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-1aa815759e69fc971033c7cce2e791fc06b4aea101b4bac1b28e6e2816740ad1 | Two videos last for \( x \) seconds and \( y \) seconds respectively, where \( x \) and \( y \) are whole numbers. Both videos start at the same time and restart immediately upon completion. How long does it take for the two videos to start at the same time for the second time? | [
"\\(\\text{lcm}(x, y)\\)"
] | [
"Two videos last for \\( x \\) seconds and \\( y \\) seconds respectively, where \\( x \\) and \\( y \\) are whole numbers. Both videos start at the same time and restart immediately upon completion. How long does it take for the two videos to start at the same time for the second time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-63205362463f908c4cf44a63c24219b520f95cb59c9ab6dba143fd3f18095720 | Two villages A and B lie on opposite sides of a straight river. The perpendicular distance from A to the river is 1 km, and that from B to the river is 3 km. The river is 1 km wide, and B is 3 km down the river from village A. If we build a bridge in an optimal position across and perpendicular to the river, what is th... | [
"6 km"
] | [
"Two villages A and B lie on opposite sides of a straight river. The perpendicular distance from A to the river is 1 km, and that from B to the river is 3 km. The river is 1 km wide, and B is 3 km down the river from village A. If we build a bridge in an optimal position across and perpendicular to the river, what ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-aa4602a47acf834a81f547eddc330c084528b855ebb77bfd9b1583020b00fb99 | Two volleyball teams play a series of 100 games, where one team wins the first game, and the other wins the second game. If the probability that each volleyball team wins is the ratio of games won by that team to the total games played, what is the probability that a team wins exactly 30 games? | [
"\\dfrac{2}{99}"
] | [
"Two volleyball teams play a series of 100 games, where one team wins the first game, and the other wins the second game. If the probability that each volleyball team wins is the ratio of games won by that team to the total games played, what is the probability that a team wins exactly 30 games?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-8751a750df69fae5d6a10302589837975c18470b7c4cb351d7b97ba819c73dd1 | Two walls are 10 ft. apart. Two ladders, one 15 ft. long and one 20 ft. long, are placed at the bottoms of the walls leaning against the opposite walls. How far from the ground is the point of intersection? | [
"6.8 feet"
] | [
"Two walls are 10 ft. apart. Two ladders, one 15 ft. long and one 20 ft. long, are placed at the bottoms of the walls leaning against the opposite walls. How far from the ground is the point of intersection?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-be060051afbb526e896ab9773abc16e9271980a3450563f77b61a3578047a850 | Two water taps together can fill a tank in $9 \frac{3}{8}$ hours. The tap with the larger diameter takes 10 minutes less than the tap with the smaller diameter to fill the tank separately. Find the time in which each tap can separately fill the tank. | [
"18.8333 hours and 18.6667 hours"
] | [
"Two water taps together can fill a tank in $9 \\frac{3}{8}$ hours. The tap with the larger diameter takes 10 minutes less than the tap with the smaller diameter to fill the tank separately. Find the time in which each tap can separately fill the tank."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 25
} | 27 | 0 | 0 |
omr-ws-7f79713d4a9a261598053253041d5436954a2342bd1b3f9cef912a0771235692 | Two weeks ago, Rick had been learning guitar for five times as long as Nic. Two days ago, Rick had been learning for twice as long as Nic. Today, what is the number of days that Rick has been learning plus the number of days that Nic has been learning? | [
"52"
] | [
"Two weeks ago, Rick had been learning guitar for five times as long as Nic. Two days ago, Rick had been learning for twice as long as Nic. Today, what is the number of days that Rick has been learning plus the number of days that Nic has been learning?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-4e14c95456bb1ddd36a5eafad640b378abb07e8fc7359e81a686e9349f9f0fe0 | Two wheels are set in rotation by an endless belt; the smaller wheel makes 400 revolutions per minute more than does the larger wheel. The larger wheel makes 5 revolutions in a time interval that is 1 second longer than that required for the smaller wheel to make 5 revolutions. How many revolutions per minute does each... | [
"The larger wheel makes 200 r.p.m., the smaller one, 600 r.p.m"
] | [
"Two wheels are set in rotation by an endless belt; the smaller wheel makes 400 revolutions per minute more than does the larger wheel. The larger wheel makes 5 revolutions in a time interval that is 1 second longer than that required for the smaller wheel to make 5 revolutions. How many revolutions per minute does... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edcd4e8cb41ad3de4c765fe70bf884945566470c71ba06894e3748e10251fbef | Two wheels have circumferences in the ratio 5:3. Two ropes are wrapped around these wheels. The difference in the lengths of the ropes exceeds the difference in the circumferences of the wheels by 280 yards. The rope on the larger wheel wraps around it 12 more times than the rope on the smaller wheel. If the larger whe... | [
"72"
] | [
"Two wheels have circumferences in the ratio 5:3. Two ropes are wrapped around these wheels. The difference in the lengths of the ropes exceeds the difference in the circumferences of the wheels by 280 yards. The rope on the larger wheel wraps around it 12 more times than the rope on the smaller wheel. If the large... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-95e6d78c9b47ce1910ebf3d2ee999217c439f8e4b76650cfea95c59e2ad2875c | Two wheels of a machine have radii \(5R\) and \(R\). If the distance between their centers is \(8R\), find the length of the chain connecting them. | [
"\\( 8R\\sqrt{3} + \\frac{22\\pi R}{3} \\)"
] | [
"Two wheels of a machine have radii \\(5R\\) and \\(R\\). If the distance between their centers is \\(8R\\), find the length of the chain connecting them."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-21f4915754003b1f67218769582eb56a9d1a645c24b476d3785e733fae8861d8 | Two white and three black cats are sitting on a line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7, and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, and to the other white cat is 15. Determine which cats are sitting on the ed... | [
"\\text{Two black cats}"
] | [
"Two white and three black cats are sitting on a line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7, and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, and to the other white cat is 15. Determine which cats are sitting on t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a2eb2e020c5fe35a0596adb793a5bf1a216bb7cb32ce3fa43704eddc3c59c651 | Two white and two black cats are sitting on a line. The sum of distances from the white cats to one black cat is 4, and to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, and to the other white cat is 9. Determine which cats are sitting on the edges. | [
"The cats sitting on the edges are the first white cat ($W_1$) and the second black cat ($B_2$)."
