B. Ship A is 50 km east of ship B. If ship A sails west at 10 km/h and ship B sails south at 20 km/h, when will these two ships be closest to each other? What is the distance between them at that time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
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"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-208f7b39cf6295e4786e51581112e7da5158b1d410ad9b3f6c2ca3a098371f0e | At 1:00 AM, Airplane A is 700 miles from the airport and flying due south at 200 mph, while Airplane B is 925 miles from the airport and flying due west at 250 mph. How far are the two airplanes from each other at 10:30 A.M.? | [
"1882.15"
] | [
"At 1:00 AM, Airplane A is 700 miles from the airport and flying due south at 200 mph, while Airplane B is 925 miles from the airport and flying due west at 250 mph. How far are the two airplanes from each other at 10:30 A.M.?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-bf38c912b024933d9e51caa1249a7eef86defeaf8f3fad3e9bab628db436c8fb | At 1:00 PM, car A is 10 km due north of Slagbury and is heading due west at 40 km/h. At the same time, car B is 8 km due south of Slagbury and is heading due east at 50 km/h. How fast is the distance between the cars changing at 1:16 PM? | [
"72 km/hr"
] | [
"At 1:00 PM, car A is 10 km due north of Slagbury and is heading due west at 40 km/h. At the same time, car B is 8 km due south of Slagbury and is heading due east at 50 km/h. How fast is the distance between the cars changing at 1:16 PM?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-f7968b4e1eafd8b4d27d597890a0d5646f0f87384acc584b012523ebfdd8eb58 | At 1:00 p.m., Sue left her house and began walking at 4 km/h toward Sandy's house. Fifteen minutes later, Sandy left her home and walked at 8 km/h toward Sue's house. If Sue lives 10 km from Sandy, at what time did they meet? Who walked farther? | [
"2:00 PM, Sandy walked farther."
] | [
"At 1:00 p.m., Sue left her house and began walking at 4 km/h toward Sandy's house. Fifteen minutes later, Sandy left her home and walked at 8 km/h toward Sue's house. If Sue lives 10 km from Sandy, at what time did they meet? Who walked farther?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-9eaf6c2471362e12d8933a648f998a975d8891290a15f7f5d39c0540b6a724fe | At 3 PM standard time, in the sky above Alabama, a honey bee heading 095 at 90 MPH and a hornet heading 005 at 90√3 MPH collided. Both were flying straight; what was the bearing from the hornet to the bee right up to the moment they collided? | [
"155"
] | [
"At 3 PM standard time, in the sky above Alabama, a honey bee heading 095 at 90 MPH and a hornet heading 005 at 90√3 MPH collided. Both were flying straight; what was the bearing from the hornet to the bee right up to the moment they collided?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-2e20fda759044ed3beb6e9b150c703a7dce5bf2b80ca35cc6041e88f2dc12473 | At 3:00 o'clock, in how many minutes will the hands of the clock first be together? | [
"\\( \\frac{180}{11} \\)"
] | [
"At 3:00 o'clock, in how many minutes will the hands of the clock first be together?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-963e33479c4d966ca11a054688ad847018edfdaed8b0c296800429791c5cd244 | At 5:57, all the digits on the clock are prime numbers. In how many minutes will all the digits on the clock be prime numbers again? | [
"85"
] | [
"At 5:57, all the digits on the clock are prime numbers. In how many minutes will all the digits on the clock be prime numbers again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-efca40a27930c48cb856e572da29f3ab216e79e4f550b4f4cfe2ded021af4708 | At 6 AM, bus station A starts to dispatch buses to station B, and station B starts to dispatch buses to station A. They each dispatch one bus to the other station every 8 minutes. The one-way trip takes 45 minutes. One passenger gets on the bus at station A at 6:16 AM. How many buses coming from station B will the pass... | [
"8"
] | [
"At 6 AM, bus station A starts to dispatch buses to station B, and station B starts to dispatch buses to station A. They each dispatch one bus to the other station every 8 minutes. The one-way trip takes 45 minutes. One passenger gets on the bus at station A at 6:16 AM. How many buses coming from station B will the... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-abac571fba62ce2de9d2c45e4ad14344416779fc91a300021f837a4e78286273 | At 6:00, the tip of the clock's hour hand was 23 cm from the tip of its minute hand. At 9:00, this distance was 17 cm. By how many cm does the minute hand's length exceed the hour hand's length? | [
"7"
] | [
"At 6:00, the tip of the clock's hour hand was 23 cm from the tip of its minute hand. At 9:00, this distance was 17 cm. By how many cm does the minute hand's length exceed the hour hand's length?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d6f14fc2ddbee30a3a3ed80c28932a64a63fac3161157bf81a252725160e996 | At 6pm, the hands of a clock are 180 degrees apart. How long will it be before the hands of the clock are 180 degrees apart again? | [
"\\( 65 + \\frac{5}{11} \\) minutes.",
"\\(\\frac{720}{11}\\)"
] | [
"At 6pm, the hands of a clock are 180 degrees apart. How long will it be before the hands of the clock are 180 degrees apart again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 54
} | 85 | 0 | 0 |
omr-ws-d301fa5fa54974adfda60e8bc32e706f6d7268891d4dea306c484d6f64118347 | At 8 AM, Black Widow and Hawkeye began moving towards each other from two cities. They were planning to meet at the midpoint between the two cities, but because Black Widow was driving 100 mi/h faster than Hawkeye, they met at a point 120 miles from the midpoint. When they met, Black Widow said, "If I knew that you dri... | [
"1200"
] | [
"At 8 AM, Black Widow and Hawkeye began moving towards each other from two cities. They were planning to meet at the midpoint between the two cities, but because Black Widow was driving 100 mi/h faster than Hawkeye, they met at a point 120 miles from the midpoint. When they met, Black Widow said, \"If I knew that y... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-935e52dc330274fda1d3b8a498d6a895b85a9d25d8908cc2d58068284fcabb5b | At 8 a.m. a fast train leaves $A$ for $B$. At the same time a passenger train and an express train leave $B$ for $A$, the speed of the former being one half that of the latter. The fast train meets the express train not earlier than $10:30$ a.m. and arrives at $B$ at $13:50$ the same day. Find the time of arrival of th... | [
"16 hours and 45 minutes"
] | [
"At 8 a.m. a fast train leaves $A$ for $B$. At the same time a passenger train and an express train leave $B$ for $A$, the speed of the former being one half that of the latter. The fast train meets the express train not earlier than $10:30$ a.m. and arrives at $B$ at $13:50$ the same day. Find the time of arrival ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74f4965d039976a24e7d62c2695f7b21f654a857f96ce7cc80b374cdec1a3fbe | At 9 a.m. a cyclist starts out from $A$ in the direction of $B$. Two hours later a motorist sets out and overtakes the cyclist not later than $12:00$ noon. The motorist continues on and reaches $B$, then immediately turns round and heads back for $A$. On the way, the motorist meets the cyclist and arrives in $A$ at $17... | [
"18 hours"
] | [
"At 9 a.m. a cyclist starts out from $A$ in the direction of $B$. Two hours later a motorist sets out and overtakes the cyclist not later than $12:00$ noon. The motorist continues on and reaches $B$, then immediately turns round and heads back for $A$. On the way, the motorist meets the cyclist and arrives in $A$ a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf6170ed85d3a80f49f98618b7d565345f0308653b023298a0f2d0447ca329ed | At 9 a.m. self-propelled barge starts from point $A$ up the river and some time later arrives at point $B$. Two hours after it arrived at $B$, the barge starts back and arrives at $A$ at 7.20 p.m. on the same day. Assuming that the speed of the river flow is 3 km/h and the inherent speed of the barge is constant, find ... | [
"14 h"
] | [
