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-c01878e3c3f155736f9fcb02fd2d32890fe712d4df6982fb4b0b339f5cf0c8ad | At the base of the pyramid with apex $S$ lies the square $ABCD$. It is known that $AS = 7$, $BS = DS = 5$. Find the length of the side edge $CS$ of such a pyramid. | [
"1"
] | [
"At the base of the pyramid with apex $S$ lies the square $ABCD$. It is known that $AS = 7$, $BS = DS = 5$. Find the length of the side edge $CS$ of such a pyramid."
] | [
"aops_c4_high_school_math"
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"cot"
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"cot": 16
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omr-ws-614ae0acab432e8b0194dbc2d303f6e9ab5192fa30d4f380283640e37a5f68a8 | At the base of the pyramid with apex $S$ lies the square $ABCD$. It is known that $AS = 7$, $BS = DS = 5$. What values can the side length of the square $ABCD$ take? | [
"\\(3\\sqrt{2} \\leq \\ell \\leq 4\\sqrt{2}\\)"
] | [
"At the base of the pyramid with apex $S$ lies the square $ABCD$. It is known that $AS = 7$, $BS = DS = 5$. What values can the side length of the square $ABCD$ take?"
] | [
"aops_c4_high_school_math"
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"cot",
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"cot": 6,
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omr-ws-e5703769de955ce3663dd56caaa7f0ec68608b9691418833688b3343162d73d4 | At the base of the quadrangular pyramid $PABCD$ lies the parallelogram $ABCD$. Points $M$ and $N$ are selected on the lateral edges $PC$ and $PD$, respectively, such that points $A, B, M,$ and $N$ lie in the same plane. The volume of the pyramid $PABMN$ is $4.5$ times less than the volume of the pyramid $PABCD$ (ratio ... | [
"\\(\\frac{\\sqrt[3]{\\frac{2}{9}}}{1 - \\sqrt[3]{\\frac{2}{9}}}\\)"
] | [
"At the base of the quadrangular pyramid $PABCD$ lies the parallelogram $ABCD$. Points $M$ and $N$ are selected on the lateral edges $PC$ and $PD$, respectively, such that points $A, B, M,$ and $N$ lie in the same plane. The volume of the pyramid $PABMN$ is $4.5$ times less than the volume of the pyramid $PABCD$ (r... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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omr-ws-dfc66ec8f0d2547f84573ce0db4e7037f64c24ce5464370be1aced2871aa5523 | At the base of the triangular pyramid $ABCD$ lies a regular triangle $ABC$ such that $AD = BC$. All plane angles at vertex $B$ are equal to each other. What might these angles be equal to? | [
"\\( 60^\\circ \\)"
] | [
"At the base of the triangular pyramid $ABCD$ lies a regular triangle $ABC$ such that $AD = BC$. All plane angles at vertex $B$ are equal to each other. What might these angles be equal to?"
] | [
"aops_c6_high_school_olympiads"
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"cot",
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"cot": 16,
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omr-ws-5572d7f6cc5d89209888d0c15d2853e760ae195afff53f7dda76e5fa5f8e657f | At the base of the triangular pyramid $ABCD$ lies an equilateral triangle $ABC$. It is known that $AD = BC$, and all flat angles at the vertex $D$ are equal to each other. What can these angles be equal to? | [
"\\(36^\\circ, 60^\\circ, 108^\\circ\\)"
] | [
"At the base of the triangular pyramid $ABCD$ lies an equilateral triangle $ABC$. It is known that $AD = BC$, and all flat angles at the vertex $D$ are equal to each other. What can these angles be equal to?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
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omr-ws-46c295627af41a7574c2be938200cc100ea2bcce2cfb43cc80190a44a43cff66 | At the base of the triangular pyramid $ABCS$ with volume $\sqrt{48}$ lies an equilateral triangle $ABC$ with side length $1$. The projection $K$ of the vertex $S$ onto the base of the pyramid lies on the altitude $BH$ of the triangle $ABC$, and $BK: KH = 6: 5$. Find the smallest cross-sectional area of the pyramid by t... | [
"\\dfrac{144}{11}"
] | [
"At the base of the triangular pyramid $ABCS$ with volume $\\sqrt{48}$ lies an equilateral triangle $ABC$ with side length $1$. The projection $K$ of the vertex $S$ onto the base of the pyramid lies on the altitude $BH$ of the triangle $ABC$, and $BK: KH = 6: 5$. Find the smallest cross-sectional area of the pyrami... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-483937a2d105b8d8442eb3308c34c5bde6208e8c7e4553f4906665345f440304 | At the beginning of a game, the numbers $1, 2, \ldots, 2004$ are written on a desk. A move consists of selecting some numbers, calculating the remainder of their sum when divided by $11$, writing this remainder on the desk, and deleting the selected numbers. After several moves, only two numbers remain on the desk, one... | [
"4"
] | [
"At the beginning of a game, the numbers $1, 2, \\ldots, 2004$ are written on a desk. A move consists of selecting some numbers, calculating the remainder of their sum when divided by $11$, writing this remainder on the desk, and deleting the selected numbers. After several moves, only two numbers remain on the des... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7de7b418a5a3e481e2cc8a39889672a1492ff4094c276103d301ee34aed38f6f | At the beginning of a semester, $55$ students have signed up for Linear Algebra; the course is offered in two sections that are taught at different times. Because of scheduling conflicts and personal preferences, $20\%$ of the students who are originally in Section $A$ switch to Section $B$ in the first few weeks of cl... | [
"25"
] | [
"At the beginning of a semester, $55$ students have signed up for Linear Algebra; the course is offered in two sections that are taught at different times. Because of scheduling conflicts and personal preferences, $20\\%$ of the students who are originally in Section $A$ switch to Section $B$ in the first few weeks... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
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omr-ws-2fa3dc0af7d24d2ada602ceee54837957989591813ea73e94826f4773ca03e7a | At the beginning of a trip, the mileage odometer read 56,200 miles. The driver filled the gas tank with 6 gallons of gasoline. During the trip, the odometer read 56,560 miles when the driver filled the tank again with 12 gallons of gasoline. At the end of the trip, the odometer read 57,060 miles when the driver filled ... | [
"22.6"
] | [
"At the beginning of a trip, the mileage odometer read 56,200 miles. The driver filled the gas tank with 6 gallons of gasoline. During the trip, the odometer read 56,560 miles when the driver filled the tank again with 12 gallons of gasoline. At the end of the trip, the odometer read 57,060 miles when the driver fi... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
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"additional_problems": 1
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omr-ws-8a397c4b8e039b1dc4c90a93511cad84a36932f9159dc9fa08a29b19294406d9 | At the beginning of a year, 1600 roubles was deposited in a savings-bank, and at the end of this year 848 roubles was withdrawn. At the close of the second year, 824 roubles turned out to be in the savings account. What is the interest rate set by the savings-bank per annum? | [
"3\\%"
] | [
"At the beginning of a year, 1600 roubles was deposited in a savings-bank, and at the end of this year 848 roubles was withdrawn. At the close of the second year, 824 roubles turned out to be in the savings account. What is the interest rate set by the savings-bank per annum?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1cfe7c75bce563eb620eb1be6cb09b1a041767b03a0417209971b0408b3ed865 | At the beginning of every period of British Literature, Mrs. Crabapple picks a random student to receive a crabapple as a gift, but really, as you might imagine, they are quite bitter and nasty. Given that there are 11 students in her class and her class meets four times a week, how many different sequences of crabappl... | [
"14,\\!641"
] | [
"At the beginning of every period of British Literature, Mrs. Crabapple picks a random student to receive a crabapple as a gift, but really, as you might imagine, they are quite bitter and nasty. Given that there are 11 students in her class and her class meets four times a week, how many different sequences of cra... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f0227d42931ae9b0e5885b8b1f512850f1c66a79fabacaf6bb121addbf57c37 | At the beginning of my bike ride I feel good, so I can travel 20 miles per hour. Later, I get tired and travel only 12 miles per hour. If I travel a total of 122 miles in a total time of 8 hours, for how many hours did I feel good? Express your answer as a common fraction. | [
"\\frac{13}{4}"
] | [
"At the beginning of my bike ride I feel good, so I can travel 20 miles per hour. Later, I get tired and travel only 12 miles per hour. If I travel a total of 122 miles in a total time of 8 hours, for how many hours did I feel good? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5141e2c9ed936c7be3bcb227c9be86293422302ede560a7fdc45451008a93a84 | At the beginning of the academic year, the Olympic Center must accept a certain number of talented students for the 2024 different classes it offers. Each student must take at least one class but cannot take all the classes. There cannot be a common class that all students take, and each class must be taken by at least... | [