] | [
"Two white and two black cats are sitting on a line. The sum of distances from the white cats to one black cat is 4, and to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, and to the other white cat is 9. Determine which cats are sitting on the edges."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-d832a4505a2d568149f3d8364147e002c5165a98640882e88cc21efa62eb0e8b | Two white pentagonal pyramids, with all side lengths the same, are glued together at their regular pentagon bases. Some of the resulting 10 faces are colored black. How many rotationally distinguishable colorings may result? | [
"120"
] | [
"Two white pentagonal pyramids, with all side lengths the same, are glued together at their regular pentagon bases. Some of the resulting 10 faces are colored black. How many rotationally distinguishable colorings may result?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-abe68a5e86266bdda3cc73b4e9e593297fe4c18ab890746bdc32d27efaea9507 | Two wine tasters rate each wine they taste on a scale of 1-5. The probabilities for both tasters' ratings of a randomly chosen wine are given in the following table: | Taster 2 | 1 | 2 | 3 | 4 | 5 | |----------|---|---|---|---|---| | 1 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | | 2 | 0.02 | 0.08 | 0.05 | 0.02 | 0.01 | | 3 | ... | [
"0.39"
] | [
"Two wine tasters rate each wine they taste on a scale of 1-5. The probabilities for both tasters' ratings of a randomly chosen wine are given in the following table:\n\n| Taster 2 | 1 | 2 | 3 | 4 | 5 |\n|----------|---|---|---|---|---|\n| 1 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 |\n| 2 | 0.02 | 0.08 | 0.... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2964a1c90e567a3d0f17591b8f77de1e07bd65a2dd8909518e17f3064abe300 | Two workers arc given a task to make a batch of identical parts. After the first had worked for 7 and the second, for 4 hours, they found out that $5/9$, of the task had been completed. Having worked together for another 4 hours, they figured out that $1/18$ of the job had yet to be done. How long would it take each wo... | [
"It would take the first worker 18 hours and the second, 24 hours to do the whole assignment"
] | [
"Two workers arc given a task to make a batch of identical parts. After the first had worked for 7 and the second, for 4 hours, they found out that $5/9$, of the task had been completed. Having worked together for another 4 hours, they figured out that $1/18$ of the job had yet to be done. How long would it take ea... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8456c37f642461718740bcdc59718c871c7cfa57725b04b7893c375803796653 | Two workers did a job in ten days, the last two of which the first worker did not work. How many days would it take the first worker to complete the job alone if it is known that for the first seven days they accomplished 80\% of the job working together? | [
"In 14 days"
] | [
"Two workers did a job in ten days, the last two of which the first worker did not work. How many days would it take the first worker to complete the job alone if it is known that for the first seven days they accomplished 80\\% of the job working together?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef9180fa00ee5649f2e8e0d7af192bbb25c2c6e7915deaa4248741b3ad91995b | Two workers fulfil some job. After 45 minutes' joint operation the first worker was given another job, and the second worker completed the remaining part of the job 2 hours and 15 minutes. How much time would it have taken each worker to complete the whole job alone if the first worker had been able to do this an hour ... | [
"3 hours and 4 hours"
] | [
"Two workers fulfil some job. After 45 minutes' joint operation the first worker was given another job, and the second worker completed the remaining part of the job 2 hours and 15 minutes. How much time would it have taken each worker to complete the whole job alone if the first worker had been able to do this an ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-49e44d294635452d027e425961ed3e66f65a4a42f80955244e0a0ab529396f29 | Two workers received the same assignment: to make a certain amount of machine parts during a certain time period. The first of them did the job in time, and the second worker accomplished only 90\% of the task in that time, having left unfinished as many parts as the first worker could make in 40 min. If the second wor... | [
"400 parts"
] | [
"Two workers received the same assignment: to make a certain amount of machine parts during a certain time period. The first of them did the job in time, and the second worker accomplished only 90\\% of the task in that time, having left unfinished as many parts as the first worker could make in 40 min. If the seco... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1119ce4b9067f6b906adb44b3f1dce53c1be9ec1e7ebf9ac328515068d3c0a3 | Two workers together can finish a job in 3 days. If the first worker completes 1/4 of the job and is then replaced by the second worker, the job is completed in 6 days. How much of the job can the second worker complete in 3 days, given that the second worker is faster than the first worker? | [
"75% of the work"
] | [
"Two workers together can finish a job in 3 days. If the first worker completes 1/4 of the job and is then replaced by the second worker, the job is completed in 6 days. How much of the job can the second worker complete in 3 days, given that the second worker is faster than the first worker?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-3a8bc8852be42137c5ebefbc4298c7e646fdb43dd883d045767f8f44b0659b51 | Two workers together complete a certain task in 8 hours. Working individually, the first worker can do the job 12 hours faster than can do the second. How many hours would it take each worker to do the job individually? | [
"The first worker can do the job individually in 12 hours, the second, in 24 hours"
] | [
"Two workers together complete a certain task in 8 hours. Working individually, the first worker can do the job 12 hours faster than can do the second. How many hours would it take each worker to do the job individually?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4e7659ac34eb8e6a51e6cb9df6cf0142f32131b60b3e11febde83b9e63a36f5 | Two workers together did a job in 12 hours. If at the beginning the first worker had done half the assignment, and then the second had completed the other half, the whole job would have been done in 25 hours. How long would it take each worker to do the whole job individually? | [
"One of the workers (either the first, or the second) can do the whole job in 20 hours, the other, in 30 hours"
] | [
"Two workers together did a job in 12 hours. If at the beginning the first worker had done half the assignment, and then the second had completed the other half, the whole job would have been done in 25 hours. How long would it take each worker to do the whole job individually?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98bc617e05c23017fb4ff2d3ee9129c2e2d50dc5e3a554508667f6c1bf8d2c2b | Two workers, one of whom began working a day and a half later than the other, worked independently and papered several rooms in 7 days, reckoning from the moment the first of them began working. If the job was entrusted to each worker separately, then it would take the first of them 3 days more to do it than the second... | [
"In 14 and 11 days"
] | [
"Two workers, one of whom began working a day and a half later than the other, worked independently and papered several rooms in 7 days, reckoning from the moment the first of them began working. If the job was entrusted to each worker separately, then it would take the first of them 3 days more to do it than the s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-065e9efa06e53815bcc22b27e2f06566c1f19f5a1e687a23bf17048d26de2d77 | Two workers, operating together, carried out a job in 12 hours. Had the first worker done half the job, and then the second worker the remaining half, the whole job would have been carried out in 25 hours. How much time would it have taken each worker to do the job separately? | [
"20 hours and 30 hours"
] | [
"Two workers, operating together, carried out a job in 12 hours. Had the first worker done half the job, and then the second worker the remaining half, the whole job would have been carried out in 25 hours. How much time would it have taken each worker to do the job separately?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-750ed4fbeb0e82e6c9892e927cf50e57ef7fefe7e57ec4a19a4662f5ef20619d | Two workers, the second one beginning working $1\frac{1}{2}$ days after the first, can complete an assignment in 7 days. If each of them had done the job individually, the first worker would have required 3 days more than would have the second. How many days would it take each worker to do the job individually? | [
"It would take the first worker 14 days and the second one, 11 days to do the job individually"
] | [
"Two workers, the second one beginning working $1\\frac{1}{2}$ days after the first, can complete an assignment in 7 days. If each of them had done the job individually, the first worker would have required 3 days more than would have the second. How many days would it take each worker to do the job individually?"
