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-3e1d03b87fc889ea04ec9133b211d85a76372c57cf841bfc7d95b5ff372bebd9 | A gambler wagers an amount \( x \) of money in a game with a 50% chance of winning, where winning results in a payout of \( 2x \). If he loses, he doubles his next wager. Calculate the expected value of the gambler's earnings using this strategy. | [
"x"
] | [
"A gambler wagers an amount \\( x \\) of money in a game with a 50% chance of winning, where winning results in a payout of \\( 2x \\). If he loses, he doubles his next wager. Calculate the expected value of the gambler's earnings using this strategy."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-01aba75d8e8a9a03dd5e436fb8c6ed14e7a785a6d08b1c0f4bddd0705baea486 | A gambler went to the casino with 1 dollar and made a series of six 1-dollar bets, winning or losing 1 dollar each time. She ended up with 1 dollar. In how many different orders (of winning/losing) could this have happened? Note that if you have no money you cannot bet. | [
"5"
] | [
"A gambler went to the casino with 1 dollar and made a series of six 1-dollar bets, winning or losing 1 dollar each time. She ended up with 1 dollar. In how many different orders (of winning/losing) could this have happened? Note that if you have no money you cannot bet."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 2
} | 25 | 0 | 0 |
omr-ws-4f2bcbe72cbd1a7043e8fd9c77dcf2113e47360993eaf4b31da4b395f91424f0 | A gambling game costs $1 to play. The payout is based on the outcome of rolling 2 dice: - If the sum of the dice is a perfect square, you win $x. - If the sum of the dice is 2, 6, 8, or 10, you win $10. - Otherwise, you lose $5. If the organizers make an average profit of $3 per game, calculate the value of x. | [
"\\(-\\frac{137}{7}\\)"
] | [
"A gambling game costs $1 to play. The payout is based on the outcome of rolling 2 dice:\n- If the sum of the dice is a perfect square, you win $x.\n- If the sum of the dice is 2, 6, 8, or 10, you win $10.\n- Otherwise, you lose $5.\nIf the organizers make an average profit of $3 per game, calculate the value of x.... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 8
} | 30 | 0 | 0 |
omr-ws-292f931366767efd9e45d547dc2746d9801a4d23190b33f27dffcfeb8aef8699 | A game board consists of 64 squares that alternate in color between black and white. A marker is placed at square $P$ in the bottom row. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7-step paths are there from $P$ to square $Q$ in the top row? | [
"35"
] | [
"A game board consists of 64 squares that alternate in color between black and white. A marker is placed at square $P$ in the bottom row. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7-step paths are there from $P$ to square $Q$ in the top row?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-f87827ad4202c794966313797fd5ea8efd20a945160ace6d6174e401164f716c | A game board consists of 64 squares that alternate in color between black and white. The figure shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P$. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7-step paths are there from... | [
"28"
] | [
"A game board consists of 64 squares that alternate in color between black and white. The figure shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P$. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7-step paths are there... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-21f2d63b806e457003bc7b7cbf7a406d7cd31ee21427ea050470479628ae8ead | A game board consists of 64 squares that alternate in color between black and white. The figure shows square \( P \) in the bottom row and square \( Q \) in the top row. A marker is placed at \( P \). A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7-step paths ar... | [
"28",
"35"
] | [
"A game board consists of 64 squares that alternate in color between black and white. The figure shows square \\( P \\) in the bottom row and square \\( Q \\) in the top row. A marker is placed at \\( P \\). A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7-st... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-f8dfd18a1b1f011d4290a554d2ecc4c490fb7af52a312b0d05de69f2c0b9dd80 | A game board consists of a \(2000 \times 2000\) grid of unit squares. One square is painted black and the rest are painted white. A step consists of selecting a white square that has at least two black neighbors (two squares are neighbors if they share an edge), and painting it black. Find the smallest positive integer... | [
"1"
] | [
"A game board consists of a \\(2000 \\times 2000\\) grid of unit squares. One square is painted black and the rest are painted white. A step consists of selecting a white square that has at least two black neighbors (two squares are neighbors if they share an edge), and painting it black. Find the smallest positive... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-c09469e6ebac1ab7a2341d5ac10b1220f173158b8312183f5143afa7bbbae011 | A game involves choosing tiles from a bag containing the letters A, A, E, E, E, I, O, U. Chris chooses two tiles at random from the bag. What is the probability that the two tiles have different letters? | [
"\\(\\frac{6}{7}\\)"
] | [
"A game involves choosing tiles from a bag containing the letters A, A, E, E, E, I, O, U. Chris chooses two tiles at random from the bag. What is the probability that the two tiles have different letters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-7e600ff10bb71d5cd21f9b5690e7e7ea9f059c4061c03d51941ac603ffd280e6 | A game is played by breaking sticks of integer length into two shorter sticks of different integer lengths. The game starts with a stick of length \( N \). Players A and B take turns, with A making the first break. The game ends when all sticks have length 1 or 2. If there are more sticks of length 1 than 2 at the end,... | [
"- For \\( N = 3 \\): First player wins.\n- For \\( N = 4 \\) or \\( N = 5 \\): Second player wins.\n- For \\( N = 6 \\): Draw.\n- For \\( N > 6 \\):\n - If \\( N \\) is even: First player wins.\n - If \\( N \\) is odd: Second player wins."