"At 9 a.m. self-propelled barge starts from point $A$ up the river and some time later arrives at point $B$. Two hours after it arrived at $B$, the barge starts back and arrives at $A$ at 7.20 p.m. on the same day. Assuming that the speed of the river flow is 3 km/h and the inherent speed of the barge is constant, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dce08c6c78f15ac4ab41d85b26c86b9ddd331b2f258fb53e109d94bbe58432be | At 9:00 A.M., Joe sees three clouds in the sky. Each hour afterwards, a new cloud appears in the sky, while each old cloud has a 40% chance of disappearing. Given that the expected number of clouds that Joe will see right after 1:00 P.M. can be written in the form $p/q$, where $p$ and $q$ are relatively prime positive ... | [
"2228"
] | [
"At 9:00 A.M., Joe sees three clouds in the sky. Each hour afterwards, a new cloud appears in the sky, while each old cloud has a 40% chance of disappearing. Given that the expected number of clouds that Joe will see right after 1:00 P.M. can be written in the form $p/q$, where $p$ and $q$ are relatively prime posi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-66c89dd518b3eb4a722936629e3453ea5a38bf99c215a50cdd0dcd3f32cc9025 | At 9:00, an analog clock has its hands perpendicular to each other. Exactly how many minutes pass before the next time that the clock’s hands are perpendicular to each other? Express your answer as a fraction in simplified form. | [
"\\dfrac{360}{11}"
] | [
"At 9:00, an analog clock has its hands perpendicular to each other. Exactly how many minutes pass before the next time that the clock’s hands are perpendicular to each other? Express your answer as a fraction in simplified form."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9457ff3cec49c3c0f4ed9c6a7925d1519c24e97daf947ede0a2aa4e55e5186c | At ARML, Santa is asked to give rubber duckies to 2013 students, numbered 1 through 2013. Students \( m \) and \( n \) are friends if and only if \( 0 \leq n - 2m \leq 1 \). Santa has four different colors of duckies and wants to give duckies of different colors to any two students who are either friends or who share a... | [
"0"
] | [
"At ARML, Santa is asked to give rubber duckies to 2013 students, numbered 1 through 2013. Students \\( m \\) and \\( n \\) are friends if and only if \\( 0 \\leq n - 2m \\leq 1 \\). Santa has four different colors of duckies and wants to give duckies of different colors to any two students who are either friends o... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69b3a7819f7caeff44d6c08c97fb4aeb1a1dedbf23829119dcadb1d29c65b76f | At ARML, there are 4 more MOPpers than moppers. The number of people who are neither MOPpers nor moppers is equal to the product of the number of MOPpers and the number of moppers. If there are $2018$ people in total, what fraction of the MOPpers are also moppers? | [
"0"
] | [
"At ARML, there are 4 more MOPpers than moppers. The number of people who are neither MOPpers nor moppers is equal to the product of the number of MOPpers and the number of moppers. If there are $2018$ people in total, what fraction of the MOPpers are also moppers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbed2016fad61cf68c9f34530bc1a589da473234d95ecb5f3abb4c28b40d7ab8 | At Academic Academy, to pass an algebra test you must score at least $80\%$. If there are 35 problems on the test, what is the greatest number you can miss and still pass? | [
"7"
] | [
"At Academic Academy, to pass an algebra test you must score at least $80\\%$. If there are 35 problems on the test, what is the greatest number you can miss and still pass?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-148acb39f3b20630739d0e71ba1c2967d59c7e4904436b06b2417acc9543f99d | At Academy Imaginative Mathletics Edisonville, a student starts walking around a school at exactly 1:23 PM. The school consists of 6 classrooms arranged in a regular hexagon with each side measuring 421 meters. The student starts in a combinatorics classroom and randomly chooses a direction at each node, with the proba... | [
"35"
] | [
"At Academy Imaginative Mathletics Edisonville, a student starts walking around a school at exactly 1:23 PM. The school consists of 6 classrooms arranged in a regular hexagon with each side measuring 421 meters. The student starts in a combinatorics classroom and randomly chooses a direction at each node, with the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-09d1f6bb19f3ffecafaca5943a99efb64d15c76418c05cb049f63507e0786c48 | At AoPS School, there are 10 people playing baseball, 5 people playing football, 5 people playing ping pong, and 5 people playing soccer. These people are seated randomly in 5 equally spaced parallel rows of 5 chairs each. A person is considered happy if they see at least 3 people playing the same sport in the same col... | [
"0"
] | [
"At AoPS School, there are 10 people playing baseball, 5 people playing football, 5 people playing ping pong, and 5 people playing soccer. These people are seated randomly in 5 equally spaced parallel rows of 5 chairs each. A person is considered happy if they see at least 3 people playing the same sport in the sam... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-033ce10e2fcc413a690cdc003a2157eb5eadd41bda6ad9bd4f1bb69739bb5dbf | At Beaumont High School, there are 12 players on the baseball team. All 12 players are taking at least one of biology or chemistry. If 7 players are taking biology and 2 players are taking both sciences, how many players are taking chemistry? | [
"7"
] | [
"At Beaumont High School, there are 12 players on the baseball team. All 12 players are taking at least one of biology or chemistry. If 7 players are taking biology and 2 players are taking both sciences, how many players are taking chemistry?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-135be8d8e7a1423110a20dd8ceb21e637c3aef2e4b667a12dccac7cf7e072e2c | At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both sciences, how many players are taking chemist... | [
"16"
] | [
"At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both sciences, how many players are taking... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e685ffec9dbcbcd61091e3ccd76eaf970cd80c6a5f4a1471078a62b4ddf74feb | At Central Middle School the $108$ students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of $15$ cookies, lists these items: $\bullet$ $1\frac{1}{2}$ cups of flour... | [
"11"
] | [
"At Central Middle School the $108$ students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of $15$ cookies, lists these items:\n\n$\\bullet$ $1\\frac{1}{2}$ cup... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-84e29387dff5c5154051543dee737f4abeeb90df132888565fffb5c9ca94ffb5 | At DcMonalds, chicken nuggets are sold in boxes of 98, 99, and 100. What is the largest possible integer value of N for which it is not possible to buy exactly N chicken nuggets? | [
"4801"
] | [
"At DcMonalds, chicken nuggets are sold in boxes of 98, 99, and 100. What is the largest possible integer value of N for which it is not possible to buy exactly N chicken nuggets?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 16,
"tir": 18
} | 57 | 0 | 0 |
omr-ws-315f09d071d170b491f68e4d5e7ae156fc4c3413910d23a4981f635855b66fd8 | At Duke, \( \frac{1}{2} \) of the students like lacrosse, \( \frac{3}{4} \) like football, and \( \frac{7}{8} \) like basketball. Let \( p \) be the proportion of students who like at least all three of these sports and let \( q \) be the difference between the maximum and minimum possible values of \( p \). If \( q \)... | [
"11"
] | [
"At Duke, \\( \\frac{1}{2} \\) of the students like lacrosse, \\( \\frac{3}{4} \\) like football, and \\( \\frac{7}{8} \\) like basketball. Let \\( p \\) be the proportion of students who like at least all three of these sports and let \\( q \\) be the difference between the maximum and minimum possible values of \... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-2b052ff137463780a39c023905c94c292b5ebdd1427a9345aa6d86acf033fbdb | At Easter Day, four children and their mothers participated in a game to find hidden chocolate eggs. The children found the following numbers of eggs: Augustine found 4 eggs, Bruno found 6 eggs, Carlos found 9 eggs, and Daniel found 12 eggs. The mothers found eggs as follows: Mrs. Gómez found the same number of eggs as... | [
"Messi is Augustine's mother, Núñez is Daniel's, Junco is Carlos's, and Gómez is Bruno's."