"2026"
] | [
"At the beginning of the academic year, the Olympic Center must accept a certain number of talented students for the 2024 different classes it offers. Each student must take at least one class but cannot take all the classes. There cannot be a common class that all students take, and each class must be taken by at ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-a68ae6ce252c07d249a88da7c06f1d78817824f785656506bb65ae75e9d14c01 | At the beginning of the n-th minute, Bessie the cow paints the interior of the rectangle with vertices $\left( \frac{1}{140}n, \frac{1}{140}n \right), \left( \frac{1}{70}n^2, \frac{1}{140}n \right), \left( \frac{1}{70}n^2, \frac{1}{70}n^2 \right)$, and $\left( \frac{1}{140}n, \frac{1}{70}n^2 \right)$, starting with the... | [
"\\(\\frac{15447}{490}\\)"
] | [
"At the beginning of the n-th minute, Bessie the cow paints the interior of the rectangle with vertices $\\left( \\frac{1}{140}n, \\frac{1}{140}n \\right), \\left( \\frac{1}{70}n^2, \\frac{1}{140}n \\right), \\left( \\frac{1}{70}n^2, \\frac{1}{70}n^2 \\right)$, and $\\left( \\frac{1}{140}n, \\frac{1}{70}n^2 \\right... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-30fa31cc82368592f065c10ef9de03e2a66ef02f15bcc578fd376377cb783e0f | At the beginning of the school year, 50% of all students in Mr. Well's math class answered "Yes" to the question "Do you love math," and 50% answered "No." At the end of the school year, 70% answered "Yes" and 30% answered "No." Altogether, \( x\% \) of the students gave a different answer at the beginning and end of t... | [
"60"
] | [
"At the beginning of the school year, 50% of all students in Mr. Well's math class answered \"Yes\" to the question \"Do you love math,\" and 50% answered \"No.\" At the end of the school year, 70% answered \"Yes\" and 30% answered \"No.\" Altogether, \\( x\\% \\) of the students gave a different answer at the begi... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-c840b0e65eac5de7ab4bb487e77b4b707fa408c178c3c204119c72d642885a0d | At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece wins the game. Ayşe will be the first player. They are playing the ga... | [
"5"
] | [
"At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece wins the game. Ayşe will be the first player. They are playing t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 7
} | 25 | 0 | 0 |
omr-ws-39031907c57628dd7f755ab246deccf2200a9cfdff3d2c49c9363571aabe6557 | At the bottom-left corner of a $2014 \times 2014$ chessboard, there are some green worms, and at the top-left corner, there are some brown worms. Green worms can move only to the right and up, and brown worms can move only to the right and down. After a while, the worms make some moves and all unit squares of the chess... | [
"1343"
] | [
"At the bottom-left corner of a $2014 \\times 2014$ chessboard, there are some green worms, and at the top-left corner, there are some brown worms. Green worms can move only to the right and up, and brown worms can move only to the right and down. After a while, the worms make some moves and all unit squares of the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1289e35290d89f532221552ac515286293f962f3dadf32318df08aacb0d78e96 | At the center of a lake with a radius of 1 km lies your enemy. Your enemy can row at a rate of 2 km/hour to any point on the shore. How fast do you need to run along the shore to be certain that you will be able to catch your enemy when he lands? | [
"2\\pi"
] | [
"At the center of a lake with a radius of 1 km lies your enemy. Your enemy can row at a rate of 2 km/hour to any point on the shore. How fast do you need to run along the shore to be certain that you will be able to catch your enemy when he lands?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81dd6b2136a81543c443e23dc112fbd8f6b4ba65861e3f8f2ea29e151a8974d4 | At the county fair, a mouse starts at position $A$ and every ten seconds has an equal probability of jumping one cup clockwise or counter-clockwise in a circle of six cups. After a minute, if the mouse returns to position $A$, you win a giant chunk of cheese. What is the probability of winning the cheese? | [
"\\(\\frac{11}{32}\\)"
] | [
"At the county fair, a mouse starts at position $A$ and every ten seconds has an equal probability of jumping one cup clockwise or counter-clockwise in a circle of six cups. After a minute, if the mouse returns to position $A$, you win a giant chunk of cheese. What is the probability of winning the cheese?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-f73af025c775100f0f6d324bbdd1465271c4f61a0a42a6cc239c7e680340accc | At the court of King Arthur, 2n knights gathered, each of whom has no more than n - 1 enemies among those present. Merlin arranges the knights around a table such that no knight sits next to an enemy. Let k be the number of ways Merlin can arrange the knights under these conditions. Calculate the remainder when k is di... | [
"440"
] | [
"At the court of King Arthur, 2n knights gathered, each of whom has no more than n - 1 enemies among those present. Merlin arranges the knights around a table such that no knight sits next to an enemy. Let k be the number of ways Merlin can arrange the knights under these conditions. Calculate the remainder when k ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-49ab56b1759bc2e7b54ee12e986b83abd0062f2cec49ae9ef3490a6c197f033a | At the court of Tsar Gorokh, a moment of time is declared favorable if on a watch with a centrally placed second hand, the minute hand is positioned between the hour hand and the second hand (counting clockwise). Does the whole day (24 hours) contain more favorable moments than unfavorable ones? | [
"No"
] | [
"At the court of Tsar Gorokh, a moment of time is declared favorable if on a watch with a centrally placed second hand, the minute hand is positioned between the hour hand and the second hand (counting clockwise). Does the whole day (24 hours) contain more favorable moments than unfavorable ones?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-c28343f0f1f5ca8318a918784b01800879ca071e52960a2cecec9664d41f96ab | At the currency exchange of the island of Luck, the exchange rates are as follows: $D = 6G$, $D = 25R$, $D = 120T$, $G = 4R$, $G = 21T$, $R = 5T$. In the morning, a broker had $80$ dinars, $100$ guilders, $100$ reals, and $50,400$ thalers. In the evening, he had the same number of dinars and thalers. What is the maximu... | [
"546"
] | [
"At the currency exchange of the island of Luck, the exchange rates are as follows: $D = 6G$, $D = 25R$, $D = 120T$, $G = 4R$, $G = 21T$, $R = 5T$. In the morning, a broker had $80$ dinars, $100$ guilders, $100$ reals, and $50,400$ thalers. In the evening, he had the same number of dinars and thalers. What is the m... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd224dca55e393e4f3f459982b29732716233a62dd278c5e40710915af4baae7 | At the end of 1994, Walter was half as old as his grandmother. The sum of the years in which they were born is 3838. How old will Walter be at the end of 1999? | [
"55"
] | [
"At the end of 1994, Walter was half as old as his grandmother. The sum of the years in which they were born is 3838. How old will Walter be at the end of 1999?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e22d8fbfe10dc49e5da5925522b83cc23130ac9b2ba7895cf0933888f5dda82 | At the end of 1994, Walter was half as old as his grandmother. The sum of the years in which they were born was 3838. How old will Walter be at the end of 1999? | [
"55"
] | [
"At the end of 1994, Walter was half as old as his grandmother. The sum of the years in which they were born was 3838. How old will Walter be at the end of 1999?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-c4260350d4f279b5ca372908873b0f9f26ea2aba13a6ce1b587a97f9ebbcb33e | At the end of a day 60 melons remained in a shop, of which 50 were ripe. A customer chooses two melons. What is the probability that they were both ripe? | [
"245/354"
] | [
"At the end of a day 60 melons remained in a shop, of which 50 were ripe. A customer chooses two melons. What is the probability that they were both ripe?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3224b913f3859e744f1a5e9b482d9749e1fb25095e2cf546476a0052103629e0 | At the end of a game, each of the five members of a basketball team shake hands with each of the five members of the other team, and all of the players shake hands with the two referees. How many handshakes occur? | [
"45"
] | [
"At the end of a game, each of the five members of a basketball team shake hands with each of the five members of the other team, and all of the players shake hands with the two referees. How many handshakes occur?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-277db3f61da70f20fb4d7a5ed25cc2fc2f3e6ceab0eec10aecf1bea2c0e4ab9a | At the end of a year a savings-bank calculated the interest due to the depositor as 6 roubles. Adding another 44 roubles, the depositor left his money for another year. At the end of the second year the due interest was calculated once again, and now the deposit together with all the due interest amounted to 257 rouble... | [