] | [
"A game is played by breaking sticks of integer length into two shorter sticks of different integer lengths. The game starts with a stick of length \\( N \\). Players A and B take turns, with A making the first break. The game ends when all sticks have length 1 or 2. If there are more sticks of length 1 than 2 at t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-8459ccf9903313a5501cecb9562876466bbc0c5a96097bd786ae6cfe3445e00f | A game is played on a board with an infinite row of holes labelled $0, 1, 2, \dots$. Initially, $2009$ pebbles are put into hole $1$; the other holes are left empty. Steps are performed according to the following scheme: - At each step, two pebbles are removed from one of the holes (if possible), and one pebble is put ... | [
"2001"
] | [
"A game is played on a board with an infinite row of holes labelled $0, 1, 2, \\dots$. Initially, $2009$ pebbles are put into hole $1$; the other holes are left empty. Steps are performed according to the following scheme:\n- At each step, two pebbles are removed from one of the holes (if possible), and one pebble ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e102a2c29621182a68225ea1fa7f9e7dbea34a72b755bab28d693fefe64db886 | A game is played on a rhombus-shaped board divided into 2n rows of equilateral triangles, with the number of triangles in each row forming the sequence 1, 3, 5, ..., 2n-1, 2n-1, ..., 5, 3, 1. Players A and B use red and blue counters respectively, starting on opposite sides of the board and alternating turns, with play... | [
"9"
] | [
"A game is played on a rhombus-shaped board divided into 2n rows of equilateral triangles, with the number of triangles in each row forming the sequence 1, 3, 5, ..., 2n-1, 2n-1, ..., 5, 3, 1. Players A and B use red and blue counters respectively, starting on opposite sides of the board and alternating turns, with... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-dc26490b9af640b20493d09c2686c23182911ad28e2da9b421a6ba48e57541cd | A game is played on an infinite chessboard where an n*n square is covered in chips. A move consists of a chip jumping over another chip onto an empty cell behind it. Find all values of n for which the game ends with one chip left over. | [
"\\( n \\not\\equiv 0 \\pmod{3} \\)"
] | [
"A game is played on an infinite chessboard where an n*n square is covered in chips. A move consists of a chip jumping over another chip onto an empty cell behind it. Find all values of n for which the game ends with one chip left over."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 1
} | 1 | 0 | 0 |
omr-ws-199491bf6e8aa44770cd0ac5b11133e22e009ad8c8f99ed817d34937411ef99b | A game is played on the vertices of the cube $(x, y, z): 0 \le x, y, z \le 1$ as follows. Beginning at $(0, 0, 0)$, the player moves along one of the edges emanating from that vertex to an adjacent vertex according to a roll of a die; if a $1$ or $2$ is rolled, she moves in the $x$-direction; if a $3$ or $4$ is rolled,... | [
"\\dfrac{3}{7}"
] | [
"A game is played on the vertices of the cube $(x, y, z): 0 \\le x, y, z \\le 1$ as follows. Beginning at $(0, 0, 0)$, the player moves along one of the edges emanating from that vertex to an adjacent vertex according to a roll of a die; if a $1$ or $2$ is rolled, she moves in the $x$-direction; if a $3$ or $4$ is ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-3fb3c95fb599acd5cf5f4f45855461de9f63a3cadf3b07b3237d7794cc298a5a | A game is played on the vertices of the cube \(\{(x,y,z) : 0 \leq x, y, z \leq 1\}\). Starting at \((0,0,0)\), the player moves to an adjacent vertex based on a die roll: 1 or 2 moves in the x-direction, 3 or 4 moves in the y-direction, and 5 or 6 moves in the z-direction. The game ends when the player reaches either \... | [
"\\dfrac{3}{7}"
] | [
"A game is played on the vertices of the cube \\(\\{(x,y,z) : 0 \\leq x, y, z \\leq 1\\}\\). Starting at \\((0,0,0)\\), the player moves to an adjacent vertex based on a die roll: 1 or 2 moves in the x-direction, 3 or 4 moves in the y-direction, and 5 or 6 moves in the z-direction. The game ends when the player rea... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-a218356965f73a5c62c1c2763ff6b26822a6b587054d178295cf9cb3961b7041 | A game is played where a standard six-sided die is rolled until a 6 occurs, and the sum of all the rolls up to and including the 6 is taken. What is the probability that this sum is even? | [
"\\(\\frac{4}{7}\\)"
] | [
"A game is played where a standard six-sided die is rolled until a 6 occurs, and the sum of all the rolls up to and including the 6 is taken. What is the probability that this sum is even?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 25
} | 73 | 0 | 0 |
omr-ws-0ceb2c029aa4ba00f401e95dd7cecbc5b6ab3ab9ae83bcbc190965ec597f46c6 | A game is played with 16 cards laid out in a row. Each card has a black side and a red side, and initially the face-up sides of the cards alternate black and red with the leftmost card black-side-up. A move consists of taking a consecutive sequence of at least two cards with the leftmost card black-side-up and the rest... | [
"176"
] | [
"A game is played with 16 cards laid out in a row. Each card has a black side and a red side, and initially the face-up sides of the cards alternate black and red with the leftmost card black-side-up. A move consists of taking a consecutive sequence of at least two cards with the leftmost card black-side-up and the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b878e1b5e40d3df76533ff2db9bb77952f94cf0b1252827aba5503058d70d59 | A game is played with \( n \) players sitting around a table, each with one penny. Players pass pennies in a clockwise fashion: the first player passes one penny, the next passes two pennies, and so on, alternating between one and two pennies. Once a player has no pennies, they are removed from the game. What is the sm... | [
"7"
] | [
"A game is played with \\( n \\) players sitting around a table, each with one penny. Players pass pennies in a clockwise fashion: the first player passes one penny, the next passes two pennies, and so on, alternating between one and two pennies. Once a player has no pennies, they are removed from the game. What is... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-710a77a0a4d1e6c5445ee7f1d9f8e9cb22668d145db257286e54c47fd5467c09 | A game is played with a coin and a six-sided die with faces numbered from 1 to 6. The game starts with a coin flip: if heads, the game ends; if tails, a die is rolled. If the die shows a 1, the game ends; otherwise, the process repeats. What is the expected number of coin flips and die rolls combined until the game end... | [
"\\(\\frac{18}{7}\\)"
] | [