] | [
"At Easter Day, four children and their mothers participated in a game to find hidden chocolate eggs. The children found the following numbers of eggs: Augustine found 4 eggs, Bruno found 6 eggs, Carlos found 9 eggs, and Daniel found 12 eggs. The mothers found eggs as follows: Mrs. Gómez found the same number of eg... | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-b4abba0d495a8d350e67dde31330f0653b00576fae20da235f7c232d7adbec30 | At Euler Middle School, the ratio of boys to girls is $1 : 1$. A teacher asked all of the students in the school to stand in a straight line such that there were two boys followed by a girl until there were no more boys. If there were $25$ girls left over, how many total students attend Euler Middle School? | [
"100"
] | [
"At Euler Middle School, the ratio of boys to girls is $1 : 1$. A teacher asked all of the students in the school to stand in a straight line such that there were two boys followed by a girl until there were no more boys. If there were $25$ girls left over, how many total students attend Euler Middle School?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a7aac2b6bc80cb3428cda023439166146d18df18f2c06794457eaf198781d8c | At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas? | [
"8",
"8 oranges"
] | [
"At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?",
"At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?"
] | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-3318c4e0c104a3ea364dbed2061cef0251bc52ee88156c5ab1d0e56469f4b014 | At Hill College House, there are $5$ residential floors. Each floor has four sections: blue, green, red, and yellow. Each section has a study lounge and a social lounge, except for the red section on the first floor, which has neither. Furthermore, there is a public study lounge and a club lounge in the atrium. How man... | [
"32"
] | [
"At Hill College House, there are $5$ residential floors. Each floor has four sections: blue, green, red, and yellow. Each section has a study lounge and a social lounge, except for the red section on the first floor, which has neither. Furthermore, there is a public study lounge and a club lounge in the atrium. Ho... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-688dfc6c4cbe5b5e54064f1e51b2da1b8adeddfd78cdcb20fec3de49ba563370 | At Idea Math, there is a group of people that satisfies the following property: for all positive integers $k$, any subset of $2k-1$ people in the group contains a subset of $k$ people in the group who are all friends with each other. If every member of the group has 40 friends in the group or fewer, find the maximum nu... | [
"82"
] | [
"At Idea Math, there is a group of people that satisfies the following property: for all positive integers $k$, any subset of $2k-1$ people in the group contains a subset of $k$ people in the group who are all friends with each other. If every member of the group has 40 friends in the group or fewer, find the maxim... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-c0e6e1d38f65a914e9a33986a4bc86fe0a11b17b412e5bcef869174aa50328f1 | At JOMO's Pizzeria, you can design your own pizza with the following toppings: Pepperoni, Salami, Onions, Mushrooms, Pineapples, and Ham. You can also choose between a thick and thin crust. A plain pizza with no toppings costs \$120HKD, and each extra topping costs an additional \$20HKD. If you have \$200HKD, how many ... | [
"420"
] | [
"At JOMO's Pizzeria, you can design your own pizza with the following toppings: Pepperoni, Salami, Onions, Mushrooms, Pineapples, and Ham. You can also choose between a thick and thin crust. A plain pizza with no toppings costs \\$120HKD, and each extra topping costs an additional \\$20HKD. If you have \\$200HKD, h... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-da0e6c37f9e872d38094215809506f48b868aa1d9e45a367e3be020bd882c012 | At John's Pizza, pizzas are circular and cut into eight slices, with each slice coming in one of three flavors: pepperoni, meatball, and chicken. If rotations are not distinct, how many different pizzas are there? | [
"834"
] | [
"At John's Pizza, pizzas are circular and cut into eight slices, with each slice coming in one of three flavors: pepperoni, meatball, and chicken. If rotations are not distinct, how many different pizzas are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-529579d64b57ecdfa6363df05a914ecda35b772869c3d8a36b91ff7dbec2447c | At Kanye Crest Academy, employees get paid in CCA Math Bananas$^{\text{TM}}$. At the end of $2018$, Professor Shian Bray was given a $10\%$ pay raise from his salary at the end of $2017$. However, inflation caused the worth of a CCA Math Banana$^{\text{TM}}$ to decrease by $1\%$. If Prof. Bray's salary at the end of $2... | [
"\\$1089000"
] | [
"At Kanye Crest Academy, employees get paid in CCA Math Bananas$^{\\text{TM}}$. At the end of $2018$, Professor Shian Bray was given a $10\\%$ pay raise from his salary at the end of $2017$. However, inflation caused the worth of a CCA Math Banana$^{\\text{TM}}$ to decrease by $1\\%$. If Prof. Bray's salary at the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-70632e62b9432e3eef445ebcbca61ae35ffcc94be088790f6e7e97f3c87b52fd | At King Arthur's Court, $2n$ knights gathered at the Round Table. Each knight has at most $n-1$ enemies among the others. Merlin the Wizard devises a seating arrangement such that no knight is seated next to any of his enemies. Let $S(n)$ be the number of distinct seating arrangements possible for $n=5$. Find the remai... | [
"440"
] | [
"At King Arthur's Court, $2n$ knights gathered at the Round Table. Each knight has at most $n-1$ enemies among the others. Merlin the Wizard devises a seating arrangement such that no knight is seated next to any of his enemies. Let $S(n)$ be the number of distinct seating arrangements possible for $n=5$. Find the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-cda18e80013e2b5932ffc62a3732a6a53e6d1cb401995fca07d70ef732cca41a | At LaGuardia Airport for a certain nightly flight, the probability that it will rain is $0.13$ and the probability that the flight will be delayed is $0.18$. The probability that it will not rain and the flight will leave on time is $0.79$. What is the probability that the flight would be delayed when it is not raining... | [
"0.0919.."