"200 roubles"
] | [
"At the end of a year a savings-bank calculated the interest due to the depositor as 6 roubles. Adding another 44 roubles, the depositor left his money for another year. At the end of the second year the due interest was calculated once again, and now the deposit together with all the due interest amounted to 257 r... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4eea298556b012fadcb2be5a336c5226572da56e590cffe3b1c215c6b98b2410 | At the end of the day, a bakery had $\frac{6}{7}$ of a pie left over. Louie, Duey and Huey split the pie, with each taking home the same amount of leftover pie. How much pie did Louie take home? | [
"\\frac{2}{7}"
] | [
"At the end of the day, a bakery had $\\frac{6}{7}$ of a pie left over. Louie, Duey and Huey split the pie, with each taking home the same amount of leftover pie. How much pie did Louie take home?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f4a8ca58fcb5580cb47250b337abeed7e478a6838c393b0fa2e090168af8dd9 | At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers? | [
"4242"
] | [
"At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eae62cce8ed0581acdd40587497d91d1cb92373e908972ae7ebb022f3350bd94 | At the ends of a finite line segment, the number 1 is written. In each subsequent step, the sum of each pair of neighboring numbers is written between them. After 2005 steps, how many times does the number 2005 appear? | [
"0"
] | [
"At the ends of a finite line segment, the number 1 is written. In each subsequent step, the sum of each pair of neighboring numbers is written between them. After 2005 steps, how many times does the number 2005 appear?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-f9bdf99ba7a6c38ef9b0c1a5919ba1afd939d8985dea12976853b9f5b23a08dc | At the faculty of mathematics, there are 40 boys and 10 girls. Each girl is acquainted with all boys who are older than her or taller than her. Let $N$ be the number of pairs of boys such that the sets of girls acquainted with each of them are the same. Find the minimum possible value of $N$. | [
"0"
] | [
"At the faculty of mathematics, there are 40 boys and 10 girls. Each girl is acquainted with all boys who are older than her or taller than her. Let $N$ be the number of pairs of boys such that the sets of girls acquainted with each of them are the same. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-f0e1afd93c060d2f01d22ed5a222a4d4be041682ea6ccc695ec04e7128c8ed69 | At the first stop, no passengers will get off the bus but there are eight other persons waiting to board the same bus. Among those waiting are three friends who must sit together. Determine the number of possible groups of five of the waiting passengers that can join the bus. | [
"11"
] | [
"At the first stop, no passengers will get off the bus but there are eight other persons waiting to board the same bus. Among those waiting are three friends who must sit together. Determine the number of possible groups of five of the waiting passengers that can join the bus."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-11be81e6b85a730abf499fae1f44a6db9e2bc2a3a8183ab29919ade834fb23b9 | At the grocery store, I bought $4$ different items. I brought $3$ identical bags, and handed them to the cashier. How many ways are there for the cashier to put the items I bought in the $3$ identical bags, assuming he might leave some of the bags empty? | [
"14"
] | [
"At the grocery store, I bought $4$ different items. I brought $3$ identical bags, and handed them to the cashier. How many ways are there for the cashier to put the items I bought in the $3$ identical bags, assuming he might leave some of the bags empty?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bbf915c647dcbd5ae7dbdb10399f0b2a5c21b7e59d1beca794e5b471564fa6a5 | At the initial moment there are $N$ bacteria in a retort. By the end of each hour the number of bacteria increases by $p\%$ as compared to their amount at the beginning of that hour; in addition, at the beginning of each hour a portion containing $n$ $(n < N)$ bacteria is taken from the retort. In how many hours will t... | [
"More than in $\\log \\left(\\frac{2Np-100n}{Np-100n}\\right)/ \\log \\left(1+\\frac{p}{100}\\right)$ h"
] | [
"At the initial moment there are $N$ bacteria in a retort. By the end of each hour the number of bacteria increases by $p\\%$ as compared to their amount at the beginning of that hour; in addition, at the beginning of each hour a portion containing $n$ $(n < N)$ bacteria is taken from the retort. In how many hours ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25c2530fb9bf97fbc28a7dde04274fae4dbc30d55190b016ccd1bc184f722459 | At the junction of a countably infinite number of roads sits a greyhound. On one of the roads, a hare runs away from the junction. The greyhound's maximal speed is $2v$, and the hare's maximal speed is $v$. The greyhound uses a strategy to catch the hare in a finite amount of time. If the hare initially runs along the ... | [
"\\frac{n}{v}"
] | [
"At the junction of a countably infinite number of roads sits a greyhound. On one of the roads, a hare runs away from the junction. The greyhound's maximal speed is $2v$, and the hare's maximal speed is $v$. The greyhound uses a strategy to catch the hare in a finite amount of time. If the hare initially runs along... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2835c5a34edec4899501de28fbfcdb6a7ca4c873917d3ab579937d8d2fc7558 | At the local Blast Store, there are sufficiently many items with a price of $n.99 for each non-negative integer n. A sales tax of 7.5% is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possible number of items in the purchase. | [
"20"
] | [
"At the local Blast Store, there are sufficiently many items with a price of $n.99 for each non-negative integer n. A sales tax of 7.5% is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possible number of items in the purchase."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6e3537375ebce395ec14d702b541596f641f72acdcf8ee4a50ab1ff2574659b7 | At the mall's food court, Crystal wants to buy a meal consisting of one entree, one drink and one dessert. The table below lists Crystal's favorite foods in the food court. How many distinct possible meals can she buy from these options? \begin{tabular}{ |c | c | c | } \hline \textbf{Entrees} & \textbf{Drinks}&\textbf{... | [
"16"
] | [
"At the mall's food court, Crystal wants to buy a meal consisting of one entree, one drink and one dessert. The table below lists Crystal's favorite foods in the food court. How many distinct possible meals can she buy from these options?\n\n\\begin{tabular}{ |c | c | c | }\n\\hline \\textbf{Entrees} & \\textbf{Dri... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-db16eebac9786c409c30b82e3e4a52ddcdd1fa8703ac4c2d5f13958730364586 | At the moment Dave crosses the finish line in a 100m freestyle competition, he is 15 meters ahead of Ann and 18 meters ahead of Martha. If Ann and Martha continue swimming at their current rates, how many meters behind will Martha be when Ann finishes? Round your answer to two decimal places. | [
"3.53"
] | [
"At the moment Dave crosses the finish line in a 100m freestyle competition, he is 15 meters ahead of Ann and 18 meters ahead of Martha. If Ann and Martha continue swimming at their current rates, how many meters behind will Martha be when Ann finishes? Round your answer to two decimal places."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-4f267d25439c3b6d0121316ac53f44dc55f699e24f1941f72890547ecdb34b51 | At the moment when Mandy crossed the finish line of a 1500-meter race, she was 300 meters ahead of Callie and 420 meters ahead of Kate. If Callie and Kate continue running at their current rates, how many meters ahead of Kate will Callie be when Callie crosses the finish line? | [
"150 meters"
] | [
"At the moment when Mandy crossed the finish line of a 1500-meter race, she was 300 meters ahead of Callie and 420 meters ahead of Kate. If Callie and Kate continue running at their current rates, how many meters ahead of Kate will Callie be when Callie crosses the finish line?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c92e988eb9414a21834f2a501a980310d5a174832f5f9c24427a336fba2c41e8 | At the opening ceremony of the All-Ukrainian Mathematical Olympiad, 100 children from 10 regions of Ukraine, with 10 children from each region, are standing in one row on the stage such that no two children from the same region are standing consecutively. Two neighboring children from different regions can be swapped i... | [
"45"
] | [
"At the opening ceremony of the All-Ukrainian Mathematical Olympiad, 100 children from 10 regions of Ukraine, with 10 children from each region, are standing in one row on the stage such that no two children from the same region are standing consecutively. Two neighboring children from different regions can be swap... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-3e64d765cc279780996403cdb824c66edcdb622f255ebe5d808ddbaf04ff1277 | At the point $(1,3)$, suppose that $f_x = f_y = 0$ and $f_{xx} > 0$, $f_{yy} > 0$, $f_{xy} = 0$. What is the shape of the graph of $f$ near the point $(1,3)$? | [
"The shape of the graph of \\(f\\) near the point \\((1,3)\\) is a bowl shape with \\((1,3)\\) as a local minimum."