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a40682f4eb8e2f875df8e0696ba5b001cbf17111366ea793a09fc5f87152d2d6 | Two years ago there were 20 trailer homes on Elm Street with an average age of 18 years. At that time, a group of brand new trailer homes was then added to Elm Street. Today, the average age of all the trailer homes on Elm Street is 14 years. How many new trailer homes were added two years ago? | [
"10"
] | [
"Two years ago there were 20 trailer homes on Elm Street with an average age of 18 years. At that time, a group of brand new trailer homes was then added to Elm Street. Today, the average age of all the trailer homes on Elm Street is 14 years. How many new trailer homes were added two years ago?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d584b111e39b4baf84ef1283bf76c9e9979af3f3d1547b73c93d9140ba225b4f | Two years ago, Aman's age was three times the square of his son's age. In three years, Aman's age will be four times his son's age. Find their present ages. | [
"Aman's age = 29, Son's age = 5"
] | [
"Two years ago, Aman's age was three times the square of his son's age. In three years, Aman's age will be four times his son's age. Find their present ages."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 31
} | 38 | 0 | 0 |
omr-ws-a8549d135530161433b356eabd9dfdc8c1c6c3c54ab55483c44d0ce7d80ee1a1 | Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room? | [
"27"
] | [
"Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5485b8c9a59f5d11670ebb31761b165bce0efc5e1da39f891a21a2bcb1b8dcb | Two-way flights are operated between 80 cities such that each city is connected to at least 7 other cities by a direct flight and any two cities are connected by a finite sequence of flights. Find the smallest $k$ such that for any such arrangement of flights, it is possible to travel from any city to any other city by... | [
"12"
] | [
"Two-way flights are operated between 80 cities such that each city is connected to at least 7 other cities by a direct flight and any two cities are connected by a finite sequence of flights. Find the smallest $k$ such that for any such arrangement of flights, it is possible to travel from any city to any other ci... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-2f6e21b41fd6dcde9c52b733789a7e49eccdc096e0fb8321d84e40f109ce13f2 | Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use? | [
"87"
] | [
"Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-79010c3938787d828db10857d618f1c95b315f1662319d4c1b62544186901236 | Tyler plays with the trapezoidal wooden block \( ABCD \), depicted below, with right angles at \( A \) and \( D \) and side lengths \( AB = 7 \), \( BC = 6 \), and \( CD = 3 \). If Tyler places side \( \overline{BC} \) flat on the floor, how high up would vertex \( A \) be? | [
"\\(\\frac{7}{3}\\sqrt{5}\\)"
] | [
"Tyler plays with the trapezoidal wooden block \\( ABCD \\), depicted below, with right angles at \\( A \\) and \\( D \\) and side lengths \\( AB = 7 \\), \\( BC = 6 \\), and \\( CD = 3 \\). If Tyler places side \\( \\overline{BC} \\) flat on the floor, how high up would vertex \\( A \\) be?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-2b80724feba798437ed6f83df67af415bcbbe792d9903d259795551f2bb0d5e9 | UVWXYZ is a regular hexagon with center O. UOWABX is also a regular hexagon with AB=12. A) Find the area of UOWABC. B) Find the area of UVWO. | [
"72\\sqrt{3}"
] | [
"UVWXYZ is a regular hexagon with center O. UOWABX is also a regular hexagon with AB=12.\nA) Find the area of UOWABC.\nB) Find the area of UVWO."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-43625ab4d426a20474b775c23cb8a8b7e5461bbf692a6896bba2aaf9c562aa82 | Ucok intends to compose a key code (password) consisting of 8 numbers that meet the following conditions: i. The numbers used are $1, 2, 3, 4, 5, 6, 7, 8$, and $9$. ii. The first number used is at least $1$, the second number is at least $2$, the third digit is at least $3$, and so on. iii. The same number can be used ... | [
"362,880 for part (a) and 16 for part (b)"
] | [
"Ucok intends to compose a key code (password) consisting of 8 numbers that meet the following conditions:\ni. The numbers used are $1, 2, 3, 4, 5, 6, 7, 8$, and $9$.\nii. The first number used is at least $1$, the second number is at least $2$, the third digit is at least $3$, and so on.\niii. The same number can ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-845854fb4b2858beb7123169707787475ca073f86a1d1c1a6debb8e770405c7b | Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4?$ | [
"\\dfrac{59}{64}"
] | [
"Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4?$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-cd91113614c2f625cd889f789eb77a438eb966799eb3b4e9ff9fab5869002152 | Uncle Donald cuts a $3 \text{ kg}$ piece of cheese into an arbitrary finite number of pieces of arbitrary weights. He distributes these pieces uniformly randomly among his nephews Huey, Dewey, and Louie. What is the probability that two of the nephews each receive strictly more than $1 \text{ kg}$ of cheese? | [
"\\(\\frac{1}{4}\\)"
] | [
"Uncle Donald cuts a $3 \\text{ kg}$ piece of cheese into an arbitrary finite number of pieces of arbitrary weights. He distributes these pieces uniformly randomly among his nephews Huey, Dewey, and Louie. What is the probability that two of the nephews each receive strictly more than $1 \\text{ kg}$ of cheese?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-f0ed18c45a1c7e18e7e92626db670d8e3a6a1fb906b5e40e9a43cb463339bcb1 | Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rota... | [
"48"
] | [
"Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-6aabb64a79d9df0e0f7ef238c20a9015edfe108b733540ddfdd6508edc709596 | Under an optimal strategy, what are the odds of winning if I start dealing a shuffled standard deck of 52 cards face-up, and you can stop me at any point before I run out of cards, winning if the card on top of the deck is an ace? | [
"\\dfrac{1}{13}"
] | [
"Under an optimal strategy, what are the odds of winning if I start dealing a shuffled standard deck of 52 cards face-up, and you can stop me at any point before I run out of cards, winning if the card on top of the deck is an ace?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-aed062a5cede3e4d567ef76a3a1b175da42bdf7fbef1acd4a509f3f844bf33d9 | Under ideal conditions, the population of a certain bacterial colony will double in 45 minutes. How much time will it take for the population to increase five-fold? | [
"104.49"
] | [
"Under ideal conditions, the population of a certain bacterial colony will double in 45 minutes. How much time will it take for the population to increase five-fold?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-2dd769fc9551ccba36eeb3f9223c87d79861eca49c8f28e043eb6c67526f7afe | Under suitable hypotheses, study the convergence in law of \( \sqrt{n} (X_{k([\alpha n], n)} - x_{\alpha}) \). | [
"\\mathcal{N}\\left(0, \\frac{\\alpha(1 - \\alpha)}{[f(x_{\\alpha})]^2}\\right)"
] | [
"Under suitable hypotheses, study the convergence in law of \\( \\sqrt{n} (X_{k([\\alpha n], n)} - x_{\\alpha}) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d7e77e1d4ecf87834f25b75a8dd737a7c7fbb03b1eb502397f323be4efd14fae | Under the additional restriction that no pair of golfers should play together for more than 2 rounds, find the best possible configuration of rounds 2 and 3 for the given initial round groups $\{0,1,2,3\}, \{4,5,6,7\}, \{8,9,10,11\}$. | [