] | [
"At LaGuardia Airport for a certain nightly flight, the probability that it will rain is $0.13$ and the probability that the flight will be delayed is $0.18$. The probability that it will not rain and the flight will leave on time is $0.79$. What is the probability that the flight would be delayed when it is not ra... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9985c73164a620812b10067695f48debf2c172dd5cff47b326d9600faac965b1 | At Largetown high school, 27% of 530 seniors and 28% of 490 juniors drive themselves to school. A simple random sample of 40 seniors and 40 juniors is taken. What is the probability that the survey will indicate more juniors drive themselves to school than seniors? | [
"0.540"
] | [
"At Largetown high school, 27% of 530 seniors and 28% of 490 juniors drive themselves to school. A simple random sample of 40 seniors and 40 juniors is taken. What is the probability that the survey will indicate more juniors drive themselves to school than seniors?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c528f60c44844e07d7ca73a0c52769921efeffc2e91db2ee3979d02fb6817d6d | At MOP, all m students share exactly one common friend, where \( m \geq 3 \). Suppose Nikhil has the largest number of friends. Determine the number of friends Nikhil has. | [
"\\( m-1 \\)"
] | [
"At MOP, all m students share exactly one common friend, where \\( m \\geq 3 \\). Suppose Nikhil has the largest number of friends. Determine the number of friends Nikhil has."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-b72586f5985b3e5ce6204c417952515e86d769f88ac182ab28c65387a90aad3a | At Mendoza High School, $35\%$ of the students are freshmen, $25\%$ are sophomores, $20\%$ are juniors, and $20\%$ are seniors. If $10\%$ of the freshmen, $20\%$ of the sophomores, $30\%$ of the juniors, and $40\%$ of the seniors attend the Mendoza Math Meet, the probability that a randomly selected participant at the ... | [
"73"
] | [
"At Mendoza High School, $35\\%$ of the students are freshmen, $25\\%$ are sophomores, $20\\%$ are juniors, and $20\\%$ are seniors. If $10\\%$ of the freshmen, $20\\%$ of the sophomores, $30\\%$ of the juniors, and $40\\%$ of the seniors attend the Mendoza Math Meet, the probability that a randomly selected partic... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-7228af24bc832b061ecd157ff99c16d1425ec78ecda500be712cfd292de58488 | At PEA, the chair of the Information Technology (IT) department has decided to set up 25 new servers in 13 academic buildings so that in each building there will be at least one new server. Outdoor cable connections are established between each pair of new servers that are in different buildings. Determine the least po... | [
"288"
] | [
"At PEA, the chair of the Information Technology (IT) department has decided to set up 25 new servers in 13 academic buildings so that in each building there will be at least one new server. Outdoor cable connections are established between each pair of new servers that are in different buildings. Determine the lea... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e59c94751e387cec474c980fca91d5a05df270eaf719cc54378a2530a287f85a | At Phillips Exeter Academy, there are $n$ student clubs with affiliations between some of them. Each day, the membership of the clubs changes according to a specific process. If a club $C$ has $m$ members and $k$ affiliated clubs, $\lfloor \frac{m}{k} \rfloor$ members will leave club $C$ and join each of the $k$ affili... | [
"2"
] | [
"At Phillips Exeter Academy, there are $n$ student clubs with affiliations between some of them. Each day, the membership of the clubs changes according to a specific process. If a club $C$ has $m$ members and $k$ affiliated clubs, $\\lfloor \\frac{m}{k} \\rfloor$ members will leave club $C$ and join each of the $k... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-965e380ef3544bb34daf9bdd735ef12979c3dbde896549d17251da1bf4b0489b | At Phillips Exeter Academy, there are $n$ student clubs with affiliations between some of them. Each day, the membership of the clubs changes according to the given process. If, for every club $c$ and day $x$, there exists a day $y > x$ such that the number of members of club $c$ is different on days $x$ and $y$, find ... | [
"2"
] | [
"At Phillips Exeter Academy, there are $n$ student clubs with affiliations between some of them. Each day, the membership of the clubs changes according to the given process. If, for every club $c$ and day $x$, there exists a day $y > x$ such that the number of members of club $c$ is different on days $x$ and $y$, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe72f76a6d9ea7f4c4df228d4bf8807ddca15cc9da52ae44c845385d97b1e62c | At Port Aventura, there are 16 secret agents, each watching one or more other agents. If agent $A$ is watching agent $B$, then $B$ is not watching $A$. It is known that any 10 agents can be ordered such that the first is watching the second, the second is watching the third, and so on, with the last watching the first.... | [
"11"
] | [
"At Port Aventura, there are 16 secret agents, each watching one or more other agents. If agent $A$ is watching agent $B$, then $B$ is not watching $A$. It is known that any 10 agents can be ordered such that the first is watching the second, the second is watching the third, and so on, with the last watching the f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-50d0f2c7d9280c32750ded35ba2539871fc7307e452d140cb445f234be1a4bb8 | At Rachelle's school, an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on four classes is computed as the total sum of points divided by 4. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She has a $\frac{1}{6}$ chance of ge... | [
"\\(\\frac{11}{24}\\)"
] | [
"At Rachelle's school, an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on four classes is computed as the total sum of points divided by 4. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She has a $\\frac{1}{6}$ chance... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-ae05e7bf487f359fe20e57032bf946de6feca0d4e0653299c6b39812c2788b93 | At Rosa's Rose Shop, a bouquet containing a dozen roses costs $\$20$. If the price of a bouquet is directly proportional to the number of roses it contains, how many dollars will a bouquet of 39 roses cost? | [
"65"
] | [
"At Rosa's Rose Shop, a bouquet containing a dozen roses costs $\\$20$. If the price of a bouquet is directly proportional to the number of roses it contains, how many dollars will a bouquet of 39 roses cost?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a78af1a0b69bde128b3af48e98021373e78b5a32dc883992101df367903b8f30 | At Rutherford's home, the hour hand of his clock moves forwards, but the minute hand runs backwards. At midnight, both hands are vertical. How many minutes later will the hands be at 30 degrees to each other for the second time that night? | [
"\\(\\frac{660}{13}\\)"
] | [
"At Rutherford's home, the hour hand of his clock moves forwards, but the minute hand runs backwards. At midnight, both hands are vertical. How many minutes later will the hands be at 30 degrees to each other for the second time that night?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-b6313a9be624eb4b6d8c06ff4547254f1b23124a56fea27da2d97c170a570711 | At Silk Road International School, 50% of the students went on the first trip, 80% went on the second trip, and 90% went on the third trip. A total of 160 students went on all three trips, and all other students went on exactly two trips. How many students are at Silk Road International School? | [
"800"
] | [
"At Silk Road International School, 50% of the students went on the first trip, 80% went on the second trip, and 90% went on the third trip. A total of 160 students went on all three trips, and all other students went on exactly two trips. How many students are at Silk Road International School?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-fb86a937079b46648351731c263a85c841543f838240885502aba6711d44b277 | At Smith College, the graduation exercises proceed as follows: Although each diploma is made out to a particular girl, all the diplomas are initially given out at random. All of the girls who do not get their own diplomas then form a circle, and each passes the diploma she has to the girl on her right. Those who now ha... | [
"\\frac{1}{k!}"
] | [
"At Smith College, the graduation exercises proceed as follows: Although each diploma is made out to a particular girl, all the diplomas are initially given out at random. All of the girls who do not get their own diplomas then form a circle, and each passes the diploma she has to the girl on her right. Those who n... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-3661425cf231f13059f84420a981c81db7601d2e0b46d9d32b8a52f911ad037c | At St. Ives High School, each student takes 7 classes, and each class contains 7 students. Every pair of students have exactly 3 classes in common. How many students go to school in St. Ives? | [