] | [
"At the point $(1,3)$, suppose that $f_x = f_y = 0$ and $f_{xx} > 0$, $f_{yy} > 0$, $f_{xy} = 0$. What is the shape of the graph of $f$ near the point $(1,3)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-d6671b5ed3ad0d784076ea5f2808f9fbdf0dbae8006f576f8cd42c042ac4f9d1 | At the point $(1,3)$, suppose that $f_x = f_y = 0$ and $f_{xx} > 0$, $f_{yy} > 0$, $f_{xy} = 0$. What would a possible contour diagram look like near the point $(1,3)$? | [
"Concentric ellipses centered at (1,3) aligned with the coordinate axes"
] | [
"At the point $(1,3)$, suppose that $f_x = f_y = 0$ and $f_{xx} > 0$, $f_{yy} > 0$, $f_{xy} = 0$. What would a possible contour diagram look like near the point $(1,3)$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36f35eaa8fb4050800eccd5f7b4dcbbc4a6dff5d9cb9d88615b8aaf6e3e0a7eb | At the significance level of $\alpha = 5\%$, compare the average number of tomatoes produced when using three types of fertilizer (A, B, and C). The data for the number of tomatoes produced are as follows: - Fertilizer A: 24, 18, 27, 28 - Fertilizer B: 21, 26, 32, 25 - Fertilizer C: 16, 22, 19, 17 Use the ANOVA test to... | [
"\\text{Fail to reject } H_0"
] | [
"At the significance level of $\\alpha = 5\\%$, compare the average number of tomatoes produced when using three types of fertilizer (A, B, and C). The data for the number of tomatoes produced are as follows:\n\n- Fertilizer A: 24, 18, 27, 28\n- Fertilizer B: 21, 26, 32, 25\n- Fertilizer C: 16, 22, 19, 17\n\nUse th... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f2823826589a45ade8a64d44321c7edd5d2768a20fa6f62319f355a3e4a9ed9 | At the start of a game, there are three boxes containing 2008, 2009, and 2010 game pieces. Anja and Bernd play in turns according to the following rule: when it is your turn, select two boxes, empty them, and then distribute the pieces from the third box to the three boxes such that no box may remain empty. If you can ... | [
"Anja has a winning strategy."
] | [
"At the start of a game, there are three boxes containing 2008, 2009, and 2010 game pieces. Anja and Bernd play in turns according to the following rule: when it is your turn, select two boxes, empty them, and then distribute the pieces from the third box to the three boxes such that no box may remain empty. If you... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 11
} | 31 | 0 | 0 |
omr-ws-fc7a8bd923290c16274113abaa25717331e46c8a71cb68f8a920c9b866f5dc74 | At the start of a weekend a player has won the fraction 0.500 of the matches he has played. After playing another four matches, three of which he wins, he has won more than the fraction 0.503 of his matches. What is the largest number of matches he could have won before the weekend? | [
"164"
] | [
"At the start of a weekend a player has won the fraction 0.500 of the matches he has played. After playing another four matches, three of which he wins, he has won more than the fraction 0.503 of his matches. What is the largest number of matches he could have won before the weekend?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-ecfbac88c9d6baf32669cca74771687343dbd8fbafc9d485a168cc33e0ee1191 | At the start of an event, there are 90 people in the audience. Over time, exactly 100 people enter and 100 people leave the auditorium. The largest audience size observed is 100. Find the largest integer \( m \) such that \( 2^m \) divides the number of different possible sequences of entries and exits given this infor... | [
"3"
] | [
"At the start of an event, there are 90 people in the audience. Over time, exactly 100 people enter and 100 people leave the auditorium. The largest audience size observed is 100. Find the largest integer \\( m \\) such that \\( 2^m \\) divides the number of different possible sequences of entries and exits given t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-49b1b6f1b66524592dbd72e093dab4632fba996c3a2ca946147b959ed1d8ba86 | At the start of the PUMaC opening ceremony, there are 90 people in the audience. Every minute, either one person enters or one person leaves. Over the course of the event, exactly 100 people enter and 100 people leave, and the largest audience size observed is 100. Find the largest integer \( m \) such that \( 2^m \) d... | [
"3"
] | [
"At the start of the PUMaC opening ceremony, there are 90 people in the audience. Every minute, either one person enters or one person leaves. Over the course of the event, exactly 100 people enter and 100 people leave, and the largest audience size observed is 100. Find the largest integer \\( m \\) such that \\( ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-5cb316b1d2559e2fdb557b33aff1660b429075c1eeea1a2893092e6532bf701b | At the start of the game, there are $r$ red and $g$ green pieces on the table. Hojoo and Kestutis take turns making moves, with Hojoo starting first. On each turn, a player chooses a color and removes $k$ pieces of that color, where $k$ must be a divisor of the current number of pieces of the other color. The player wh... | [
"The first player can force the victory if the even parts of \\( r \\) and \\( g \\) are not equal; otherwise, the second player can force the victory."