"A game is played with a coin and a six-sided die with faces numbered from 1 to 6. The game starts with a coin flip: if heads, the game ends; if tails, a die is rolled. If the die shows a 1, the game ends; otherwise, the process repeats. What is the expected number of coin flips and die rolls combined until the gam... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-603d11988eccb3bb7310d25497d89ec4aac54dd1be99289132a5b26333861d2d | A game is played with a special fair cubic die which has one red side, two blue sides, and three green sides. The game continues until two consecutive red results are obtained. The player who rolls the second of the consecutive reds wins. If Whip has the first roll and they alternate rolls, find the probability that Wa... | [
"\\(\\frac{36}{71}\\)"
] | [
"A game is played with a special fair cubic die which has one red side, two blue sides, and three green sides. The game continues until two consecutive red results are obtained. The player who rolls the second of the consecutive reds wins. If Whip has the first roll and they alternate rolls, find the probability th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76eaa5bfff45952f93ef09631c6c61be999f93c44a1850328b82460589072c7d | A game machine works according to the following procedure: i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers) without showing them on screen. ii. It shows the product of the seven chosen numbers on screen. iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the... | [
"Disagree"
] | [
"A game machine works according to the following procedure:\ni. It picks 7 numbers randomly from 1 to 9 (these numbers are integers) without showing them on screen.\nii. It shows the product of the seven chosen numbers on screen.\niii. It shows a calculator menu (it does not function as a calculator) on screen and ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b99946559d8a96404706f97631167876f5bfafa0f0227c3905ad24adbfbdbf7 | A game master divides a group of 12 players into two teams of six. Each player receives a card with the names of two other players, one of whom is a teammate and the other is not. Can the master write the names on the cards in such a way that the players can determine the teams by working together? | [
"Yes"
] | [
"A game master divides a group of 12 players into two teams of six. Each player receives a card with the names of two other players, one of whom is a teammate and the other is not. Can the master write the names on the cards in such a way that the players can determine the teams by working together?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f25a529843225c7a89f6cd1df8b9643ced54baba1618a9e90186e6dd69381f4a | A game master divides a group of 40 players into four teams of ten. Each player receives a card with the names of two other players: one teammate and one non-teammate, but the player does not know which is which. Can the master write the names on the cards in such a way that the players can determine the teams by worki... | [
"Yes"
] | [
"A game master divides a group of 40 players into four teams of ten. Each player receives a card with the names of two other players: one teammate and one non-teammate, but the player does not know which is which. Can the master write the names on the cards in such a way that the players can determine the teams by ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8bfa6e5effda3ae2a6bb700467838b932ce10fce662e2daf23b47e130b57705 | A game of mafia is down to 4 players: 1 Mafia: The antagonist. He has one kill per night, and wins if he is either the only player left, or if he is 1 of 2 players left. 1 Vigilante: A protagonist. He has one kill per night, and wins if all the mafia are dead. 1 Townie: A protagonist. He has no special ability. He wins... | [
"\\dfrac{4}{9}"
] | [
"A game of mafia is down to 4 players:\n1 Mafia: The antagonist. He has one kill per night, and wins if he is either the only player left, or if he is 1 of 2 players left.\n1 Vigilante: A protagonist. He has one kill per night, and wins if all the mafia are dead.\n1 Townie: A protagonist. He has no special ability.... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-184e1e747194657b146b2c8883889e12bf4448fcb3de1d9afe75e7b51a54ce99 | A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time and accumulates a penalty based on the following rules: - If a blue card is played, the penalty is the number of white cards still in the player's hand. - If a white card is played, the pena... | [
"\\min\\{BW, 2RW, 3RB\\}"
] | [
"A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time and accumulates a penalty based on the following rules:\n- If a blue card is played, the penalty is the number of white cards still in the player's hand.\n- If a white card is played, t... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-6425543c64599b25f17f3d459a07cab49709d9f6a5b08de66d4a4a8602b74e01 | A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time, accumulating a penalty as follows: - If a blue card is played, the penalty is the number of white cards still in hand. - If a white card is played, the penalty is twice the number of red ca... | [
"\\(\\min(BW, 2WR, 3RB)\\)"
] | [
"A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time, accumulating a penalty as follows:\n- If a blue card is played, the penalty is the number of white cards still in hand.\n- If a white card is played, the penalty is twice the number of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 8,
"tir": 7
} | 36 | 0 | 0 |
omr-ws-a70b22ecb10bd7089319e121166b8a666cd19deddccb273f16035a8fecfc67dc | A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is cha... | [
"Minimum penalty: \\(\\min(BW, 2WR, 3RB)\\)\n\nNumber of optimal strategies:\n- \\(1\\) if \\(BW \\neq 2WR \\neq 3RB\\)\n- \\(W+1\\) if \\(BW = 2WR < 3RB\\)\n- \\(R+1\\) if \\(2WR = 3RB < BW\\)\n- \\(B+1\\) if \\(3RB = BW < 2WR\\)\n- \\(R+B+W\\) if \\(BW = 2WR = 3RB\\)"
] | [
"A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he i... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-a5900220e46280b906c892df52730220554d66ca8e9ad2d3d80b552c547bd89e | A game show is held in which a contestant selects an integer between 1 and 1000, inclusive. Let \( n \) be the number of integers between 1 and 1000, inclusive, that are not relatively prime to the selected number. The contestant wins \( n \) dollars. The maximum number of dollars the contestant could win is \( k \). F... | [
"4"
] | [
"A game show is held in which a contestant selects an integer between 1 and 1000, inclusive. Let \\( n \\) be the number of integers between 1 and 1000, inclusive, that are not relatively prime to the selected number. The contestant wins \\( n \\) dollars. The maximum number of dollars the contestant could win is \... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 1