"15"
] | [
"At St. Ives High School, each student takes 7 classes, and each class contains 7 students. Every pair of students have exactly 3 classes in common. How many students go to school in St. Ives?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3fff361c6fbba58858472efb60dc9fcf2152c84c346f7d8f45465a5ed0e71e7b | At \( t = 0 \), Arthur the ant starts on the top left corner of a \( 2 \times 2 \) grid. In one second, he can move right 1 unit, down 1 unit, diagonally down right \(\sqrt{2}\) units, or stay put. At \( t = 6 \), Arthur is at the bottom right of the grid. How many ways can this happen? | [
"225"
] | [
"At \\( t = 0 \\), Arthur the ant starts on the top left corner of a \\( 2 \\times 2 \\) grid. In one second, he can move right 1 unit, down 1 unit, diagonally down right \\(\\sqrt{2}\\) units, or stay put. At \\( t = 6 \\), Arthur is at the bottom right of the grid. How many ways can this happen?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 6
} | 9 | 0 | 0 |
omr-ws-1505f05777c681ff78937450c0ecaecdc11cf49db5af6dd3ab9d2aed6f536243 | At a 10000-meter ski race, the first skier starts out and is followed shortly by a second one, the rate of the second skier being 1 m/sec more than that of the first one. When the second skier catches up with the first, the latter increases his rate by 2 m/sec, while the rate of the second skier remains unchanged. As a... | [
"2 minutes"
] | [
"At a 10000-meter ski race, the first skier starts out and is followed shortly by a second one, the rate of the second skier being 1 m/sec more than that of the first one. When the second skier catches up with the first, the latter increases his rate by 2 m/sec, while the rate of the second skier remains unchanged.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51279d93b903b003453c56f34b1a9892c9f83bebf8ee715695c8540645c859b2 | At a BDMO event with 36 participants, each participant recorded the number of handshakes they made. No two participants with the same number of handshakes shook hands with each other. Determine the maximum number of handshakes that could have occurred under these conditions. | [
"546"
] | [
"At a BDMO event with 36 participants, each participant recorded the number of handshakes they made. No two participants with the same number of handshakes shook hands with each other. Determine the maximum number of handshakes that could have occurred under these conditions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-23e1347d24c1ea5e2e1a420cbca608ed33480cc690a9a92a4a34bfef1de559ba | At a Round Table, 50 knights each have a glass of either red or white wine, with at least one glass of each type. After the first clap, every knight with a glass of red wine takes a glass from their left neighbor. After the second clap, every knight with a glass of white wine (and possibly something more) gives their g... | [
"2"
] | [
"At a Round Table, 50 knights each have a glass of either red or white wine, with at least one glass of each type. After the first clap, every knight with a glass of red wine takes a glass from their left neighbor. After the second clap, every knight with a glass of white wine (and possibly something more) gives th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2bc8906b985c5d6748071e955d4f362b64f82d18fd0118f268905f696c666698 | At a banquet with $2n$ guests, each person has at most $n-1$ enemies among the other guests. Determine the number of ways to arrange these $2n$ people around a table such that no one is sitting next to an enemy, assuming that rotations and reflections of the same arrangement are not considered distinct. Provide your an... | [
"(n-1)!^2"
] | [
"At a banquet with $2n$ guests, each person has at most $n-1$ enemies among the other guests. Determine the number of ways to arrange these $2n$ people around a table such that no one is sitting next to an enemy, assuming that rotations and reflections of the same arrangement are not considered distinct. Provide yo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-cdc3d3b4466648ac801923441f7c4edcc92a6d3e5b54d5b8a803a6a7f5fa9edb | At a banquet with an even number $n$ of guests seated at a round table, each guest has a card with their name on their plate. Determine the minimum number of distinct seating arrangements (rotations considered the same) such that at least two guests are seated according to the cards on their plates for each arrangement... | [
"\\dfrac{n}{2}"
] | [
"At a banquet with an even number $n$ of guests seated at a round table, each guest has a card with their name on their plate. Determine the minimum number of distinct seating arrangements (rotations considered the same) such that at least two guests are seated according to the cards on their plates for each arrang... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-19f23745fdc28f738fcb176684c6f3fc7fc6faa6cd4550dc07423d703c2246c4 | At a banquet with an odd number $n$ of guests seated at a round table, each guest has a card on their plate indicating their seat number. Determine the number of distinct seating arrangements (considering rotations as identical) such that no guest is seated according to the number on their card. Let $n = 9$. Find the r... | [
"833"
] | [
"At a banquet with an odd number $n$ of guests seated at a round table, each guest has a card on their plate indicating their seat number. Determine the number of distinct seating arrangements (considering rotations as identical) such that no guest is seated according to the number on their card. Let $n = 9$. Find ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-239622f378040ab9ce7910535b1f15ec9df4f62a9c4ab87e7beac81641fc18fa | At a big international congress, each participant shakes hands with others. Let $E(n)$ be the number of participants who have exchanged an odd number of handshakes after $n$ minutes. Find the parity of $E(n)$ for any positive integer $n$. | [
"even"
] | [
"At a big international congress, each participant shakes hands with others. Let $E(n)$ be the number of participants who have exchanged an odd number of handshakes after $n$ minutes. Find the parity of $E(n)$ for any positive integer $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68ccdeaaf2d9a63de90902fca2cbb59cebb83727ef422e488a22010e9ef5298f | At a birthday party, everyone who gets an invitation card can come alone or bring a friend. Everyone invited was present at the event. The ratio of the number of invitation cards to the number of guests attending is $3:5$. What is the ratio of the number of invitation cards to the number of guests who bring friends? | [
"$\\frac{3}{2}$"
] | [
"At a birthday party, everyone who gets an invitation card can come alone or bring a friend. Everyone invited was present at the event. The ratio of the number of invitation cards to the number of guests attending is $3:5$. What is the ratio of the number of invitation cards to the number of guests who bring friend... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-26c9140a2549eecbba3dadd660d9fb5cc5c7387fa6a128a7f4169a0ef39b90b2 | At a blind taste competition, a contestant is offered 3 cups of each of the 3 samples of tea in a random arrangement of 9 marked cups. If each contestant tastes 4 different cups of tea, what is the probability that a contestant does not taste all of the samples? | [
"\\(\\frac{5}{14}\\)"
] | [
"At a blind taste competition, a contestant is offered 3 cups of each of the 3 samples of tea in a random arrangement of 9 marked cups. If each contestant tastes 4 different cups of tea, what is the probability that a contestant does not taste all of the samples?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-98c793888e92a38b3a8b3b77856cb4aebd2773f61a47ee1b24c507e2e1799ca8 | At a business conference with 100 people from 20 companies (5 people per company), each person knows exactly 2 other people from other companies. Calculate the total number of distinct pairs of people who know each other, assuming this scenario were possible. | [
"100"
] | [
"At a business conference with 100 people from 20 companies (5 people per company), each person knows exactly 2 other people from other companies. Calculate the total number of distinct pairs of people who know each other, assuming this scenario were possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8662f4b94d5b447e5846680920a020ad1000d90867eabad63f8ef2cbd9534e6a | At a campground near Lake Michigan, the temperature outside a cabin is given by \(0.8t + 18\) degrees Celsius, and the temperature inside is given by \(1.5t + 15\) degrees Celsius, where \(t\) is the number of hours after 6 am. To the nearest minute, at what time will the temperatures inside and outside be the same? | [