] | [
"At the start of the game, there are $r$ red and $g$ green pieces on the table. Hojoo and Kestutis take turns making moves, with Hojoo starting first. On each turn, a player chooses a color and removes $k$ pieces of that color, where $k$ must be a divisor of the current number of pieces of the other color. The play... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-5f216062c49fb8274363330f6d140eeb0fbdd0a1391e95fb3144ccaf6c6f2b52 | At the surface of a polygon \( P \) with area 13, 10 smaller polygons each with area 6 are placed. Calculate the minimum possible common area shared by any 4 of these smaller polygons. Express your answer as a fraction in simplest form. | [
"\\dfrac{3}{14}"
] | [
"At the surface of a polygon \\( P \\) with area 13, 10 smaller polygons each with area 6 are placed. Calculate the minimum possible common area shared by any 4 of these smaller polygons. Express your answer as a fraction in simplest form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-76c747ec037ec5413b23e793debd1c61e92f9e332cc9b6170b8b26b6bdd6bfca | At the time of writing this question, the Philly Varsity Math League just got its first league results. The first place team had $100\%$, the second place team had $95\%$, the third place team had $90\%$, the fourth place team had $85\%$, the fifth place team had $80\%$, and the sixth place team had $75\%$. If this pat... | [
"21"
] | [
"At the time of writing this question, the Philly Varsity Math League just got its first league results. The first place team had $100\\%$, the second place team had $95\\%$, the third place team had $90\\%$, the fourth place team had $85\\%$, the fifth place team had $80\\%$, and the sixth place team had $75\\%$. ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc7eb5fffd92777341d7c0176d00c8ecca757a95eee9de1c67aea587a7a2edd9 | At the time-point \( t = 0 \), three people are at each corner of a triangle. At each subsequent time-point \( t = 1, 2, \ldots \), each person moves to one of the other two corners chosen at random. Let \( M \) be the waiting time (the number of steps) until the three people meet at the same corner. Calculate the valu... | [
"\\dfrac{153}{2048}"
] | [
"At the time-point \\( t = 0 \\), three people are at each corner of a triangle. At each subsequent time-point \\( t = 1, 2, \\ldots \\), each person moves to one of the other two corners chosen at random. Let \\( M \\) be the waiting time (the number of steps) until the three people meet at the same corner. Calcul... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d83b0e61223ac62278976b1e1205a78d61a7fe070e964237acf9d20eef6660a | At the time-point \( t = 0 \), three people are at each corner of a triangle. At each subsequent time-point \( t = 1, 2, \ldots \), each person moves to one of the other two corners chosen at random. Let \( M \) be the waiting time (the number of steps) until the three people meet at the same corner. Calculate the valu... | [
"\\dfrac{1113}{16384}"
] | [
"At the time-point \\( t = 0 \\), three people are at each corner of a triangle. At each subsequent time-point \\( t = 1, 2, \\ldots \\), each person moves to one of the other two corners chosen at random. Let \\( M \\) be the waiting time (the number of steps) until the three people meet at the same corner. Calcul... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-195ad8baa87391f00bae25037b17a9e8948259d1b500919e6a24fd9733551d04 | At the top of a piece of paper is written a list of distinct natural numbers. To continue the list, you must choose 2 numbers from the existing ones and write in the list the least common multiple of them, provided it isn’t written yet. The list is closed if there are no other solutions left (for example, the list 2, 3... | [
"1023"
] | [
"At the top of a piece of paper is written a list of distinct natural numbers. To continue the list, you must choose 2 numbers from the existing ones and write in the list the least common multiple of them, provided it isn’t written yet. The list is closed if there are no other solutions left (for example, the list... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-86699c96076e6ac0bb71206e20a598c72746df9d7d62d3d47fca3c5860188b3a | At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves where he picks a vertex and replaces the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Let $S$ be the minimum possible su... | [
"1"
] | [
"At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves where he picks a vertex and replaces the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Let $S$ be the minimum possib... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-750ce0995ffa4be2f335c71ecfbfebdabec1097de86a8d693f795b8a40c7d9a8 | At the vertices of a regular hexagon are written six nonnegative integers whose sum is \(2003^{2003}\). Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Let ... | [
"0"
] | [
"At the vertices of a regular hexagon are written six nonnegative integers whose sum is \\(2003^{2003}\\). Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertice... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-835c33d5a55c99ba39cd9c1fd2ec9cb2140a58e623fc5a4878873c16da09efac | At the vertices of a regular octagon are written eight nonnegative integers whose sum is odd. Peter is allowed to make moves where he picks a vertex and replaces the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Let $S$ be the sum of the number... | [
"1"
] | [
"At the vertices of a regular octagon are written eight nonnegative integers whose sum is odd. Peter is allowed to make moves where he picks a vertex and replaces the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Let $S$ be the sum of the n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-3f3ded52054b2a2c1062a9daa3c3e2db229e5f5809dc27a0ec5781d0cdea59b9 | At time $0$, a particle is at the point $0$ on the real line. At time $1$, the particle divides into two, with one moving $1$ unit to the left and the other moving $1$ unit to the right. At each subsequent time $t$, each particle divides into two, with one moving $1$ unit to the left and the other moving $1$ unit to th... | [
"2"
] | [
"At time $0$, a particle is at the point $0$ on the real line. At time $1$, the particle divides into two, with one moving $1$ unit to the left and the other moving $1$ unit to the right. At each subsequent time $t$, each particle divides into two, with one moving $1$ unit to the left and the other moving $1$ unit ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac1f50363405789fe9700d2d86c9dd838dc2e7b5808012e89072747d93562087 | At time $0$, a particle is at the point $0$ on the real line. At time $1$, the particle divides into two, with one moving $1$ unit to the left and the other moving $1$ unit to the right. At time $2$, each of these particles divides into two, with one of the new particles moving $1$ unit to the left and the other moving... | [
"4"
] | [
"At time $0$, a particle is at the point $0$ on the real line. At time $1$, the particle divides into two, with one moving $1$ unit to the left and the other moving $1$ unit to the right. At time $2$, each of these particles divides into two, with one of the new particles moving $1$ unit to the left and the other m... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88b8c141ab282c28ac8001452a369f172d1ff8ab568de9084599345aaf33d448 | At time $T$ am, a mirror image of an analog clock shows a time $X$ am that is 5 hours and 28 minutes later than $T$. Compute $T$ in terms of hours and minutes. | [
"3:16"
] | [
"At time $T$ am, a mirror image of an analog clock shows a time $X$ am that is 5 hours and 28 minutes later than $T$. Compute $T$ in terms of hours and minutes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-86b627f051867b2db30d292672558d246056a3e8e15a670aa28dddfd0b963085 | At time $t = 0$, a particle starts on the number line at position $0$. At any time, the particle has a $\frac{1}{2}$ probability of moving to the left by one unit and a $\frac{1}{2}$ probability of moving to the right by one unit. What is the expected value of the first time where the particle reaches position $n$? | [
"\\(\\infty\\)"
] | [
"At time $t = 0$, a particle starts on the number line at position $0$. At any time, the particle has a $\\frac{1}{2}$ probability of moving to the left by one unit and a $\\frac{1}{2}$ probability of moving to the right by one unit. What is the expected value of the first time where the particle reaches position $... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-2a34c072a6b43c51b24a539207ad4cf0c94f15eb11ffae7c340b6a16821c799d | At time $t$ seconds, the radius of a spherical balloon is $r$ cm. The balloon is being inflated so that the rate of increase of its radius is inversely proportional to the square root of its radius. When $t=5$, $r=9$ and, at this instant, the radius is increasing at $1.08$ cm $s^{-1}$. How much air is in the balloon in... | [