} | 24 | 0 | 0 |
omr-ws-d187e0db8ca23f0a394f9dddec86157f84f79dd85d0b0d796d2088b16b545e96 | A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the dig... | [
"420"
] | [
"A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, th... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-e106dab58c56b451290afe55c3e15baefc7fccd022fd0014e3d3cf00a89cb615 | A game works as follows: the player pays 2 tokens to enter the game. Then, a fair coin is flipped. If the coin lands on heads, they receive 3 tokens; if the coin lands on tails, they receive nothing. A player starts with 2 tokens and keeps playing this game until they do not have enough tokens to play again. What is th... | [
"\\(\\frac{3 - \\sqrt{5}}{2}\\)"
] | [
"A game works as follows: the player pays 2 tokens to enter the game. Then, a fair coin is flipped. If the coin lands on heads, they receive 3 tokens; if the coin lands on tails, they receive nothing. A player starts with 2 tokens and keeps playing this game until they do not have enough tokens to play again. What ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
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] | {
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} | 40 | 0 | 0 |
omr-ws-6a6119dc5f48d08bd9c5e94b63146f48cc1a6d9b7b03512d095d3ed3e6cd4913 | A garage has dimensions 30 feet by 50 feet by 30 feet. A steel support beam 56 feet in length is positioned in the center of the garage. Another beam will be attached to the top of this center beam to support the roof. What is the angle of elevation of the new beam, or the pitch of the roof? | [
"60^\\circ"
] | [
"A garage has dimensions 30 feet by 50 feet by 30 feet. A steel support beam 56 feet in length is positioned in the center of the garage. Another beam will be attached to the top of this center beam to support the roof. What is the angle of elevation of the new beam, or the pitch of the roof?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-03e96b66e3392eafdcac93cd3a2d2dd94a995db86020e903314cbf2ec2fa270f | A garden urn contains 18 colored beetles: 6 red beetles numbered from 1 to 6, and 12 yellow beetles numbered from 1 to 12. Beetles wander out of the urn in random order, one at a time, without any going back in. What is the probability that the sequence of numbers on the first four beetles to wander out is steadily inc... | [
"\\(\\frac{157}{4896}\\)"
] | [
"A garden urn contains 18 colored beetles: 6 red beetles numbered from 1 to 6, and 12 yellow beetles numbered from 1 to 12. Beetles wander out of the urn in random order, one at a time, without any going back in. What is the probability that the sequence of numbers on the first four beetles to wander out is steadil... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-c49790d9ab2a6db43d892bb75db0930fa97eef0e5b8b52d7bd812ed21d6d4407 | A garden urn contains 18 colored beetles: 6 red beetles numbered from 1 to 6, and 12 yellow beetles numbered from 1 to 12. Beetles wander out of the urn in random order, one at a time, without any going back in. What is the probability that the sequence of numbers on the first four beetles to wander out is steadily inc... | [
"\\(\\frac{157}{4896}\\)"
] | [
"A garden urn contains 18 colored beetles: 6 red beetles numbered from 1 to 6, and 12 yellow beetles numbered from 1 to 12. Beetles wander out of the urn in random order, one at a time, without any going back in. What is the probability that the sequence of numbers on the first four beetles to wander out is steadil... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 42 | 0 | 0 |
omr-ws-0f537f97ee514cd98f58d19c17e8bef7629a58d99897b30f7eb43166af434759 | A garden urn contains 18 colored beetles: 6 red beetles numbered from 1 to 6, and 12 yellow beetles numbered from 1 to 12. Beetles wander out of the urn one at a time without replacement. What is the probability that the sequence of numbers on the first four beetles to wander out is steadily increasing, with no repeate... | [
"\\dfrac{157}{18 \\times 17 \\times 16}"
] | [
"A garden urn contains 18 colored beetles: 6 red beetles numbered from 1 to 6, and 12 yellow beetles numbered from 1 to 12. Beetles wander out of the urn one at a time without replacement. What is the probability that the sequence of numbers on the first four beetles to wander out is steadily increasing, with no re... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-180f4839f27a36501fa9a73609ab629a1f17c36887ce2268f815b333501e0e1f | A gardener plants 3 maple trees, 4 oak trees, and 5 birch trees in a row. He plants them in random order, each arrangement being equally likely. Find the probability that no two birch trees are next to each other, assuming the trees of the same type are distinguishable. | [
"\\dfrac{7}{99}"
] | [
"A gardener plants 3 maple trees, 4 oak trees, and 5 birch trees in a row. He plants them in random order, each arrangement being equally likely. Find the probability that no two birch trees are next to each other, assuming the trees of the same type are distinguishable."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd8edbc447371d62e2e44d721a80e71a5ed79bb5be27a36cd41e84d5882bc9a2 | A gardener plants 3 maple trees, 4 oak trees, and 5 birch trees in a row. He plants them in random order, each arrangement being equally likely. Find the probability that no two birch trees are next to each other, assuming the trees of the same type are indistinguishable. | [
"\\dfrac{7}{99}"
] | [
"A gardener plants 3 maple trees, 4 oak trees, and 5 birch trees in a row. He plants them in random order, each arrangement being equally likely. Find the probability that no two birch trees are next to each other, assuming the trees of the same type are indistinguishable."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-818bca7c3c23be16e873f244c97b384d27af97052110ce869bec6bcee704f408 | A gardener plants three maple trees, four oak trees, and five birch trees in a row in random order. Find the probability that no two birch trees are next to one another. | [
"\\dfrac{7}{99}"
] | [
"A gardener plants three maple trees, four oak trees, and five birch trees in a row in random order. Find the probability that no two birch trees are next to one another."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7f17c344eccd25803a6ad2d3987a117c2d6315233616ed12e20b31e149ff21e0 | A gardener plants three maple trees, four oak trees, and five birch trees in a row. The trees are planted in random order, with each arrangement being equally likely. Find the probability that no two birch trees are next to one another, and express this probability as $\frac{m}{n}$ in lowest terms. Find $m + n$. | [
"106"