"10:17 am"
] | [
"At a campground near Lake Michigan, the temperature outside a cabin is given by \\(0.8t + 18\\) degrees Celsius, and the temperature inside is given by \\(1.5t + 15\\) degrees Celsius, where \\(t\\) is the number of hours after 6 am. To the nearest minute, at what time will the temperatures inside and outside be t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-8561b6b94a62d07914ada0687abf17a004dc3270601f478b7bd2e6abcf847480 | At a ceremony, there are $n$ guests, one of whom might be a famous mathematician. Each guest can only answer whether they know another guest. Carl, the journalist, can ask up to $3n-3$ questions of the form "Does guest A know guest B?" to determine if there is a famous mathematician and identify them. Assuming Carl use... | [
"3n-3"
] | [
"At a ceremony, there are $n$ guests, one of whom might be a famous mathematician. Each guest can only answer whether they know another guest. Carl, the journalist, can ask up to $3n-3$ questions of the form \"Does guest A know guest B?\" to determine if there is a famous mathematician and identify them. Assuming C... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-2aec7c13bc5bdf30df7f376a54000b928b14b0a98ebd0c87c51ec9a93b3f0521 | At a certain college, there are 10 clubs and some number of students. For any two different students, there is some club such that exactly one of the two belongs to that club. For any three different students, there is some club such that either exactly one or all three belong to that club. What is the largest possible... | [
"512"
] | [
"At a certain college, there are 10 clubs and some number of students. For any two different students, there is some club such that exactly one of the two belongs to that club. For any three different students, there is some club such that either exactly one or all three belong to that club. What is the largest pos... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 25,
"genselect": 2
} | 27 | 0 | 0 |
omr-ws-f2ddb4b1b4d87a0b036f73fe9ecb94912644d5a578db8b5c6e04310d2f33c856 | At a certain intersection, cars arriving from the west go straight ahead 10% of the time, turn left 70% of the time, and turn right 20% of the time. Drivers use their signals properly 80% of the time. You are heading into the intersection from the west and are in a car behind a driver that does not have their signal on... | [
"0.5"
] | [
"At a certain intersection, cars arriving from the west go straight ahead 10% of the time, turn left 70% of the time, and turn right 20% of the time. Drivers use their signals properly 80% of the time. You are heading into the intersection from the west and are in a car behind a driver that does not have their sign... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 9
} | 14 | 0 | 0 |
omr-ws-d04a64563093762e90f72a31ad7bddcf5348b50a367b97135683a0a2af09b183 | At a certain moment a watch is 2 min behind although it is running fast. If it was 3 min behind but was 1/2 min faster in 24 hours than it is now, it would show the correct time a day earlier than it would do now. By how many minutes a day is the watch fast? | [
"By half a minute"
] | [
"At a certain moment a watch is 2 min behind although it is running fast. If it was 3 min behind but was 1/2 min faster in 24 hours than it is now, it would show the correct time a day earlier than it would do now. By how many minutes a day is the watch fast?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41671c491aa195e5a05a6171d82e736cdb25918cc2a5eab3ef27230818382f93 | At a certain moment a watch shows a 2-minutes lag although it is fast. If it showed a 3-minutes lag at that moment but gained half a minute more a day than it does it would show true time one day sooner than it actually does. How many minutes a day does the watch gain? | [
"0.5"
] | [
"At a certain moment a watch shows a 2-minutes lag although it is fast. If it showed a 3-minutes lag at that moment but gained half a minute more a day than it does it would show true time one day sooner than it actually does. How many minutes a day does the watch gain?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8925599a4de350018fdb6c27e9a626fc34e7626082199931906825ed2d67c7f | At a certain orphanage, every pair of orphans are either friends or enemies. For every three of an orphan's friends, an even number of pairs of them are enemies. Assign each orphan two parents such that every pair of friends shares exactly one parent, but no pair of enemies does, and no three parents are in a love tria... | [
"7"
] | [
"At a certain orphanage, every pair of orphans are either friends or enemies. For every three of an orphan's friends, an even number of pairs of them are enemies. Assign each orphan two parents such that every pair of friends shares exactly one parent, but no pair of enemies does, and no three parents are in a love... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-7c5def4823688cdc6c58f01bc3df90bd5df84d553183e0c2654c3a4f4c515ae6 | At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Let $N$ be the number of people at the party such that the departure of any two of them leaves no 3-clique remaining. Find the maximum possible value of $N$. | [
"4"
] | [
"At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Let $N$ be the number of people at the party such that the departure of any two of them leaves no 3-clique remaining. Find the maximum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b16213fcb9317deb93f4f930871c0a5c770338bdb09899fbaa05864cb87851f8 | At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from ea... | [
"088",
"88"
] | [
"At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors fr... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 5
} | 37 | 0 | 0 |
omr-ws-0c3e5a952b8397df7cf1eef54cb9655e036aa5c7f64a6a57bb9ae7ceb6163540 | At a chess tournament, each player played two matches with every other player (one with white pieces and one with black pieces). Points were awarded as follows: 1 point for a victory and 0.5 points for a tie. At the end of the tournament, all players had the same number of points. If there are 10 players in the tournam... | [
"45"
] | [
"At a chess tournament, each player played two matches with every other player (one with white pieces and one with black pieces). Points were awarded as follows: 1 point for a victory and 0.5 points for a tie. At the end of the tournament, all players had the same number of points. If there are 10 players in the to... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e81e5b6f29d85a68ea5e8f3e687c8ad6fdad293368826123394772f44a40ca23 | At a chess tournament, every pair of contestants played each other at most once. If any two contestants, $A$ and $B$, failed to play each other, then exactly two other contestants, $C$ and $D$, played against both $A$ and $B$ during the tournament. Moreover, no $4$ contestants played exactly $5$ games between them. Let... | [
"20"
] | [
"At a chess tournament, every pair of contestants played each other at most once. If any two contestants, $A$ and $B$, failed to play each other, then exactly two other contestants, $C$ and $D$, played against both $A$ and $B$ during the tournament. Moreover, no $4$ contestants played exactly $5$ games between them... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-1266df214a79db5bb125ef58c3d9f495d7be5373c27bed58f287f9d871008981 | At a circular table for 100 people, how many ways can you select 4 people so that no two selected people are sitting next to each other? | [
"3866975"
] | [
"At a circular table for 100 people, how many ways can you select 4 people so that no two selected people are sitting next to each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-938f88561a7e9aee4cdd61fd954cbf19d04a514f4d38c74bdeee66c7be3b9321 | At a circular table with \( n \) students, how many ways are there to assign 4 different versions of an exam such that no two neighboring students receive the same version? | [
"3^n + 3(-1)^n"
] | [
"At a circular table with \\( n \\) students, how many ways are there to assign 4 different versions of an exam such that no two neighboring students receive the same version?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f64a8af8e235242f9bb6f3e813227eb7a1535695c1f1689678b4f95be16f7c0e | At a circular table, there are 8 women and 25 men seated such that there are at least 2 men between every pair of consecutive women. In how many ways can people be seated at the table if two configurations are considered equal if one can be obtained by rotating the other? | [
"\\dbinom{16}{7} \\times 7! \\times 25!"