"\\( 30.5 \\text{ cm}^3 \\)"
] | [
"At time $t$ seconds, the radius of a spherical balloon is $r$ cm. The balloon is being inflated so that the rate of increase of its radius is inversely proportional to the square root of its radius. When $t=5$, $r=9$ and, at this instant, the radius is increasing at $1.08$ cm $s^{-1}$. How much air is in the ballo... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 15
} | 16 | 0 | 0 |
omr-ws-6641abd396465ac97bfb7ab6e1bccb405ab4998b94fba01017bc027ae58de8a8 | At time $t$ seconds, the radius of a spherical balloon is $r$ cm. The balloon is being inflated so that the rate of increase of its radius is inversely proportional to the square root of its radius. When $t=5$, $r=9$ and, at this instant, the radius is increasing at $1.08$ cm $s^{-1}$. Write down a differential equatio... | [
"\\( r = (4.86t + 2.7)^{\\frac{2}{3}} \\)"
] | [
"At time $t$ seconds, the radius of a spherical balloon is $r$ cm. The balloon is being inflated so that the rate of increase of its radius is inversely proportional to the square root of its radius. When $t=5$, $r=9$ and, at this instant, the radius is increasing at $1.08$ cm $s^{-1}$. Write down a differential eq... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 28
} | 41 | 0 | 0 |
omr-ws-95a5a5ec07f0ca42f503e72ae5a8598de9413194aa68a086b2cc204bbf93718c | At time $t=0,$ a ball is thrown downward at 24 feet per second from a height of 160 feet above the ground. The equation $h = -16t^2 - 24t +160$ describes the height (in feet) of the ball. In how many seconds will the ball hit the ground? Express your answer as a decimal. | [
"2.5"
] | [
"At time $t=0,$ a ball is thrown downward at 24 feet per second from a height of 160 feet above the ground. The equation $h = -16t^2 - 24t +160$ describes the height (in feet) of the ball. In how many seconds will the ball hit the ground? Express your answer as a decimal."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-686392c69775a1853d0ee96caf391cb23864e2f2d619718d819452cdea0ffde6 | At time $t=0,$ point $A$ starts at $(3,0)$ and moves around the circle with equation $x^2+y^2=9,$ counterclockwise, at a constant rate of $1$ revolution every $10$ seconds. At the same time, point $B$ starts at $(5,0)$ and moves around the circle with equation $x^2+y^2=25,$ counterclockwise, at a constant rate of $1$ r... | [
"\\(\\frac{20}{3}\\)"
] | [
"At time $t=0,$ point $A$ starts at $(3,0)$ and moves around the circle with equation $x^2+y^2=9,$ counterclockwise, at a constant rate of $1$ revolution every $10$ seconds. At the same time, point $B$ starts at $(5,0)$ and moves around the circle with equation $x^2+y^2=25,$ counterclockwise, at a constant rate of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-db1d178459454f28be55a6c180570f154259636ee52746464384db8b96b86610 | At time 0, a particle is at the point 0 on the real line. At time 1, the particle divides into two, with one moving 1 unit to the left and the other moving 1 unit to the right. At each subsequent time step, each particle divides into two, with one moving 1 unit to the left and the other moving 1 unit to the right. When... | [
"4"
] | [
"At time 0, a particle is at the point 0 on the real line. At time 1, the particle divides into two, with one moving 1 unit to the left and the other moving 1 unit to the right. At each subsequent time step, each particle divides into two, with one moving 1 unit to the left and the other moving 1 unit to the right.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ffdb5dd0587041745dfe98b4e34155117896b4569ac0b29f003dbf32fcf72c10 | At time 0, a particle is at the point 0 on the real line. At time 1, the particle divides into two, with one moving 1 unit to the left and the other moving 1 unit to the right. At each subsequent time, each particle divides into two, with one moving 1 unit to the left and the other moving 1 unit to the right. Whenever ... | [
"4"
] | [
"At time 0, a particle is at the point 0 on the real line. At time 1, the particle divides into two, with one moving 1 unit to the left and the other moving 1 unit to the right. At each subsequent time, each particle divides into two, with one moving 1 unit to the left and the other moving 1 unit to the right. When... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-bd7893c48acc98779fead28561602daeb5a6668490efc1b338331c233489bc28 | At time 0, a particle is at the point 0 on the real line. At time 1, the particle divides into two, with one particle moving one unit to the left and the other moving one unit to the right. At each subsequent time, each particle divides into two, with one moving one unit to the left and the other moving one unit to the... | [
"4"
] | [
"At time 0, a particle is at the point 0 on the real line. At time 1, the particle divides into two, with one particle moving one unit to the left and the other moving one unit to the right. At each subsequent time, each particle divides into two, with one moving one unit to the left and the other moving one unit t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e261ec6747c644eefce39e22e1933be78671b05703c00126cd8916bdc67d5367 | At time = 0 minutes, there are 2 red, 2 green, and 2 blue balls in the bin. What is the expected number of minutes before the configuration of the color of the balls is 2 red, 2 green, and 2 blue again? | [
"\\(\\frac{81}{10}\\)"
] | [
"At time = 0 minutes, there are 2 red, 2 green, and 2 blue balls in the bin. What is the expected number of minutes before the configuration of the color of the balls is 2 red, 2 green, and 2 blue again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-e04cc13210dcc7597535a1e1ae66168e1095230a15c0db1e730cf48ade425646 | At time \( n = 0 \), two ladybirds are placed at vertices \( i \) and \( j \) of a regular hexagon, whose vertices are labeled 1 through 6. At each time \( n = 1, 2, 3, \ldots \), each ladybird moves independently to one of the two adjacent vertices with probability \( \frac{1}{2} \). Denote \( X_n \) as the 'distance'... | [
"\\[\n\\begin{pmatrix}\n1/2 & 0 & 1/2 & 0\\\\\n0 & 3/4 & 0 &1/4\\\\\n1/4&0&3/4&0\\\\\n0&1/2&0&1/2\\\\\n\\end{pmatrix}\n\\]"
] | [
"At time \\( n = 0 \\), two ladybirds are placed at vertices \\( i \\) and \\( j \\) of a regular hexagon, whose vertices are labeled 1 through 6. At each time \\( n = 1, 2, 3, \\ldots \\), each ladybird moves independently to one of the two adjacent vertices with probability \\( \\frac{1}{2} \\). Denote \\( X_n \\... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-2ad1f9b641dc06dd1aa8dc157451debd66d168ddf9b0a03c09b49912b586cabc | At time \( t = 0 \), a lion \( L \) is standing at point \( O \) and a horse \( H \) is at point \( A \) running with speed \( v \) perpendicular to \( OA \). The speed and direction of the horse do not change. The lion's strategy is to run with constant speed \( u \) at an angle \( 0 < \varphi < \pi/2 \) to the line \... | [
"\\( \\frac{v}{u} < 1 \\)"
] | [
"At time \\( t = 0 \\), a lion \\( L \\) is standing at point \\( O \\) and a horse \\( H \\) is at point \\( A \\) running with speed \\( v \\) perpendicular to \\( OA \\). The speed and direction of the horse do not change. The lion's strategy is to run with constant speed \\( u \\) at an angle \\( 0 < \\varphi <... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-6cd00c51603efb965eb4a09e48ac329300552b64206ffc9b55c59a3bbd554bd0 | At time \( t = 0 \), a navy ship is at point \( O \), and an enemy ship is at point \( A \) cruising with speed \( v \) perpendicular to \( OA = a \). The speed and direction of the enemy ship do not change. The navy ship travels with constant speed \( u \) at an angle \( 0 < \phi < \pi / 2 \) to the line \( OA \). Wha... | [
" v = u \\sin \\phi "
] | [
"At time \\( t = 0 \\), a navy ship is at point \\( O \\), and an enemy ship is at point \\( A \\) cruising with speed \\( v \\) perpendicular to \\( OA = a \\). The speed and direction of the enemy ship do not change. The navy ship travels with constant speed \\( u \\) at an angle \\( 0 < \\phi < \\pi / 2 \\) to t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-10fc71ee77ed0ec0b55f709075edb8fa87bbf4bdaaf3fcdbc7dd00790b0a9e4c | At time \( t = 0 \), a particle is at point 0 on the real number line. At time \( t = 1 \), the particle divides into two, with one moving one unit to the left and the other moving one unit to the right. At each subsequent time \( t = 2, 3, \ldots \), each particle divides into two, with one moving one unit to the left... | [