" \n\\begin{aligned} \n&\\text{Round 2: } \\{0,4,8,3\\}, \\{1,5,9,7\\}, \\{2,6,10,11\\} \\\\ \n&\\text{Round 3: } \\{0,1,6,9\\}, \\{2,5,8,11\\}, \\{3,4,7,10\\} \n\\end{aligned} \n"
] | [
"Under the additional restriction that no pair of golfers should play together for more than 2 rounds, find the best possible configuration of rounds 2 and 3 for the given initial round groups $\\{0,1,2,3\\}, \\{4,5,6,7\\}, \\{8,9,10,11\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-bb08af3ec26f5903aeebecbad6735cdd031fb193f10d9d4ba7aa2d6d76817694 | Under the assumption that the opinions of each expert are independent, if each expert has a probability $p$ of being correct, what is the probability of a consensus prediction being correct? The formula given is $\frac{p^3}{p^3+(1-p)^3}$. | [
"\\(\\frac{p^3}{p^3 + (1-p)^3}\\)"
] | [
"Under the assumption that the opinions of each expert are independent, if each expert has a probability $p$ of being correct, what is the probability of a consensus prediction being correct? The formula given is $\\frac{p^3}{p^3+(1-p)^3}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-d36e86cebc4df5e8eaa208529784f37205870cfa472e5c7c737f46186a9f4ab5 | Under the assumption that the orbits of Venus and Earth are circular with the Sun at the center, and that the radii of these orbits are $108,200,000$ km for Venus and $149,600,000$ km for Earth, what is the maximum possible angle (to the nearest degree) between Venus and the Sun as viewed from Earth? | [
"46"
] | [
"Under the assumption that the orbits of Venus and Earth are circular with the Sun at the center, and that the radii of these orbits are $108,200,000$ km for Venus and $149,600,000$ km for Earth, what is the maximum possible angle (to the nearest degree) between Venus and the Sun as viewed from Earth?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-426b47631bba011ec8a9a2defe9ce19a9597197d361b96d872d29f0cdb48da1e | Under the assumptions of the Picard–Lindelöf theorem, let $\mathbf{y(\cdot)}$ and $\mathbf{z(\cdot)}$ be solutions of the Cauchy problems $(\mathbf{f}, t_0, \mathbf{y_0})$ and $(\mathbf{f}, t_0, \mathbf{z_0})$ over an interval $J$ containing $t_0$. If $\mathbf{y_0} = (1, 2)$, $\mathbf{z_0} = (3, 4)$, and $L = 2$, find ... | [
"2\\sqrt{2}e^{2}"
] | [
"Under the assumptions of the Picard–Lindelöf theorem, let $\\mathbf{y(\\cdot)}$ and $\\mathbf{z(\\cdot)}$ be solutions of the Cauchy problems $(\\mathbf{f}, t_0, \\mathbf{y_0})$ and $(\\mathbf{f}, t_0, \\mathbf{z_0})$ over an interval $J$ containing $t_0$. If $\\mathbf{y_0} = (1, 2)$, $\\mathbf{z_0} = (3, 4)$, and... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a306a920b8c5efcfba17c9c27af054f5fb7f796ae5f01f1193965e20f624c7a4 | Under the condition $6x^2 + 12xy - 5z^2 = 1$, find the maximum and minimum values of $x^2 - xy + 2z^2$. | [
"-\\dfrac{2}{5}"
] | [
"Under the condition $6x^2 + 12xy - 5z^2 = 1$, find the maximum and minimum values of $x^2 - xy + 2z^2$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-449e0cb472a3db27cb3455384a06d3cdafe944ebe29071cefa16f3fc6d8fe20d | Under the condition \( 6x^2 + 12xy - 5y^2 = 1 \), find the maximum and minimum of \( x^2 - 2xy + 2y^2 \). | [
"\\dfrac{1}{22}"
] | [
"Under the condition \\( 6x^2 + 12xy - 5y^2 = 1 \\), find the maximum and minimum of \\( x^2 - 2xy + 2y^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-43d97c6b4c5ad324929683c72c99e299ade12df97023f1d840ef225b130488d8 | Under the condition \( x^2 + xy + 2y^2 = 1 \), find the maximum and minimum of \( 2x^2 - y^2 \). | [
"Maximum = \\(\\frac{6 + 2\\sqrt{23}}{7}\\), Minimum = \\(\\frac{6 - 2\\sqrt{23}}{7}\\)"
] | [
"Under the condition \\( x^2 + xy + 2y^2 = 1 \\), find the maximum and minimum of \\( 2x^2 - y^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-6cab060ad69ceaa6b59e02eee6a1e111903bc081146daa6349730ffaf6bb44aa | Under the condition \( x^4 + y^4 + z^4 = 1 \), find the maximum and minimum values of \( x^2 + y^2 + z^2 \) without using Lagrange multipliers. | [
"\\(\\text{Max} = \\sqrt{3}, \\text{Min} = \\frac{\\sqrt{3}}{3}\\)"
] | [
"Under the condition \\( x^4 + y^4 + z^4 = 1 \\), find the maximum and minimum values of \\( x^2 + y^2 + z^2 \\) without using Lagrange multipliers."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9be69559a175ee132dfcb609364f3c035bad708b7a4ae603f9eba8fa42b7f624 | Under the condition \( x^4 + y^4 + z^4 = 1 \), find the maximum and minimum values of \( x^2 + y^2 + z^2 \). | [
"Min = 1, Max = \\(\\sqrt{3}\\)"
] | [
"Under the condition \\( x^4 + y^4 + z^4 = 1 \\), find the maximum and minimum values of \\( x^2 + y^2 + z^2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 27
} | 29 | 0 | 0 |
omr-ws-d2f1cceffeb0e93fff9ecb7918d31e2582bb175d8d26566fc54ff41cfb074495 | Under the condition \(\sum_{i=1}^{100} \left( \sum_{j=1}^{100} |\overrightarrow{A_iA_j}|^2 \right) = 4000\), find the maximum value of the number of points \(A_i\) such that \(|\overrightarrow{CA_i}| \geq 2\). | [
"5"
] | [
"Under the condition \\(\\sum_{i=1}^{100} \\left( \\sum_{j=1}^{100} |\\overrightarrow{A_iA_j}|^2 \\right) = 4000\\), find the maximum value of the number of points \\(A_i\\) such that \\(|\\overrightarrow{CA_i}| \\geq 2\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-e4a9c4b9f45f63d04ba7d5ff6dd405fe3ebb580fb82cf85a1ca0061d069fdd5c | Under the mapping $I$, the point $(x,y)$ is rotated $45^{\circ}$ counter-clockwise and then translated right $2$ units. Where is the point $(1,0)$ after $1000$ such mappings of $I$? | [
"(1,0)"
] | [
"Under the mapping $I$, the point $(x,y)$ is rotated $45^{\\circ}$ counter-clockwise and then translated right $2$ units. Where is the point $(1,0)$ after $1000$ such mappings of $I$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-692cd4d1da6992c878dbbb494da6a0fc1ca93aa79696578d01425ee8d3487f29 | Under the new AMC 10, 12 scoring method, 6 points are given for each correct answer, 2.5 points are given for each unanswered question, and no points are given for an incorrect answer. Some scores between 0 and 150 can be obtained in exactly three ways. What is the sum of these scores? | [
"188.5"
] | [
"Under the new AMC 10, 12 scoring method, 6 points are given for each correct answer, 2.5 points are given for each unanswered question, and no points are given for an incorrect answer. Some scores between 0 and 150 can be obtained in exactly three ways. What is the sum of these scores?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 26
} | 38 | 0 | 0 |
omr-ws-09c03d1359d4e715ffe2afd775cd35c8b4388216ba3cefda17b2368ee292f5b5 | Under the new AMC 10,12 scoring method, 6 points are given for each correct answer, 2.5 points are given for each unanswered question, and no points are given for an incorrect answer. Some scores between 0 and 150 can be obtained in exactly three ways. What is the sum of these scores? | [
"188.5.\"\n\nTherefore, the sum of the scores that can be obtained in exactly three ways is:"
] | [
"Under the new AMC 10,12 scoring method, 6 points are given for each correct answer, 2.5 points are given for each unanswered question, and no points are given for an incorrect answer. Some scores between 0 and 150 can be obtained in exactly three ways. What is the sum of these scores?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 20
} | 31 | 0 | 0 |
omr-ws-214920f5e31339ba20f1cee0ff2a89853e24c26a8f52287b5a645b48e7ae227d | Under the same conditions as Problem 1, for what value of $x$ will the time $w(x)$ be the longest? | [
"\\( x = 5 \\)"
] | [
"Under the same conditions as Problem 1, for what value of $x$ will the time $w(x)$ be the longest?"