] | [
"A gardener plants three maple trees, four oak trees, and five birch trees in a row. The trees are planted in random order, with each arrangement being equally likely. Find the probability that no two birch trees are next to one another, and express this probability as $\\frac{m}{n}$ in lowest terms. Find $m + n$."... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-8b21a73cdbd300d06c78f92d957df0cd5f2b5a08fe0efcdd0ef545f0f39dddbe | A gardener plants three maple trees, four oak trees, and five birch trees in a row. The trees are planted in random order, with each arrangement being equally likely. Find the probability that no two birch trees are next to one another, expressed as $\frac{m}{n}$ in lowest terms, and determine $m+n$. | [
"106"
] | [
"A gardener plants three maple trees, four oak trees, and five birch trees in a row. The trees are planted in random order, with each arrangement being equally likely. Find the probability that no two birch trees are next to one another, expressed as $\\frac{m}{n}$ in lowest terms, and determine $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1b479f496ea75829db8b2f0ae4c26237bf8e09a666f7c1fa2322ea0eca5950b | A gardener plants three maple trees, four oak trees, and five birch trees in a row. What is the probability that no two birch trees are next to one another? | [
"\\(\\frac{7}{99}\\)"
] | [
"A gardener plants three maple trees, four oak trees, and five birch trees in a row. What is the probability that no two birch trees are next to one another?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-0eac3856d1563810cb2eae10607c02713a43291003c44950b490e63efef7b4a4 | A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Find the probability that no two birch trees are next to one another, and express this probability as $\frac{m}{n}$ in lowest terms. Find $m+n$. | [
"106"
] | [
"A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Find the probability that no two birch trees are next to one another, and express this probability as $\\frac{m}{n}$ in lowest terms. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 35
} | 67 | 0 | 0 |
omr-ws-d8dc1c5f5b8496d2bf406d0b313759091fb4bd8e51cba50ab750a60b1bc850a4 | A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$. | [
"106"
] | [
"A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f04c7f697bcd58d82d452bb1201b0a4e0c7247dee248437d47f832a9de460684 | A gardener plants three maple trees, four oaks, and five birch trees in a row. The trees are planted in random order, with each arrangement being equally likely. Find the probability that no two birch trees are next to one another, and express this probability as $\frac{m}{n}$ in lowest terms. Find $m+n$. | [
"496"
] | [
"A gardener plants three maple trees, four oaks, and five birch trees in a row. The trees are planted in random order, with each arrangement being equally likely. Find the probability that no two birch trees are next to one another, and express this probability as $\\frac{m}{n}$ in lowest terms. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ba6211e5320bb6207a6cb1f3ba0f2fd322189e05d541c21671fc2ba16678be4 | A gardener throws 18 seeds onto an equilateral triangle-shaped plot of land with sides of length one meter. Let $d$ be the minimum distance such that there are at least two seeds within a distance of $d$ centimeters from each other. Find the maximum possible value of $d$. | [
"20"
] | [
"A gardener throws 18 seeds onto an equilateral triangle-shaped plot of land with sides of length one meter. Let $d$ be the minimum distance such that there are at least two seeds within a distance of $d$ centimeters from each other. Find the maximum possible value of $d$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-9f865d001dc7b1b218af039943bd2f0e5ddf4e8344194884d3a7b307041a1023 | A gardener wants to plant 100 trees (oaks and birches) along an avenue in the park. The number of trees between any two oaks must not be equal to five. What is the greatest number of oaks that the gardener can plant? | [
"52"
] | [
"A gardener wants to plant 100 trees (oaks and birches) along an avenue in the park. The number of trees between any two oaks must not be equal to five. What is the greatest number of oaks that the gardener can plant?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-0f2a0074d1ddd1cb0e4d476335971237b29c32c4cdbbfde2c7147ac22d186865 | A gardener wishes to plant three rose bushes, four hydrangeas, and three azaleas in a row such that no two rose bushes are next to each other. How many ways can the bushes be planted if bushes of the same type are not considered distinct? | [
"1960"
] | [
"A gardener wishes to plant three rose bushes, four hydrangeas, and three azaleas in a row such that no two rose bushes are next to each other. How many ways can the bushes be planted if bushes of the same type are not considered distinct?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-531fd2b4ac4fc34af1b4e8ab35367ec620b6310bf7261d05d9a6e0becc5b6e5a | A gas pipeline is to be constructed from a storage tank to a house. A road runs straight from the tank, passing at its closest 300 feet away from the house at a point 600 feet from the tank. Pipe laid along the road costs $8 per foot, while pipe laid off the road costs $10 per foot. What is the minimum cost for which t... | [
"6600"
] | [
"A gas pipeline is to be constructed from a storage tank to a house. A road runs straight from the tank, passing at its closest 300 feet away from the house at a point 600 feet from the tank. Pipe laid along the road costs $8 per foot, while pipe laid off the road costs $10 per foot. What is the minimum cost for wh... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e922708f18fed91863df7220bc8c735bc1bc1a430426ac0edacb501084aa0aa3 | A gasoline tank is $\frac78$ full. After $12$ gallons have been used, it is half full. How many gallons does this tank hold when it is full? | [
"32"
] | [
"A gasoline tank is $\\frac78$ full. After $12$ gallons have been used, it is half full. How many gallons does this tank hold when it is full?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-79935771ba2ca06d73d7805c0991daa949da8142cc051c53d02212e7d4c8f7ba | A gear with radius 34.5 rolls around the circumference of a larger gear with radius 103.5. How many revolutions around the larger gear would the smaller gear have completed by the time it makes 3 complete rotations? | [
"3/4"
] | [
"A gear with radius 34.5 rolls around the circumference of a larger gear with radius 103.5. How many revolutions around the larger gear would the smaller gear have completed by the time it makes 3 complete rotations?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 14
} | 62 | 0 | 0 |