"2"
] | [
"At time \\( t = 0 \\), a particle is at point 0 on the real number line. At time \\( t = 1 \\), the particle divides into two, with one moving one unit to the left and the other moving one unit to the right. At each subsequent time \\( t = 2, 3, \\ldots \\), each particle divides into two, with one moving one unit... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ddfae77acbad7db348471d9bb2905c2010319e9d7f4b04d8fe391597bf302b5 | At time \( t = 0 \), a particle is located at the point \( (1, 2) \). It travels in a straight line to the point \( (4, 1) \), has speed 2 at \( (1, 2) \), and constant acceleration \( 3\mathbf{i} - \mathbf{j} \). Find an equation for the position vector \( \mathbf{r}(t) \) of the particle at time \( t \). | [
"\\( \\mathbf{r}(t) = \\left( \\frac{3t^2}{2} + 2t + 1 \\right) \\mathbf{i} + \\left( -\\frac{t^2}{2} + 2 \\right) \\mathbf{j} \\)"
] | [
"At time \\( t = 0 \\), a particle is located at the point \\( (1, 2) \\). It travels in a straight line to the point \\( (4, 1) \\), has speed 2 at \\( (1, 2) \\), and constant acceleration \\( 3\\mathbf{i} - \\mathbf{j} \\). Find an equation for the position vector \\( \\mathbf{r}(t) \\) of the particle at time \... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4ccbdee7eebb36fd850c2563cd5fec3171894c3b424f092237f786537c40ad51 | At time \( t = 0 \), a particle is located at the point \((1, 2)\). It travels in a straight line to the point \((4, 1)\), has speed 2 at \((1, 2)\), and constant acceleration \( 3\mathbf{i} - \mathbf{j} \). Find an equation for the position vector \(\mathbf{r}(t)\) of the particle at time \( t \). | [
"\\mathbf{r}(t) = \\left(1 + \\frac{3\\sqrt{10}}{5}t + \\frac{3}{2}t^2\\right)\\mathbf{i} + \\left(2 - \\frac{\\sqrt{10}}{5}t - \\frac{1}{2}t^2\\right)\\mathbf{j}"
] | [
"At time \\( t = 0 \\), a particle is located at the point \\((1, 2)\\). It travels in a straight line to the point \\((4, 1)\\), has speed 2 at \\((1, 2)\\), and constant acceleration \\( 3\\mathbf{i} - \\mathbf{j} \\). Find an equation for the position vector \\(\\mathbf{r}(t)\\) of the particle at time \\( t \\)... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34dbfe8cbfa6380a1d24c463093ce8dd33f6a829ce95ae14b36ce23d649931ae | At time \( t = 0 \), a tank contains \( Q_0 = 1800 \) grams of salt dissolved in 300 liters of water. Water containing 5 grams of salt per liter is entering the tank at a rate of 2 liters per minute, and the well-stirred mixture is draining from the tank at a rate of 3 liters per minute. Set up the initial value proble... | [
"\\( Q(t) = 5(300 - t) + \\frac{(300 - t)^3}{300^2} \\)"
] | [
"At time \\( t = 0 \\), a tank contains \\( Q_0 = 1800 \\) grams of salt dissolved in 300 liters of water. Water containing 5 grams of salt per liter is entering the tank at a rate of 2 liters per minute, and the well-stirred mixture is draining from the tank at a rate of 3 liters per minute. Set up the initial val... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-99d0f7e1edf1c58d77c9746483d8e12749bb5f917538a722d7532cdab9e9f987 | At time \( t = 0 \), an angry peasant arises from each of the 90 cottages. Every 8 minutes (480 seconds) thereafter, another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all t... | [
"1920"
] | [
"At time \\( t = 0 \\), an angry peasant arises from each of the 90 cottages. Every 8 minutes (480 seconds) thereafter, another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages unti... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f19faaf65a4baa00bc39512e4798d5eaef6e70ce24c90d8f70768d98fb345d6e | At time \( t \), a particle is moving vertically downwards with speed \( v \) in a medium which exerts a resistance to the motion proportional to the square of the speed of the particle. If its terminal velocity is \( V \), find the equation of the motion of the particle in terms of \( \frac{dv}{dt} \), \( g \), \( v^2... | [
"\\frac{dv}{dt} = g\\left(1 - \\frac{v^2}{V^2}\\right)"
] | [
"At time \\( t \\), a particle is moving vertically downwards with speed \\( v \\) in a medium which exerts a resistance to the motion proportional to the square of the speed of the particle. If its terminal velocity is \\( V \\), find the equation of the motion of the particle in terms of \\( \\frac{dv}{dt} \\), \... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-57f1facfc38d48b1baa4f132436ce1318baa6e41893e49aa52afd6ee2d1bb67b | At time \( t=0 \), an ant is at \((1,0)\) and a spider is at \((-1,0)\). The ant starts walking counterclockwise along the unit circle, and the spider starts moving to the right along the x-axis. The ant's horizontal speed is always half of the spider's. What is the shortest distance ever between the ant and the spider... | [
"\\(\\frac{\\sqrt{56}}{9}\\)"
] | [
"At time \\( t=0 \\), an ant is at \\((1,0)\\) and a spider is at \\((-1,0)\\). The ant starts walking counterclockwise along the unit circle, and the spider starts moving to the right along the x-axis. The ant's horizontal speed is always half of the spider's. What is the shortest distance ever between the ant and... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-0bf6aac002dfb44fadbdaea2dc11d56c04f99cacb9a72cb3f4ceddac06e80466 | At time \( t=0 \), four insects A, B, C, and D stand at the corners of a square with side length \( a \). For \( t > 0 \), each insect crawls with constant speed \( v \) in the direction of the next insect in cyclic order (A towards B, B towards C, etc.). Calculate the total distance traveled by insect A before the ins... | [
"a"
] | [
"At time \\( t=0 \\), four insects A, B, C, and D stand at the corners of a square with side length \\( a \\). For \\( t > 0 \\), each insect crawls with constant speed \\( v \\) in the direction of the next insect in cyclic order (A towards B, B towards C, etc.). Calculate the total distance traveled by insect A b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e8d687345e198f181f20ce6bb56bbb8ed461dab6f5bba0e01d78e6dcebf5a79 | At time zero, an ant is placed on every vertex of a unit cube and starts moving across the edges at a speed of one unit per minute. Each ant alternates between turning right and left at every vertex it reaches (with the first turn being random). If two or more ants meet, they die. After three minutes, if at least one a... | [
"6"
] | [
"At time zero, an ant is placed on every vertex of a unit cube and starts moving across the edges at a speed of one unit per minute. Each ant alternates between turning right and left at every vertex it reaches (with the first turn being random). If two or more ants meet, they die. After three minutes, if at least ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-864d46119e1efabd12312459df52f03b3faa9d8df056a5229346d9c55ca68e25 | At what angle $\theta$ does opening both doors of a double door set to the same angle $\theta$ create a wider opening than opening a single door to that same angle $\theta$? The opening is defined by the distance between the two interior/center edges of the double doors. | [
"\\(\\theta \\ge 60^\\circ\\)"
] | [
"At what angle $\\theta$ does opening both doors of a double door set to the same angle $\\theta$ create a wider opening than opening a single door to that same angle $\\theta$? The opening is defined by the distance between the two interior/center edges of the double doors."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 3
} | 14 | 0 | 0 |
omr-ws-e39991a361f45db0fc2289ba8bf46f958ae8d4a73a2f281b9e90f230ccd43419 | At what angle above an inclined ramp (inclined at an angle $\theta$ above the ground) should Ricky kick a soccer ball to maximize the distance it travels up the ramp? (Assume that air resistance is negligible.) | [
"\\frac{\\pi}{4} - \\frac{\\theta}{2}"
] | [
"At what angle above an inclined ramp (inclined at an angle $\\theta$ above the ground) should Ricky kick a soccer ball to maximize the distance it travels up the ramp? (Assume that air resistance is negligible.)"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d0fb43400b7a6defebb353ca2eb1f36582ae6d7fe7ef22ca7c21e4598bac7e4e | At what angle can the circle $x^2+y^2=16$ be seen from the point $(+8,0)$? | [
"$\\varphi =\\pi /3$"
] | [
"At what angle can the circle $x^2+y^2=16$ be seen from the point $(+8,0)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08ef944b09f370591444e9785492a226e9dea80ab951c1146aeb21fa990b99fb | At what angle is the tangent inclined to the $Ox$ axis if it is drawn to the curve $y=2x^3-x$ at the point of intersection of that curve with the $Oy$ axis? | [