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-037c7417c097f03b16543fe45a334ba268e4a8351e7d49cb882ead2cfd5ae6a5 | Under the same conditions as Problem 1, let $P(x)$ be the final polynomial on the board after a sequence of steps such that $P(x)$ has exactly 2021 real roots. If the initial polynomial is $x^{2022} - 1$, find the sum of the coefficients of $P(x)$. | [
"2022"
] | [
"Under the same conditions as Problem 1, let $P(x)$ be the final polynomial on the board after a sequence of steps such that $P(x)$ has exactly 2021 real roots. If the initial polynomial is $x^{2022} - 1$, find the sum of the coefficients of $P(x)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-d27ab8d3d6ada0a5a7600c661abca3ce6ed0fdfe2a2f40d4ac23031181a73d96 | Under the same conditions as Problem 1, suppose \(BE\) and \(CF\) intersect \((O)\) again at \(L\) and \(K\) respectively, and \(AH \cap KL = P\). On \(EF\), take \(Q\) such that \(QP = QI\). Let \(J\) be a point on the circumcircle of triangle \(IBC\) such that \(IJ \perp IQ\). If \(M\) is the midpoint of \(IJ\), find... | [
"\\dfrac{R}{2}"
] | [
"Under the same conditions as Problem 1, suppose \\(BE\\) and \\(CF\\) intersect \\((O)\\) again at \\(L\\) and \\(K\\) respectively, and \\(AH \\cap KL = P\\). On \\(EF\\), take \\(Q\\) such that \\(QP = QI\\). Let \\(J\\) be a point on the circumcircle of triangle \\(IBC\\) such that \\(IJ \\perp IQ\\). If \\(M\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-84bbfd8fd8fa1ad9718d2b4b4be8134d23e865fce421a6964d337613fd59c0b7 | Under the same conditions as Problem 1, suppose a person is asked three times and gives the same answer each time (either east or west). Let $\alpha$ be the probability that the address is east given that the person answers east all three times (EEE), and let $\beta$ be the probability that the address is east given th... | [
"1"
] | [
"Under the same conditions as Problem 1, suppose a person is asked three times and gives the same answer each time (either east or west). Let $\\alpha$ be the probability that the address is east given that the person answers east all three times (EEE), and let $\\beta$ be the probability that the address is east g... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74f173f223e5ad9998cac7dea521b5b1801339fa689f0b2556c2612f97b2b1ab | Under the transformation \( w = \frac{1}{z} \), where \( z = x + iy \) and \( w = u + iv \), find the expression for \( v \) when \( y = 0 \). Simplify your answer to a single expression involving \( u \) and \( x \). | [
"0"
] | [
"Under the transformation \\( w = \\frac{1}{z} \\), where \\( z = x + iy \\) and \\( w = u + iv \\), find the expression for \\( v \\) when \\( y = 0 \\). Simplify your answer to a single expression involving \\( u \\) and \\( x \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62489c26a48101a8e5e6519e0a39cfea01cfffd00a684c408727d86728f40b3a | Under the transformation \( w = \frac{1}{z} \), where \( z = x + yi \) and \( w = u + vi \), find the equation of the circle that is the image of the line \( y = x - 1 \). Express your answer in the form \( u^2 + v^2 + au + bv + c = 0 \) and provide the value of \( a + b + c \). | [
"-2"
] | [
"Under the transformation \\( w = \\frac{1}{z} \\), where \\( z = x + yi \\) and \\( w = u + vi \\), find the equation of the circle that is the image of the line \\( y = x - 1 \\). Express your answer in the form \\( u^2 + v^2 + au + bv + c = 0 \\) and provide the value of \\( a + b + c \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-312a332b2c956d4f80c5214361d2c63629ecc088bbbe3321049d00e28e38164c | Under trapezoidal approximation for the area under the curve \( l_x \), calculate the value of \( e^0_x \) given \( q_x = 0.1 \), \( l_x = 5 \), \( l_{x+i} = \frac{1}{i+1} \), and \( q_{x+i} = \frac{1}{(i+1)^2} \) for \( i = 1, 2, 3, \ldots \). Express your answer as a simplified fraction. | [
"\\dfrac{11}{20}"
] | [
"Under trapezoidal approximation for the area under the curve \\( l_x \\), calculate the value of \\( e^0_x \\) given \\( q_x = 0.1 \\), \\( l_x = 5 \\), \\( l_{x+i} = \\frac{1}{i+1} \\), and \\( q_{x+i} = \\frac{1}{(i+1)^2} \\) for \\( i = 1, 2, 3, \\ldots \\). Express your answer as a simplified fraction."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-28a04a57b0fc4457eac4dfe10e93242c540d1d414216576c949d7ac7349ec3bf | Under what (necessary and sufficient) additional conditions on the function \( f \), which changes sign exactly once, does the function \[ F(t) := \int_{-\infty}^{\infty} \exp\left(\frac{|t-x|}{\varepsilon}\right) f(x) \, dx \] have exactly one root? | [
"f \\text{ is monotonic}"
] | [
"Under what (necessary and sufficient) additional conditions on the function \\( f \\), which changes sign exactly once, does the function \n\\[\nF(t) := \\int_{-\\infty}^{\\infty} \\exp\\left(\\frac{|t-x|}{\\varepsilon}\\right) f(x) \\, dx\n\\]\nhave exactly one root?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-64c18fa25528f20cea5dacc1c2fcf71d49e0e8302ed6fad9c224b536d3630791 | Under what additional conditions on $n \in \mathbb{N}$ and $\lambda \in \mathbb{C} \setminus \{0\}$ is $J_k(\lambda)^n$ similar to $J_k(\lambda^n)$, where $J_k(\lambda)$ is the Jordan block corresponding to $\lambda$ of size $k$? | [
"No additional conditions needed."
] | [
"Under what additional conditions on $n \\in \\mathbb{N}$ and $\\lambda \\in \\mathbb{C} \\setminus \\{0\\}$ is $J_k(\\lambda)^n$ similar to $J_k(\\lambda^n)$, where $J_k(\\lambda)$ is the Jordan block corresponding to $\\lambda$ of size $k$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-8d6c8d8075f2b7abc80f1ec46d7e4327c709b870615c6f2f9df7c98a6bbc8659 | Under what circumstances can the pair of equations \[ x^{2}-y^{2}-u^{3}+v^{2}=-4 \] \[ 2xy+y^{2}-2u^{2}+3v^{4}=-8 \] be solved to give \((u,v)\) in terms of \((x,y)\)? | [
"u \\neq 0, v \\neq 0, \\text{ and } 9uv^2 \\neq 2"
] | [
"Under what circumstances can the pair of equations\n\n\\[ x^{2}-y^{2}-u^{3}+v^{2}=-4 \\]\n\\[ 2xy+y^{2}-2u^{2}+3v^{4}=-8 \\]\n\nbe solved to give \\((u,v)\\) in terms of \\((x,y)\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-24febc397ac56cd3058cca0ea18cc78cff49f78f67a2ce0030aaa4833a3d4ed7 | Under what circumstances does the expression \(x^p + y^q\) run over all residues modulo \(2pq + 1\) for integers \(x\) and \(y\), given that \(p\), \(q\), and \(2pq + 1\) are odd primes? | [
"p = q = 3"
] | [
"Under what circumstances does the expression \\(x^p + y^q\\) run over all residues modulo \\(2pq + 1\\) for integers \\(x\\) and \\(y\\), given that \\(p\\), \\(q\\), and \\(2pq + 1\\) are odd primes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-c5b56526186b0fbdeabe78674d3c6c7788007a12bfd6d64852b19d010bebb17f | Under what circumstances does the function \( f(x, y, z) = 0 \) in set \( D \) satisfy the condition \( \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1 \)? | [
"\\text{When } f_x, f_y, \\text{ and } f_z \\text{ are all non-zero in } D"
] | [
"Under what circumstances does the function \\( f(x, y, z) = 0 \\) in set \\( D \\) satisfy the condition \\( \\frac{\\partial x}{\\partial y} \\frac{\\partial y}{\\partial z} \\frac{\\partial z}{\\partial x} = -1 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-22ffef0eadc1a64a5bd3c3ca09450d9afabbdba35257786972058cfade9bb76d | Under what circumstances is the set $\sum$ nonempty, where $\sum$ is the set of all circles $y$ that meet each side of a scalene triangle $ABC$ in two (possibly coincident) points and satisfy the parallelism conditions $TS \parallel BC$, $PU \parallel CA$, and $RQ \parallel AB$? | [
"\\text{The set } \\sum \\text{ is nonempty for any scalene triangle } ABC."