] | [
"At a circular table, there are 8 women and 25 men seated such that there are at least 2 men between every pair of consecutive women. In how many ways can people be seated at the table if two configurations are considered equal if one can be obtained by rotating the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-10fee10477ca0062bae01f024465fe46b13db33e73159c67b7f292d9b423ab72 | At a circular track, $2n$ cyclists start from the same point at the same time and travel in the same direction with different constant speeds. If any two cyclists meet at the same point again, they are considered to have met. No three or more cyclists meet at the same time. Let $M$ be the total number of meetings that ... | [
"45"
] | [
"At a circular track, $2n$ cyclists start from the same point at the same time and travel in the same direction with different constant speeds. If any two cyclists meet at the same point again, they are considered to have met. No three or more cyclists meet at the same time. Let $M$ be the total number of meetings ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-6f9915a7d35b85b62f858b22860c4b648c03b975d907dbae6a896d88247046d5 | At a circular track, 10 cyclists start from the same point at the same time and travel in the same direction with different constant speeds. If any two cyclists meet at the same point again, they are considered to have met. No three or more cyclists meet at the same time. Calculate the minimum total number of meetings ... | [
"45"
] | [
"At a circular track, 10 cyclists start from the same point at the same time and travel in the same direction with different constant speeds. If any two cyclists meet at the same point again, they are considered to have met. No three or more cyclists meet at the same time. Calculate the minimum total number of meet... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-165654beaedda3e4170c3467964b05741cdf7187adca97ed695fe94499f9d207 | At a circular track, 10 cyclists start from the same point at the same time and travel in the same direction with different constant speeds. If any two cyclists meet at the same point again, they are said to have met. No three or more cyclists meet at the same time. Calculate the minimum total number of meetings that o... | [
"45"
] | [
"At a circular track, 10 cyclists start from the same point at the same time and travel in the same direction with different constant speeds. If any two cyclists meet at the same point again, they are said to have met. No three or more cyclists meet at the same time. Calculate the minimum total number of meetings t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-fc02d0e2859170bf75f8abb6f95b071644996698bb675197b7f60ec0e72ef12c | At a classroom costume party, the average age of the \( b \) boys is \( g \), and the average age of the \( g \) girls is \( b \). If the average age of everyone at the party (all these boys and girls, plus their 42-year-old teacher) is \( b + g \), what is the value of \( b + g \)? | [
"8"
] | [
"At a classroom costume party, the average age of the \\( b \\) boys is \\( g \\), and the average age of the \\( g \\) girls is \\( b \\). If the average age of everyone at the party (all these boys and girls, plus their 42-year-old teacher) is \\( b + g \\), what is the value of \\( b + g \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-55de19969ad0ec2c7acf0a91e16d885126d502f24a9a68b0d17a1797f62124f5 | At a company, workers are divided into 100 buses such that no two enemies are in the same bus. The chief wants to assign these workers to brigades of at least two people, ensuring no two enemies are in the same brigade. Let $N$ be the minimum number of brigades needed to satisfy these conditions. If it is impossible to... | [
"100"
] | [
"At a company, workers are divided into 100 buses such that no two enemies are in the same bus. The chief wants to assign these workers to brigades of at least two people, ensuring no two enemies are in the same brigade. Let $N$ be the minimum number of brigades needed to satisfy these conditions. If it is impossib... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-df3122fad18257f076f9b723eecdb0ab88ebb6a7a339f2061098a1148bdfdd9d | At a competition with 6 problems, there are $n$ students. Each problem is solved by exactly 100 students. For any two students, there is at least one problem that neither of them solved. Find the least value of $n$. | [
"150"
] | [
"At a competition with 6 problems, there are $n$ students. Each problem is solved by exactly 100 students. For any two students, there is at least one problem that neither of them solved. Find the least value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b652874c29fb25af0c706dfc44722862164414feaa422f7566fcb864e6a872b | At a competition with \( N \) players, the number of players given elite status is equal to \( 2^{1+\lfloor \log_{2} (N-1) \rfloor} - N \). Suppose that 19 players are given elite status. What is the sum of the two smallest possible values of \( N \)? | [
"154"
] | [
"At a competition with \\( N \\) players, the number of players given elite status is equal to \\( 2^{1+\\lfloor \\log_{2} (N-1) \\rfloor} - N \\). Suppose that 19 players are given elite status. What is the sum of the two smallest possible values of \\( N \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-668485b6cdce894fbadb26faed2b079f8ec4fee5dd8b654cbc4858ffdb8da80e | At a conference attended by 512 mathematicians sharing 256 twin rooms, each mathematician has a distinct set of interests in nine subjects. The mathematicians are seated in one big circle such that everyone is seated next to their roommate, and if two non-roommates are seated next to each other, their sets of interests... | [
"(255! \\times 2^{256}) \\mod (10^9 + 7)",
"185794560"
] | [
"At a conference attended by 512 mathematicians sharing 256 twin rooms, each mathematician has a distinct set of interests in nine subjects. The mathematicians are seated in one big circle such that everyone is seated next to their roommate, and if two non-roommates are seated next to each other, their sets of inte... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-243699d1d94bde98963708fb622095a50a3aa437d0fa5846f16a8aa5234bb956 | At a conference with $n$ mathematicians, each mathematician knows exactly $k$ other mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians who are all acquainted with each other. | [
"\\( \\left\\lfloor \\frac{n}{2} \\right\\rfloor + 1 \\)"
] | [
"At a conference with $n$ mathematicians, each mathematician knows exactly $k$ other mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians who are all acquainted with each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-a825adcbe7140237cff3473c53cac42f363998bf873fa43ba917c56fe02cd5c4 | At a conference with $n=6$ people, each group of three people wishes to meet exactly once. On each of the $d$ days of the conference, every person participates in at most one meeting with two other people. Determine the minimum number of days, $\min d$, required for all groups of three to meet exactly once. | [
"10"
] | [
"At a conference with $n=6$ people, each group of three people wishes to meet exactly once. On each of the $d$ days of the conference, every person participates in at most one meeting with two other people. Determine the minimum number of days, $\\min d$, required for all groups of three to meet exactly once."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-d49829fe2f28e03e13e35feb4812440c739dae7149f9dd1143a398d1443fa455 | At a conference with $n=8$ people, each group of three people wishes to meet exactly once. On each of the $d$ days of the conference, every person participates in at most one meeting with two other people. Determine the minimum number of days, $\min d$, required for all groups of three to meet exactly once. | [
"28"
] | [
"At a conference with $n=8$ people, each group of three people wishes to meet exactly once. On each of the $d$ days of the conference, every person participates in at most one meeting with two other people. Determine the minimum number of days, $\\min d$, required for all groups of three to meet exactly once."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70a6dcd403b48c9080ea93d184d5f6bf0dae5e385ffe92760c0d4409c407d231 | At a conference with 100 representatives (15 French and 85 Germans), each French representative is acquainted with at least 70 representatives, and each German representative is acquainted with no more than 10 representatives. The representatives are divided into 21 rooms. Calculate the minimum number of rooms in which... | [
"6"
] | [
"At a conference with 100 representatives (15 French and 85 Germans), each French representative is acquainted with at least 70 representatives, and each German representative is acquainted with no more than 10 representatives. The representatives are divided into 21 rooms. Calculate the minimum number of rooms in ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-552cd8b6715f50b2a46b9518438dae4e2f13178198349bac7ad89254371615d3 | At a conference with 15 persons sitting at a round table, after a break, they sit down again in different positions. Let $k$ be the number of persons between two specific individuals, Alice and Bob, before and after the break. Calculate the sum of all possible values of $k$. | [
"21"
] | [