"$3\\pi /4$. {\\bf 6.19.**} {\\it Solution.} We differentiate each equation considering $y$ to be a function of $x$. We have $y+y'x=0$ and $2x-2yy'=0$. Finding the expression for $y'$ in each equation, we have $y'=-y/x$, $y'=x/y$, respectively. Consequently, at any point $M(x_0,y_0)$ which is a point of intersectio... | [
"At what angle is the tangent inclined to the $Ox$ axis if it is drawn to the curve $y=2x^3-x$ at the point of intersection of that curve with the $Oy$ axis?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1944f7949b84ff0bb7aa67af4cb0a99d8b66a287be178985551074395b636b78 | At what angle must a cylinder with radius $4$ cm and height $15$ cm be tilted so that it will just fit inside a $12$ cm high box? | [
"\\(\\arcsin \\left(\\frac{180 - 8 \\sqrt{145}}{289}\\right)\\) or approximately \\(16.828^\\circ\\)"
] | [
"At what angle must a cylinder with radius $4$ cm and height $15$ cm be tilted so that it will just fit inside a $12$ cm high box?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3acc3353e9e3056513f181ab8146ed3f19ccec43a0fc7c354bfac4308b4b87b6 | At what exact time after 1:00 o'clock will the hour and minute hands of a clock be in the same position? | [
"\\(1:05\\frac{5}{11}\\)"
] | [
"At what exact time after 1:00 o'clock will the hour and minute hands of a clock be in the same position?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-c6885b409a581de943ba1cbdd112d40ddbd021a404a76dc5ba9582ee85c69dfd | At what fraction of an hour after 3 o'clock are the minute and hour hands of a twelve-hour clock pointing in the same direction? | [
"\\(\\frac{3}{11}\\)"
] | [
"At what fraction of an hour after 3 o'clock are the minute and hour hands of a twelve-hour clock pointing in the same direction?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-aabdca2d8bde8ab13cbbb00b0c8e55598e02aca803aa27c7286d55bec7b237d2 | At what index does the harmonic series have a fractional part of $1/12$? | [
"4"
] | [
"At what index does the harmonic series have a fractional part of $1/12$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fadae13d5e398cca62c4768a8d12587711f769855d904deadb613ad73d16ca0f | At what interest rate, compounded semi-annually, would Elisabeth need to invest 500 pounds for 8 years to double the value of her investment? Use the equation \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the final amount, \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) i... | [
"\\( r \\approx 0.08854 \\)"
] | [
"At what interest rate, compounded semi-annually, would Elisabeth need to invest 500 pounds for 8 years to double the value of her investment? Use the equation \\( A = P \\left(1 + \\frac{r}{n}\\right)^{nt} \\), where \\( A \\) is the final amount, \\( P \\) is the principal amount, \\( r \\) is the annual interest... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d21ab400fe57627a5afc07752bcc17e6d1559693237ec8a9c42418fa8aa0882d | At what minimum value of $k$ is the statement true: any set of $100$ weights that Bambula could previously carry in $50$ runs (with each run carrying up to three weights totaling no more than $200$ kg) can now be carried in no more than $k$ runs, given that he can now carry no more than two weights per run (still total... | [
"50"
] | [
"At what minimum value of $k$ is the statement true: any set of $100$ weights that Bambula could previously carry in $50$ runs (with each run carrying up to three weights totaling no more than $200$ kg) can now be carried in no more than $k$ runs, given that he can now carry no more than two weights per run (still ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-32de78ef12844c8595804fd25f7a547199c109070c666386d18cb69caf4a0cff | At what numerical values of $a$ does the equation $$x^2+2ax\sqrt {a^2-3}+4=0$$ have equal roots? {\bf 180a.} In what interval must the number $m$ vary so that both roots of the equation $$x^2-2mx+m^2-1=0$$ lie between $-2$ and 4? | [
"$-1<m<3$"
] | [
"At what numerical values of $a$ does the equation $$x^2+2ax\\sqrt {a^2-3}+4=0$$ have equal roots? {\\bf 180a.} In what interval must the number $m$ vary so that both roots of the equation $$x^2-2mx+m^2-1=0$$ lie between $-2$ and 4? "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5b297e049ff40c1ff20fccceeccfeb5c4785cd78950fa1d988c8920cc901b98 | At what point $M_0$ of the curve $y=\sqrt 2x^{3/2}$ is the tangent perpendicular to the straight line $4x + 3y + 2 = 0$? | [
"$(1/8, 1/16)$"
] | [
"At what point $M_0$ of the curve $y=\\sqrt 2x^{3/2}$ is the tangent perpendicular to the straight line $4x + 3y + 2 = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-441b624ad40cf161f7381217333564d1eaf2d3566f403d3988d58364c8664154 | At what point does the curve defined by $x=1-2\cos^2 t$ and $y=(\tan t)(1-2\cos^2 t)$ cross itself? Find the equations of both tangents at that point. | [
"\\((0,0)\\), \\(y = x\\), \\(y = -x\\)"
] | [
"At what point does the curve defined by $x=1-2\\cos^2 t$ and $y=(\\tan t)(1-2\\cos^2 t)$ cross itself? Find the equations of both tangents at that point."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-2bc0c7033c078a6c209f8b9f9e98aec52a4991ae4a5ab39e4966f23753588683 | At what point does the curve defined by \( x = 1 - 2\cos^2 t \) and \( y = x \tan t \) cross itself? Find the equations of both tangents at that point. | [
"\\((0,0)\\)"
] | [
"At what point does the curve defined by \\( x = 1 - 2\\cos^2 t \\) and \\( y = x \\tan t \\) cross itself? Find the equations of both tangents at that point."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b16aab3e95f2804d6058e3442aa66722bc7c3554254a7f38cabdf7421870b7b1 | At what point does the line $3y-4x=12$ intersect the $x$-axis? Express your answer as an ordered pair. | [
"(-3,0)"
] | [
"At what point does the line $3y-4x=12$ intersect the $x$-axis? Express your answer as an ordered pair."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-486237e693d746e41b90c3d4a00fec5a2f361069872d34ded1303a0c4d3b468b | At what point does the line containing the points $(1, 7)$ and $(3, 11)$ intersect the $y$-axis? Express your answer as an ordered pair. | [
"(0,5)"
] | [
"At what point does the line containing the points $(1, 7)$ and $(3, 11)$ intersect the $y$-axis? Express your answer as an ordered pair."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c08c13a1801ff76a9b2e5b069d243b749ed63024fe7b1b45c9b98a56a02f6bc8 | At what point of the curve $y=ax^2+bx+c$ should a tangent line to that curve be drawn for the tangent to pass through the origin? Find the values of $a$, $b$ and $c$ for which the problem has a solution. | [
"$\\left(\\sqrt {\\frac{c}{a}},b\\sqrt {\\frac{c}{a}}+2c\\right)$, $\\left(-\\sqrt {\\frac{c}{a}},2c-b\\sqrt {\\frac{c}{a}}\\right)$ for $ac>0$, $(0,0)$ for $c=0$; for $ac<0$ there is no solution"
] | [
"At what point of the curve $y=ax^2+bx+c$ should a tangent line to that curve be drawn for the tangent to pass through the origin? Find the values of $a$, $b$ and $c$ for which the problem has a solution."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0471761babd9559426145212b2ee574994ffb4633020bb9853fa4517e28f593 | At what point of the curve $y=x^2-5x+6$ should a tangent line be drawn for it to pass through the point $M (a, b)$? Find the values of $a$ and $b$ for which the problem has a solution. | [
"$\\left(a\\!+\\!\\sqrt {a^2-(5a+b-6)};2a^2\\!+\\!\\sqrt {a^2\\!-\\!(5a+b-6)}(2a-5) \\!-\\!10a\\!-\\!b\\right)$, $\\left(a-\\sqrt {a^2-(5a+b-6)};2a^2-\\sqrt {a^2-(5a+b-6)}(2a-5)-10a-b\\right)$,\\\\ if $a^2-(5a+b-6)>0$; $(a,2a^2-10a-b)$, if $a^2-(5a+b-6)=0$;\\\\ now if $a^2-(5a+b-6)<0$, then there is no solution"
] | [
"At what point of the curve $y=x^2-5x+6$ should a tangent line be drawn for it to pass through the point $M (a, b)$? Find the values of $a$ and $b$ for which the problem has a solution."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7487f320fbcaaa7153da325b1591edcce59225aca0b3a33b01e9112de49eafd2 | At what point on the graph of \( y = 0.5x^2 \) is the tangent line parallel to the line \( 2x - 4y = 3 \)? | [
"\\(\\left( \\frac{1}{2}, \\frac{1}{8} \\right)\\)"
] | [
"At what point on the graph of \\( y = 0.5x^2 \\) is the tangent line parallel to the line \\( 2x - 4y = 3 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-cdc838476d030e9f55a504b031f2146e46b71f3c147cbc73e771fda39745dd5f | At what points does the function \( f(x) = \left| \sin x \right| \) have no derivatives? | [
"\\( x = k\\pi \\) for \\( k \\in \\mathbb{Z} \\)"
] | [
"At what points does the function \\( f(x) = \\left| \\sin x \\right| \\) have no derivatives?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
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