] | [
"Under what circumstances is the set $\\sum$ nonempty, where $\\sum$ is the set of all circles $y$ that meet each side of a scalene triangle $ABC$ in two (possibly coincident) points and satisfy the parallelism conditions $TS \\parallel BC$, $PU \\parallel CA$, and $RQ \\parallel AB$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-c8b274f36066394e76f6328c30176e7abc113dabfd0d63ca1b6a9cde21c636f0 | Under what circumstances is there equality in the inequality that the sum of the squares of all distances between pairs of points on the surface of a sphere of radius 1 is $\le n^2$? | [
"$\\sum_{i=1}^{n} {\\bf v_i} = {\\bf 0}$"
] | [
"Under what circumstances is there equality in the inequality that the sum of the squares of all distances between pairs of points on the surface of a sphere of radius 1 is $\\le n^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 14
} | 34 | 0 | 0 |
omr-ws-1d841d189039d29b835385ded669c75a786c33094c8dfdaceb7abb2e137652b1 | Under what circumstances will the function \( g(z) = u - iv \) be analytic if \( f(z) = u + iv \) is analytic? | [
"\\( f(z) \\) must be constant."
] | [
"Under what circumstances will the function \\( g(z) = u - iv \\) be analytic if \\( f(z) = u + iv \\) is analytic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-b3a71755656fff10d392bbfa5beaef3e6fb7528bf6727da1b2518c6a543b808d | Under what circumstances will the two triangles $\triangle ABC$ and $\triangle ABD$ be equal in area, where two circles of radii $r_{1}$ and $r_{2}$ intersect at points $A$ and $B$, and a common tangent touches them at points $C$ and $D$ respectively? | [
"Always"
] | [
"Under what circumstances will the two triangles $\\triangle ABC$ and $\\triangle ABD$ be equal in area, where two circles of radii $r_{1}$ and $r_{2}$ intersect at points $A$ and $B$, and a common tangent touches them at points $C$ and $D$ respectively?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 8
} | 19 | 0 | 0 |
omr-ws-7d8d3e602b7e111c1a14f230c28987c515c17a9d88776932dd4d87a471ec2820 | Under what condition does the equality hold for the inequality \(\sqrt[m]{\frac{a_{1}^{m}+a_{2}^{m}+\cdots+a_{x}^{m}}{k}} \geq \sqrt[n]{\frac{a_{1}^{n}+a_{2}^{n}+\cdots+a_{x}^{n}}{k}}\) given that \(m > n\) and \(a_{i}\) are all positive numbers? | [
"\\(a_1 = a_2 = \\cdots = a_x\\)",
"a_1 = a_2 = \\cdots = a_x"
] | [
"Under what condition does the equality hold for the inequality \\(\\sqrt[m]{\\frac{a_{1}^{m}+a_{2}^{m}+\\cdots+a_{x}^{m}}{k}} \\geq \\sqrt[n]{\\frac{a_{1}^{n}+a_{2}^{n}+\\cdots+a_{x}^{n}}{k}}\\) given that \\(m > n\\) and \\(a_{i}\\) are all positive numbers?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 5
} | 35 | 0 | 0 |
omr-ws-d2c6c4b3512740d34ad3a4181a76b29b8fd4e4b29125ecf93bffae979ff5312d | Under what condition does the inequality \( a + b + c \ge ab + bc + ac \) hold for \( a, b, c \in \mathbb{R}^+ \)? | [
"\\( a, b, c \\in (0, 1] \\)"
] | [
"Under what condition does the inequality \\( a + b + c \\ge ab + bc + ac \\) hold for \\( a, b, c \\in \\mathbb{R}^+ \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-0a83c8db761196e9d381406dd31c93999d161fd55b9097753b05b0f1e2698d3b | Under what condition does the inequality \( a^{2^n} < \frac{(2^{n-2}!)^2}{2^{n-1}!} \) hold for large enough \( n \)? | [
"\\( a < \\frac{1}{\\sqrt{2}} \\)"
] | [
"Under what condition does the inequality \\( a^{2^n} < \\frac{(2^{n-2}!)^2}{2^{n-1}!} \\) hold for large enough \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-23fb1a9bf721c692e5112fb8c3209ab741b3e47c65e0eacb90b7d6749d5100b0 | Under what condition does the inequality \(IA + IB + IC < R\) hold? Try to find a sufficient condition that is as good as possible. | [
"\\text{The triangle has an angle greater than } 120^\\circ"
] | [
"Under what condition does the inequality \\(IA + IB + IC < R\\) hold? Try to find a sufficient condition that is as good as possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-0488367c9e70cbd3561aec1bb752a844c753906631b3bb774b74cfedf4e49af2 | Under what condition does the polynomial $ax^3 + bx^2 + cx + d$ have extremal points? | [
"\\( b^2 > 3ac \\)"
] | [
"Under what condition does the polynomial $ax^3 + bx^2 + cx + d$ have extremal points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 10
} | 65 | 0 | 0 |
omr-ws-f20ff295ccd086275158259f02905ce0716991e749b069afbb22a773df1249f8 | Under what condition is $\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers? | [
"\\(\\textbf{(D)} \\text{ It is true if and only if } ab = 0 \\text{ and } a + b \\geq 0\\)"
] | [
"Under what condition is $\\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-1abdd5a30cb44e6f31f5ab7e5f189caf0bc3a6e813d0de2e2e112991916bf1eb | Under what condition is \( ab - n \) divisible by \( (a - n)(b - n) \) where \( a > n \) and \( b > n \)? | [
"\\( n = 0 \\)"
] | [
"Under what condition is \\( ab - n \\) divisible by \\( (a - n)(b - n) \\) where \\( a > n \\) and \\( b > n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23cbc5f12449b8c31df583c3ccfd15a393744d06a94df8f4753e23dfde9552ef | Under what condition is a convex function \( f \) on an open convex set \( \omega \) in \( \mathbb{R}^n \), which has partial derivatives, differentiable? | [
"f \\text{ is differentiable on } \\omega"
] | [
"Under what condition is a convex function \\( f \\) on an open convex set \\( \\omega \\) in \\( \\mathbb{R}^n \\), which has partial derivatives, differentiable?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-9afe00c864606f55930a60c8e106cb4cdbc6e4e6e53fb35ced5a5795ec8070ef | Under what condition is there only one prime, up to multiplication by a unit, in $ \frac{\mathbb{Z}}{m\mathbb{Z}}$? | [
"\\( m = p^k \\) for some prime \\( p \\) and integer \\( k \\geq 1 \\)."