omr-ws-59aeffdc93a9e46aff0403213f7225121b0cdcb4db74249424945d4438c0ff18 | A geiger counter detects radioactive decay events according to a Poisson process with rate 0.5. After each detection, it will not detect another event until it has gone a full two seconds with no decays. Find the expected value of the time between events detected by the counter. | [
"2e"
] | [
"A geiger counter detects radioactive decay events according to a Poisson process with rate 0.5. After each detection, it will not detect another event until it has gone a full two seconds with no decays. Find the expected value of the time between events detected by the counter."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-adac6b9d4b4b29cab4c90d7ca74c38d3dfc816bd2507a396c54f31adad076474 | A general has an army of 1000 soldiers. He lines them up in rows of 11, 13, and 17 and counts the remainders, which are 3, 4, and 9 respectively. How can he determine the number of soldiers using these remainders? | [
"927"
] | [
"A general has an army of 1000 soldiers. He lines them up in rows of 11, 13, and 17 and counts the remainders, which are 3, 4, and 9 respectively. How can he determine the number of soldiers using these remainders?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-cb1075c7891352ba97604a06b18fc643aa1c74dce3405c610b0b8164fe871175 | A genie has tied and blindfolded you, and has placed four coins on the vertices of a square plate - the coins do not start out all heads or all tails. You may turn any number of coins and say "ALAKAZAM" to indicate you are ready for the genie to inspect the coins. The genie will inspect the coins and either free you if... | [
"7"
] | [
"A genie has tied and blindfolded you, and has placed four coins on the vertices of a square plate - the coins do not start out all heads or all tails. You may turn any number of coins and say \"ALAKAZAM\" to indicate you are ready for the genie to inspect the coins. The genie will inspect the coins and either free... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d22eeb886a12406eaebdde72bbd227f823822dd9b6f012726b56dce711a2e930 | A geometric figure has four vertices, A, B, C, and D, with the following distances between the vertices: |AB|=1, |AC|=1, |AD|=11/20, |BC|=1, |BD|=11/20, and |CD|=11/20. Identify the type of geometric figure this is. | [
"tetrahedron"
] | [
"A geometric figure has four vertices, A, B, C, and D, with the following distances between the vertices: |AB|=1, |AC|=1, |AD|=11/20, |BC|=1, |BD|=11/20, and |CD|=11/20. Identify the type of geometric figure this is."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-80ee9587eb09e703702681e9ce9c2e9f984b4ff3c05085bfa815601f56d7e358 | A geometric progression consists of 37 positive integers where the first and the last terms are relatively prime numbers. Let the 19th term of the progression be expressed as \(a^{18}\) for some positive integer \(a\). If the first term is 2 and the last term is 127, find the value of \(a\). | [
"\\sqrt[36]{254}"
] | [
"A geometric progression consists of 37 positive integers where the first and the last terms are relatively prime numbers. Let the 19th term of the progression be expressed as \\(a^{18}\\) for some positive integer \\(a\\). If the first term is 2 and the last term is 127, find the value of \\(a\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d38d686535a6ba2b7caaa4ae7c3d5cc00542fde403ecfe7cd89fca532a73bd6c | A geometric progression consists of an even number of terms. The sum of all the terms is three times that of the odd terms. Determine the common ratio of the progression. | [
"$q=2$"
] | [
"A geometric progression consists of an even number of terms. The sum of all the terms is three times that of the odd terms. Determine the common ratio of the progression. "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d4f64ffa918692d7eb0fc93fd944d30ea5bf84bf1447893711c87d89e1bc80d | A geometric progression consists of five terms; their sum less the first term is equal to $19\frac{1}{2}$, and that less the last one equal 13. Compute the extremes of the progression. | [
"$u_1=1.6$ and $u_5=8.1$"
] | [
"A geometric progression consists of five terms; their sum less the first term is equal to $19\\frac{1}{2}$, and that less the last one equal 13. Compute the extremes of the progression."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3fa6dc6f7f423f70468da2a06b405825227ea5ed11393c41a9be29ab39415363 | A geometric progression of positive integers has \( n \) terms; the first term is \( 10^{2020} \) and the last term is an odd positive integer. How many possible values of \( n \) are there? | [
"12"
] | [
"A geometric progression of positive integers has \\( n \\) terms; the first term is \\( 10^{2020} \\) and the last term is an odd positive integer. How many possible values of \\( n \\) are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 46,
"genselect": 8
} | 54 | 0 | 0 |
omr-ws-a2abf735418f1607fab66cb3068ebdfaa8e1f22aec26bd6fcc6205206746d42f | A geometric sequence begins \( \sqrt[7]{12}, \sqrt[3]{18}, \ldots \). Find the first term of this sequence which is an integer. | [
"13,122"
] | [
"A geometric sequence begins \\( \\sqrt[7]{12}, \\sqrt[3]{18}, \\ldots \\). Find the first term of this sequence which is an integer."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-24bf0f722f2c4a9b49cd89c9565d3a24cb7f6cb94d54a21460f369acc23da4e4 | A geometric sequence of positive integers \(a\), \(b\), and \(c\) with integral common ratio \(r\) is called \textit{bogus} if the numbers \(d(a)\), \(d(b)\), and \(d(c)\) form an arithmetic sequence, where \(d(x)\) is the number of divisors of \(x\). Find the number of values of \(r\) where \(1 \le r \le 100\) such th... | [
"36"
] | [
"A geometric sequence of positive integers \\(a\\), \\(b\\), and \\(c\\) with integral common ratio \\(r\\) is called \\textit{bogus} if the numbers \\(d(a)\\), \\(d(b)\\), and \\(d(c)\\) form an arithmetic sequence, where \\(d(x)\\) is the number of divisors of \\(x\\). Find the number of values of \\(r\\) where \... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8ab2a63d600366128c2d2381d4a3b435355bfef5ee906ee8e9934a1489a32a0 | A geometric sequence of positive integers is formed for which the first term is 2 and the fifth term is 162. What is the sixth term of the sequence? | [
"486"
] | [
"A geometric sequence of positive integers is formed for which the first term is 2 and the fifth term is 162. What is the sixth term of the sequence?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e85bf5914bfdd902a3a59d2cace2ed2a0b6768ecda23fcb7ecff6e54399ebe11 | A geometric sequence of positive integers is formed for which the first term is 3 and the fourth term is 192. What is the third term of the sequence? | [