"At a conference with 15 persons sitting at a round table, after a break, they sit down again in different positions. Let $k$ be the number of persons between two specific individuals, Alice and Bob, before and after the break. Calculate the sum of all possible values of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-ffd9d3a4e411445adb51e3a6071baec117bfa982607616c4c320e8526320b703 | At a conference with 1985 people, in every group of three people, at least two speak the same language. Find the maximum number of people who can be selected such that all of them speak the same language. | [
"993"
] | [
"At a conference with 1985 people, in every group of three people, at least two speak the same language. Find the maximum number of people who can be selected such that all of them speak the same language."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-063ed00c942ac28a883d899121b469fd8369ee728d6f5ede774abd37374cf528 | At a conference with 2022 persons sitting at a round table, after a break, they sit down again in different positions. Let $k$ be the number of pairs of persons such that the number of persons between them remains the same. Find the minimum possible value of $k$. | [
"1011"
] | [
"At a conference with 2022 persons sitting at a round table, after a break, they sit down again in different positions. Let $k$ be the number of pairs of persons such that the number of persons between them remains the same. Find the minimum possible value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-84172955548d622a947d99069c72ccb7fe1acc4e7902396a373f0dbb6a3fed96 | At a conference with 40 persons, every 19 persons share exactly one common idol. Let \( T \) be a set of 20 persons chosen from the conference. Define \( f(T) \) as the number of persons in \( T \) who are not the common idol of the remaining 19 persons in \( T \). Find the maximum possible value of \( f(T) \). | [
"20"
] | [
"At a conference with 40 persons, every 19 persons share exactly one common idol. Let \\( T \\) be a set of 20 persons chosen from the conference. Define \\( f(T) \\) as the number of persons in \\( T \\) who are not the common idol of the remaining 19 persons in \\( T \\). Find the maximum possible value of \\( f(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0df677fd1b4af1db927caf97741f23651c908a840e88538faa15bde2cafdf9d6 | At a conference, the 2016 participants are registered from P1 to P2016. Each participant from P1 to P2015 shook hands with exactly the same number of participants as the number on their registration tag. How many hands did the 2016th participant shake? | [
"1008"
] | [
"At a conference, the 2016 participants are registered from P1 to P2016. Each participant from P1 to P2015 shook hands with exactly the same number of participants as the number on their registration tag. How many hands did the 2016th participant shake?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 12
} | 34 | 0 | 0 |
omr-ws-35364d1275dbff2cb4235d1663e50ffa15721ab6e39419993bf7a465c68f8866 | At a conference, the number of participants is a multiple of $15$. The participants collectively know $5$ languages. For any pair of these languages, $2/5$ of the participants know both languages, and for any triplet of these languages, $1/5$ of the participants know all three languages. If $N$ is the total number of p... | [
"15"
] | [
"At a conference, the number of participants is a multiple of $15$. The participants collectively know $5$ languages. For any pair of these languages, $2/5$ of the participants know both languages, and for any triplet of these languages, $1/5$ of the participants know all three languages. If $N$ is the total number... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-8d56e6dd9f64e9d52164e3fb77ce3beaacd5a2e2711572e11ca3dbfbb0990f1a | At a conference, there are 9 delegates, each speaking at most three languages. Given any three delegates, at least two speak a common language. Find the minimum number of distinct languages spoken by the delegates such that there are three delegates who speak a common language. | [
"8"
] | [
"At a conference, there are 9 delegates, each speaking at most three languages. Given any three delegates, at least two speak a common language. Find the minimum number of distinct languages spoken by the delegates such that there are three delegates who speak a common language."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-24cd0343c2d3957eafd8870fb836a3a7e7c5f02342bbf3efcb7011e5904a523a | At a contest, $n$ problems were proposed, each graded from $0$ to $k-1$. It is known that no two participants have the same score for more than $m \leq n$ problems. Determine all possible numbers of participants in the contest. | [
"k^{m+1}"
] | [
"At a contest, $n$ problems were proposed, each graded from $0$ to $k-1$. It is known that no two participants have the same score for more than $m \\leq n$ problems. Determine all possible numbers of participants in the contest."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-98dfdc52efc94f04d3e38416651346191b3725e9eff0e822fb8ff838014f3a38 | At a dance party, a group of boys and girls exchange dances as follows: one boy dances with $5$ girls, a second boy dances with $6$ girls, and so on, with the last boy dancing with all the girls. If $b$ represents the number of boys and $g$ the number of girls, determine the relationship between $b$ and $g$. | [
"\\( b = g - 4 \\)"
] | [
"At a dance party, a group of boys and girls exchange dances as follows: one boy dances with $5$ girls, a second boy dances with $6$ girls, and so on, with the last boy dancing with all the girls. If $b$ represents the number of boys and $g$ the number of girls, determine the relationship between $b$ and $g$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-6dabd80951cf4922777864f81f9cd5ec1c674c30af4ae21f519351d1d48e0c05 | At a dance party, the first boy dances with 5 girls, the second boy with 6 girls, and so on. Let \( b \) represent the number of boys and \( g \) represent the number of girls. Find \( b \) in terms of \( g \). | [
"\\( b = \\sqrt{2g + \\frac{81}{4}} - \\frac{9}{2} \\)"
] | [
"At a dance party, the first boy dances with 5 girls, the second boy with 6 girls, and so on. Let \\( b \\) represent the number of boys and \\( g \\) represent the number of girls. Find \\( b \\) in terms of \\( g \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-f22aa80c2f2a2b2d7725619c8f86583aecf3a4d77613fc23753af97a017f5e50 | At a dance, Abhinav starts from point $(a, 0)$ and moves along the negative $x$-direction with speed $v_a$, while Pei-Hsin starts from $(0, b)$ and glides in the negative $y$-direction with speed $v_b$. What is the distance of closest approach between the two? | [
"\\(\\sqrt{a^2 + b^2 - \\frac{(av_a+bv_b)^2}{v_a^2+v_b^2}}\\)"
] | [
"At a dance, Abhinav starts from point $(a, 0)$ and moves along the negative $x$-direction with speed $v_a$, while Pei-Hsin starts from $(0, b)$ and glides in the negative $y$-direction with speed $v_b$. What is the distance of closest approach between the two?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2592a8cbb7a1eb21337d08104539f5d490bccfaa6bee1de67bed604e62139221 | At a dance, a group of girls exchange dances as follows: the first boy dances with 5 girls, the second boy dances with 6 girls, and so on, with the last boy dancing with all the girls. If \( b \) represents the number of boys and \( g \) represents the number of girls, find \( b \) in terms of \( g \). | [
"\\( b = g - 4 \\)"
] | [
"At a dance, a group of girls exchange dances as follows: the first boy dances with 5 girls, the second boy dances with 6 girls, and so on, with the last boy dancing with all the girls. If \\( b \\) represents the number of boys and \\( g \\) represents the number of girls, find \\( b \\) in terms of \\( g \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7bafadf23b437fb7390ab68ef5b8a2fab31b9c9a44b30a037c56445f833fba2c | At a dance-hop, there are 100 men and 20 women. Each man selects a group of \(a_i\) women as potential dance partners (his dance list). It is required that for any group of 20 men, it is always possible to pair each of these 20 men with 20 women such that each man is paired with a woman on his dance list. What is the s... | [
"1620"
] | [
"At a dance-hop, there are 100 men and 20 women. Each man selects a group of \\(a_i\\) women as potential dance partners (his dance list). It is required that for any group of 20 men, it is always possible to pair each of these 20 men with 20 women such that each man is paired with a woman on his dance list. What i... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 2
} | 34 | 0 | 0 |
omr-ws-dc5cb6a1a82d716e5041cad0b7fb8edce75745195cff0302a497c72d391e9749 | At a dancing club, there are \( n \) boys and \( n \) girls, where \( n > k > 1 \). Each boy knows exactly \( k \) girls, and each girl knows exactly \( k \) boys. Let \( P(n, k) \) be the number of ways to form \( n \) pairs, each consisting of a boy and a girl who know each other. Find the value of \( P(10, 3) \). | [
"12096"
] | [
"At a dancing club, there are \\( n \\) boys and \\( n \\) girls, where \\( n > k > 1 \\). Each boy knows exactly \\( k \\) girls, and each girl knows exactly \\( k \\) boys. Let \\( P(n, k) \\) be the number of ways to form \\( n \\) pairs, each consisting of a boy and a girl who know each other. Find the value of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
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