] | [
"Under what condition is there only one prime, up to multiplication by a unit, in $ \\frac{\\mathbb{Z}}{m\\mathbb{Z}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-adcf456207aef8d26b7cbedb8cc38cec8f6b094ad58083246479240b02376561 | Under what condition on an $L^2$ function $f$ does the Fredholm alternative guarantee the existence of a weak solution to the problem \[L[u]=f \text{ on } 1<x<e^{2\pi/3}, \text{ with } u(1)=u(e^{2\pi/3})=0,\] where the operator $L$ is given by \[L[u]:=-x^2 u''-6xu'-8u?\] | [
"\\( \\int_1^{e^{2\\pi/3}} f(x) v(x) \\, dx = 0 \\) for all \\( v \\) such that \\( L^*[v] = 0 \\)."
] | [
"Under what condition on an $L^2$ function $f$ does the Fredholm alternative guarantee the existence of a weak solution to the problem\n\\[L[u]=f \\text{ on } 1<x<e^{2\\pi/3}, \\text{ with } u(1)=u(e^{2\\pi/3})=0,\\]\nwhere the operator $L$ is given by\n\\[L[u]:=-x^2 u''-6xu'-8u?\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-be01a82c7fd9614665709e2c633f765831662507f5dbb185ec882e73d6141ee0 | Under what condition on the function \( f \) defined on \( S \subset \mathbb{R} \) do we have \(\bigcap_{n=1}^{\infty} f(E_n) = f\left(\bigcap_{n=1}^{\infty} E_n\right)\) for any sequence of sets \( S \supset E_1 \supset E_2 \supset \cdots \)? | [
"f \\text{ is injective}"
] | [
"Under what condition on the function \\( f \\) defined on \\( S \\subset \\mathbb{R} \\) do we have \\(\\bigcap_{n=1}^{\\infty} f(E_n) = f\\left(\\bigcap_{n=1}^{\\infty} E_n\\right)\\) for any sequence of sets \\( S \\supset E_1 \\supset E_2 \\supset \\cdots \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-6a38f8cd7f8591f3e81c407a148df4a930de23cf7b692ec838e9b6d589ef217d | Under what condition on triangle \(ABC\) are the points \(A\), the Feuerbach point \(Fe\), and the Nagel point \(Na\) collinear? | [
"\\(AB = AC\\) or \\(2BC = AB + AC\\)"
] | [
"Under what condition on triangle \\(ABC\\) are the points \\(A\\), the Feuerbach point \\(Fe\\), and the Nagel point \\(Na\\) collinear?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-f4742b0dd26126a071aae78debfe0f4382526315fdeb46d5180bf605ed41c151 | Under what conditions are $(X*Y)*Z$ and $X* ( Y * Z)$ distinct? | [
"\\(\\lambda \\neq 0\\) and \\(\\lambda \\neq 1\\)"
] | [
"Under what conditions are $(X*Y)*Z$ and $X* ( Y * Z)$ distinct?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-20d39ac7e344a2b2809e0244e5dd8f93f1daf96bb6a22d30ff6b3b7f392b644a | Under what conditions are the ranks of the matrices $[B \ AB \ \ldots \ A^{n-1}B]$ and $[AB \ A^2B \ \ldots \ A^nB]$ equal? | [
"A \\text{ is invertible}"
] | [
"Under what conditions are the ranks of the matrices $[B \\ AB \\ \\ldots \\ A^{n-1}B]$ and $[AB \\ A^2B \\ \\ldots \\ A^nB]$ equal?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-7649af282214d93914c70a8e628014a00fe356c7a8342d152fc8fea0f0778578 | Under what conditions can \( A = m(2n + 1) \) be a prime number? | [
"\\( m = 1 \\) and \\( 2n + 1 \\) is a prime number."
] | [
"Under what conditions can \\( A = m(2n + 1) \\) be a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-60a80258a727292f9dcddfb344ea49ca64d458fbb21fd60b7bf420e285bff0a0 | Under what conditions can a floor of size $n \times n$ be completely laid using an equal number of $2 \times 2$ and $3 \times 1$ tiles, without cutting or overlapping the tiles? | [
"\\( n \\) must be divisible by 7."
] | [
"Under what conditions can a floor of size $n \\times n$ be completely laid using an equal number of $2 \\times 2$ and $3 \\times 1$ tiles, without cutting or overlapping the tiles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 10
} | 20 | 0 | 0 |
omr-ws-23f6eaf39268a7afb9b19491fee80efc5a9fedfd109f7597ce9d7ebc1988504d | Under what conditions can an integer matrix \( R \) be diagonalized by two orthogonal matrices \( N \) and \( M \), i.e., \( MRN = D \) for some diagonal matrix \( D \)? | [
"The matrix \\( R \\) can be diagonalized by two orthogonal matrices \\( N \\) and \\( M \\) if and only if the resulting matrix \\( D \\) has at most one non-zero entry in any row or column after the transformation \\( MRN \\)."
] | [
"Under what conditions can an integer matrix \\( R \\) be diagonalized by two orthogonal matrices \\( N \\) and \\( M \\), i.e., \\( MRN = D \\) for some diagonal matrix \\( D \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-b204080659460ae2e4b06388bc363887dc0955db446ea282562bfe1ccfbd66a5 | Under what conditions can the approximation \(f(x) \approx u(x)\) and \(g(x) \approx v(x)\) be used to evaluate the limit \(\lim_{x \rightarrow 0} \frac{f(x) + g(x)}{h(x)}\) as \(\lim_{x \rightarrow 0} \frac{u(x) + v(x)}{h(x)}\)? | [
"The approximation can be used if the Taylor expansions of \\( f(x) \\) and \\( g(x) \\) around \\( x = 0 \\) are accurately captured by \\( u(x) \\) and \\( v(x) \\) respectively, ensuring that the error terms do not affect the limit."
] | [
"Under what conditions can the approximation \\(f(x) \\approx u(x)\\) and \\(g(x) \\approx v(x)\\) be used to evaluate the limit \\(\\lim_{x \\rightarrow 0} \\frac{f(x) + g(x)}{h(x)}\\) as \\(\\lim_{x \\rightarrow 0} \\frac{u(x) + v(x)}{h(x)}\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-158fd888025558dfba28a74f92bd378d704f11fdd307131fc07693e2f68c7cca | Under what conditions can the congruence $x^2 \equiv 1 \pmod{n}$ be simplified to $x \equiv 1 \pmod{n}$? | [
"\\( n = 2 \\) or \\( x \\equiv 1 \\pmod{n} \\) or \\( n \\) is prime and \\( x \\not\\equiv -1 \\pmod{n} \\)"
] | [
"Under what conditions can the congruence $x^2 \\equiv 1 \\pmod{n}$ be simplified to $x \\equiv 1 \\pmod{n}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc76e5b9fb10401699192b7c725b6ea61e7738465ce1ec5bf96a237bd1caf865 | Under what conditions can the following system of equations be solved to give \((u, v)\) explicitly in terms of \((x, y)\)? \[ x^2 - y^2 - u^3 + v^2 = -4 \] \[ 2xy + y^2 - 2u^2 + 3v^4 = -8 \] | [
"u \\neq 0, v \\neq 0, \\text{ and } 9uv^2 \\neq 2"
] | [
"Under what conditions can the following system of equations be solved to give \\((u, v)\\) explicitly in terms of \\((x, y)\\)?\n\\[ x^2 - y^2 - u^3 + v^2 = -4 \\]\n\\[ 2xy + y^2 - 2u^2 + 3v^4 = -8 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8284de4406d1c424dae9bf966f0c38235756434c88dfcccd6e7b0fef9b4632e | Under what conditions can the linear operator \( F: \mathbb{R}^n \rightarrow \mathbb{R}^n \) with a matrix having numbers \( \lambda_1, \ldots, \lambda_n \) on the secondary diagonal and zeros elsewhere be diagonalized over \( \mathbb{R} \)? | [
"\\(\\lambda_k \\lambda_{n+1-k} \\ge 0\\) for all \\( k \\)"
] | [
"Under what conditions can the linear operator \\( F: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n \\) with a matrix having numbers \\( \\lambda_1, \\ldots, \\lambda_n \\) on the secondary diagonal and zeros elsewhere be diagonalized over \\( \\mathbb{R} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 9
} | 21 | 0 | 0 |
omr-ws-9997d341f1d72f6b178cfc1b7e23397e4ad86a208b7ee7a4ec5ae40bf9ba9116 | Under what conditions can the polynomial \( f(x) = ax^4 + bx^3 + cx^2 + dx + e \) be factored into the product of two quadratic polynomials with real coefficients, i.e., \( f(x) = (mx^2 + nx + p)(qx^2 + rx + s) \), where \( m, n, p, q, r, s \in \mathbb{R} \)? | [
"Always possible."