"48"
] | [
"A geometric sequence of positive integers is formed for which the first term is 3 and the fourth term is 192. What is the third term of the sequence?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed117adacaa7226a7549ebb801d0fe1d3d3b1546f03e52804cbab6c2fe41c37c | A geometric sequence starts $16$, $-24$, $36$, $-54$. What is the common ratio of this sequence? | [
"-\\frac{3}{2}"
] | [
"A geometric sequence starts $16$, $-24$, $36$, $-54$. What is the common ratio of this sequence?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5cccc20cba5372197000cc155b97ebccbbdbbcd9832f852d561fcee1dc47e17 | A ghost leg is a game with some vertical lines and some horizontal lines. A player starts at the top of a vertical line and goes downwards, and always walks through a horizontal line if he encounters one. We define a layer as some horizontal line with the same height and has no duplicated endpoints. Find the smallest n... | [
"n"
] | [
"A ghost leg is a game with some vertical lines and some horizontal lines. A player starts at the top of a vertical line and goes downwards, and always walks through a horizontal line if he encounters one. We define a layer as some horizontal line with the same height and has no duplicated endpoints. Find the small... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-8fd47fda771b9bcdf64269ea579f7756973bd7717f13a2ffa2c8d58a78d60577 | A giant, circular and perfectly flat pizza must be shared by 22 people. What is the least number of cuts to be made on the pizza so that each person can have a piece of pizza? (The pieces are not necessarily the same shape or with the same area.) | [
"6"
] | [
"A giant, circular and perfectly flat pizza must be shared by 22 people. What is the least number of cuts to be made on the pizza so that each person can have a piece of pizza? (The pieces are not necessarily the same shape or with the same area.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-61f111c040bcd8725b0fb17ec2f1e9551428e4cd6e8536040bfe60945ff96375 | A gift shop offers free gift wrapping. They have eight varieties of wrapping paper, three colors of ribbon, and four types of gift cards. Using one variety of wrapping paper, one color of ribbon and one type of gift card, how many combinations are possible? | [
"96"
] | [
"A gift shop offers free gift wrapping. They have eight varieties of wrapping paper, three colors of ribbon, and four types of gift cards. Using one variety of wrapping paper, one color of ribbon and one type of gift card, how many combinations are possible?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f63f926fa6b988eb0598c44d4139cc5ff18adef15fa4700ddb4c37c3eba8dfb1 | A girl 1.5m tall is running toward a street light 8m tall at a rate of 2m/s. At what rate is her shadow decreasing in length? At what rate is the tip of her shadow moving? | [
"-0.46, -5m/s"
] | [
"A girl 1.5m tall is running toward a street light 8m tall at a rate of 2m/s. At what rate is her shadow decreasing in length? At what rate is the tip of her shadow moving?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f05169c650538f53494877de7093ae49039fd37100f4c66c9c6e70015255ae5 | A girl and a guy are going to arrive at a train station. The girl will arrive between 8 AM and 9 AM with equal probability, and the guy will arrive between 7 AM and 8:30 AM with equal probability. If they arrive within 10 minutes of each other, they will fall in love. Let \(\frac{p}{q}\) for \(p, q\) coprime be the pro... | [
"10"
] | [
"A girl and a guy are going to arrive at a train station. The girl will arrive between 8 AM and 9 AM with equal probability, and the guy will arrive between 7 AM and 8:30 AM with equal probability. If they arrive within 10 minutes of each other, they will fall in love. Let \\(\\frac{p}{q}\\) for \\(p, q\\) coprime ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-d1a329a59abbebc63c1884525f1dfbbc001b8970b4d8bdccb7ec8343e0424c5f | A girl of height 5 feet is walking towards a light post of height 20 feet at a speed of 4 feet/second. Her little brother is following her with a constant distance of 4 feet between them. Determine the speed at which the end of the shadow is moving, i.e., $\frac{dy}{dt}$. When the girl is far from the light post, she c... | [
"\\(\\frac{16}{3}\\) feet/second when \\( x \\geq 30 \\) feet, and \\(\\frac{80}{17}\\) feet/second when \\( x < 30 \\) feet."
] | [
"A girl of height 5 feet is walking towards a light post of height 20 feet at a speed of 4 feet/second. Her little brother is following her with a constant distance of 4 feet between them. Determine the speed at which the end of the shadow is moving, i.e., $\\frac{dy}{dt}$. When the girl is far from the light post,... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1df4f4afdda78b9f2d0572054957479e5da0e73f06c51c83e3c9b5b38af3d5a9 | A girl spends 1/2 of her money on clothes, then 1/3 of the remaining money on games, and then 1/4 of the remaining money on toys. After these purchases, she has $7777 left. How much money did she originally have, and assuming she only gets money by allowance, how much does she earn every month? | [
"\\(\\frac{7777}{3}\\)"
] | [
"A girl spends 1/2 of her money on clothes, then 1/3 of the remaining money on games, and then 1/4 of the remaining money on toys. After these purchases, she has $7777 left. How much money did she originally have, and assuming she only gets money by allowance, how much does she earn every month?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b7ba81b7585a7122ce2dfdb6c7f3acfda1af2ea885628ac73ed4d3cd9a0f1a6 | A girl spends 1/2 of her money on clothes, then 1/3 of the remaining money on games, and then 1/4 of the remaining money on toys. After these purchases, she has $7777 left. How much money did she originally have? Assuming she only gets money by allowance, how much money does she earn every month? | [
"2592\\frac{1}{3}"
] | [
"A girl spends 1/2 of her money on clothes, then 1/3 of the remaining money on games, and then 1/4 of the remaining money on toys. After these purchases, she has $7777 left. How much money did she originally have? Assuming she only gets money by allowance, how much money does she earn every month?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efcdee93e6d0c02adf217bbe9ab37d4012316d27d0efc6a0766a4c4e26ec67a5 | A girl was crossing a railroad bridge when, halfway across, she saw a train 50 meters away moving toward her. She immediately turned and ran back, and the train missed her by the narrowest of margins. If she had continued to cross the bridge, the train would have struck her one meter before she could have reached the e... | [
"\\( 1 + \\sqrt{101} \\) meters or approximately \\( 11.05 \\) meters."