] | [
"Under what conditions can the polynomial \\( f(x) = ax^4 + bx^3 + cx^2 + dx + e \\) be factored into the product of two quadratic polynomials with real coefficients, i.e., \\( f(x) = (mx^2 + nx + p)(qx^2 + rx + s) \\), where \\( m, n, p, q, r, s \\in \\mathbb{R} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-abb6caf72e5be3efb4e9ed8e9642f27b521d78ae2cf1a4da673f5d972f795b7c | Under what conditions can two different quadrilaterals have the same product of their side lengths? | [
"The product of the side lengths of one quadrilateral must equal the product of the side lengths of the other quadrilateral."
] | [
"Under what conditions can two different quadrilaterals have the same product of their side lengths?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 20
} | 40 | 0 | 0 |
omr-ws-f20a4fddea06fa6ce3db4a9ad9b84cf1be6916651562cc0e0b0afc634565052a | Under what conditions can we conclude that $\sup_{\boldsymbol{\mu} \in \Phi(\boldsymbol{x}_0,\delta)}\|\boldsymbol{\mu}-\boldsymbol{\mu}_0\| \to 0$ as $\delta \to 0$, where $\Phi(\boldsymbol{x}_0,\delta)$ denotes the collection of all subgradients of the function part squeezed between the supporting hyperplane $A=\{(\b... | [
"Strict convexity in a neighborhood of \\(\\boldsymbol{x}_0\\)."
] | [
"Under what conditions can we conclude that $\\sup_{\\boldsymbol{\\mu} \\in \\Phi(\\boldsymbol{x}_0,\\delta)}\\|\\boldsymbol{\\mu}-\\boldsymbol{\\mu}_0\\| \\to 0$ as $\\delta \\to 0$, where $\\Phi(\\boldsymbol{x}_0,\\delta)$ denotes the collection of all subgradients of the function part squeezed between the suppor... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e4fc53e5f428d8a7d7428484fb46482ce819422ed1ae395942a5f6529b89b1c | Under what conditions do four given points in a plane lie on a circle such that the circle is equidistant from all four points? | [
"The points must be concyclic."
] | [
"Under what conditions do four given points in a plane lie on a circle such that the circle is equidistant from all four points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-3d3395992942a9071fa30ecd7db48cae90948e7ba4800c67c58d885ff3843aa4 | Under what conditions do the functions \( f(x) \) and \( g(x) \) satisfy \( f \circ g = g \circ f \)? | [
"f(g(x)) = g(f(x)) \\text{ for all } x"
] | [
"Under what conditions do the functions \\( f(x) \\) and \\( g(x) \\) satisfy \\( f \\circ g = g \\circ f \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55a91f4ac544600c9c66b07d53f7a04b928f933dcf4595ed761070ea7bdf17e1 | Under what conditions do the spaces $L^{r}(\mu)$ and $L^{s}(\mu)$ contain the same functions, where $\mu$ is a positive measure on $X$ with $\mu(X) = 1$ and $0 < r < s$? | [
"\\mu \\text{ is purely atomic with finitely many atoms}"
] | [
"Under what conditions do the spaces $L^{r}(\\mu)$ and $L^{s}(\\mu)$ contain the same functions, where $\\mu$ is a positive measure on $X$ with $\\mu(X) = 1$ and $0 < r < s$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-53febb0964a0cca683a3806625edb7a2125599b6d66dd8bbc67dec7d5b94e193 | Under what conditions does \( a^m \equiv 1 \pmod{p} \) hold for \( a < \phi(p) \)? | [
"\\gcd(a, p) = 1 \\text{ and } m \\text{ is a multiple of the order of } a \\text{ modulo } p"
] | [
"Under what conditions does \\( a^m \\equiv 1 \\pmod{p} \\) hold for \\( a < \\phi(p) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-826fbd191121204e27059516294fd500134038fba05719814d97fcb037e97d0b | Under what conditions does \( f(2n) = f(n) + 1 \) hold? | [
"\\( k = 1 \\) or \\( k = \\ldots112_3 \\) (a block of 1's followed by a 2)"
] | [
"Under what conditions does \\( f(2n) = f(n) + 1 \\) hold?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a4fc6fcb92dd350fbf711eecd7271d510db4ad076d917e13744736c35d2c7bd7 | Under what conditions does \( f(3n) = f(2n) \) hold? | [
"The base 3 representation of \\( n \\) contains only 1's and 0's."
] | [
"Under what conditions does \\( f(3n) = f(2n) \\) hold?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2050de402a89d455038052121b081ebfd9089c636b528fb13df633d6e232f5d | Under what conditions does a connected bipartite graph obtained by a hex-comb have a perfect matching, given that the graph has an equal number of vertices of each color? | [
"The graph must satisfy Hall's condition."
] | [
"Under what conditions does a connected bipartite graph obtained by a hex-comb have a perfect matching, given that the graph has an equal number of vertices of each color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 11
} | 31 | 0 | 0 |
omr-ws-a1594fc3a42327dbd2aecbf56b00add46fbba0aec52f5cf1366dc9f09c5664d7 | Under what conditions does a function \( f \) defined on an open interval \((a, b)\) have a primitive, i.e., a differentiable function \( F \) on the same interval such that \( F' = f \)? | [
"f \\text{ is continuous on } (a, b)"
] | [
"Under what conditions does a function \\( f \\) defined on an open interval \\((a, b)\\) have a primitive, i.e., a differentiable function \\( F \\) on the same interval such that \\( F' = f \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-0c674759ed79926added22f1843728b1353841aa7a0f1f49ba732ee2cc3fd5cf | Under what conditions does a linear transformation \( T : V \rightarrow V \) have a proper \( T \)-invariant subspace \( W \) (i.e., a proper subspace \( W \) such that \( T(W) \subseteq W \))? | [
"The characteristic polynomial of \\( T \\) is not irreducible."
] | [
"Under what conditions does a linear transformation \\( T : V \\rightarrow V \\) have a proper \\( T \\)-invariant subspace \\( W \\) (i.e., a proper subspace \\( W \\) such that \\( T(W) \\subseteq W \\))?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-07aedd85e638b26c0141d8bd6977f4cd3b1b884c77e03954044df9d2c630b256 | Under what conditions does a polynomial \( P(x) \) with real coefficients and degree \( n \) have a root with multiplicity \( m \) where \( m \in \mathbb{N} \)? | [
"1 \\leq m \\leq n"
] | [
"Under what conditions does a polynomial \\( P(x) \\) with real coefficients and degree \\( n \\) have a root with multiplicity \\( m \\) where \\( m \\in \\mathbb{N} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
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