] | [
"A girl was crossing a railroad bridge when, halfway across, she saw a train 50 meters away moving toward her. She immediately turned and ran back, and the train missed her by the narrowest of margins. If she had continued to cross the bridge, the train would have struck her one meter before she could have reached ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-7914f1168c7a00eaa524deddad58f85e50d3a4259f81dcdfcb5b713c42a913cb | A girls' camp is located 300 rods from a straight road. On this road, a boys' camp is located 500 rods from the girls' camp. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. What is the distance of the canteen from each of the camps? | [
"312.5 rods"
] | [
"A girls' camp is located 300 rods from a straight road. On this road, a boys' camp is located 500 rods from the girls' camp. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. What is the distance of the canteen from each of the camps?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-d2e24acc391eede2358dadbd4d25ed016c939c6e302323b1e7fd726d1c0df00f | A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region, we write \( 1 \) or \( -1 \). In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines... | [
"1"
] | [
"A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region, we write \\( 1 \\) or \\( -1 \\). In one operation, we can choose any triangle made of three of the lines (which may be cut by ot... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-74079ab0e5944efb6180817c86553a3b53164ee9c6896ef4463dce9917dbd43f | A given point is at distances $\sqrt{2}$, $2$, and $\sqrt{3}-1$ from the vertices of a triangle. Find the maximum possible area of the triangle. | [
"\\dfrac{5\\sqrt{2} - \\sqrt{6}}{2}"
] | [
"A given point is at distances $\\sqrt{2}$, $2$, and $\\sqrt{3}-1$ from the vertices of a triangle. Find the maximum possible area of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-0b2fae86da564b235b9fe20b87b5383d9753e2813d04dac398cf2a528a5e1213 | A given rectangle $R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. What is the minimum number of appropriately selected rectangles' areas that should be known in order to determine the area of $R$? | [
"\\( m + n - 1 \\)"
] | [
"A given rectangle $R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. What is the minimum number of appropriately selected rectangles' areas that should be known in order to determine the area of $R$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ff7e38cbe995533d19e09f6181995d01d9c1adfa618a0885847f193e8dcd0b3e | A given rectangle \( R \) is divided into \( nm \) small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of \( R \)? | [
"\\( n + m - 1 \\)"
] | [
"A given rectangle \\( R \\) is divided into \\( nm \\) small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of \\( R \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9d8eabd93c9744ee3ed628412e2a8f59f8b335872bdfdd1f36d848517db6a1fd | A given regular truncated triangular pyramid can contain a sphere which touches all the faces and a sphere which touches all the edges. Find the sides of the base of the pyramid if the lateral edges are equal to $b$. | [
"$b\\left(1+\\sqrt {\\frac{2}{3}}\\right),\\ b\\left(1-\\sqrt {\\frac{2}{3}}\\right)$"
] | [
"A given regular truncated triangular pyramid can contain a sphere which touches all the faces and a sphere which touches all the edges. Find the sides of the base of the pyramid if the lateral edges are equal to $b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-752fa795d17aa88ae20ca35428bce0652d7bd3ce1c9dc1c50df9f9635e476b68 | A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the thir... | [
"931"
] | [
"A given sequence $r_1, r_2, \\dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more \"bubble passes\". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then com... | [
"MATH_training_set"
] | [
"additional_problems",
"cot",
"genselect"
] | {
"additional_problems": 1,
"cot": 2,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-eff7a47f3305dd3b33fac5fbb0d98bf9b5bb91fa72e0623820652d7925e70190 | A glass cube is inserted inside a fish tank with a worm at its center. Three fish start at the same point inside the tank and swim to different vertices of the cube as follows: - The first fish swims 3 units to the right and 2 units into the depth. - The second fish swims 5 units to the right and 4 units upwards. - The... | [
"\\( 3\\sqrt{2} \\)"
] | [
"A glass cube is inserted inside a fish tank with a worm at its center. Three fish start at the same point inside the tank and swim to different vertices of the cube as follows:\n- The first fish swims 3 units to the right and 2 units into the depth.\n- The second fish swims 5 units to the right and 4 units upwards... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-f22c8c890d3a05d9cb00a4e26aa630288103ba339549a1c7c32635ace057c367 | A glass in the shape of a truncated cone holds 1 liter. To measure half a liter, it is filled in height by \( \frac{2}{3} \). How are the diameters of the bases of the glass related? | [
"\\(\\frac{13 + \\sqrt{1269}}{22}\\)"
] | [
"A glass in the shape of a truncated cone holds 1 liter. To measure half a liter, it is filled in height by \\( \\frac{2}{3} \\). How are the diameters of the bases of the glass related?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-90508f4f74c87f66fba702f112ab660d6b73ee06df3dea9240d1e8c2b01e74df | A glass of iced water at $3^{\circ}C$ is left in the shade outside at $27^{\circ}C$. After 5 minutes, the water’s temperature is $7^{\circ}C$. Find the water’s temperature after another ten minutes. | [
"\\( \\frac{118}{9} \\) or \\( 13.\\overline{1} \\)"
] | [
"A glass of iced water at $3^{\\circ}C$ is left in the shade outside at $27^{\\circ}C$. After 5 minutes, the water’s temperature is $7^{\\circ}C$. Find the water’s temperature after another ten minutes."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-19ce356e69c174c321babfd8b576a0f371462277b552224a5575f694b785a750 | A glass sphere with radius $20$ cm is tangent to the ground at point $T$. A light bulb is installed at the center of the sphere (point $O$). Points $A$ and $B$ are marked on the surface of the sphere such that $\angle AOB = 60^\circ$. The heights of points $A$ and $B$ from the ground are $10$ cm and $5$ cm, respectivel... | [
"\\( 10\\sqrt{3} \\) cm and \\( 5\\sqrt{7} \\) cm"
] | [
"A glass sphere with radius $20$ cm is tangent to the ground at point $T$. A light bulb is installed at the center of the sphere (point $O$). Points $A$ and $B$ are marked on the surface of the sphere such that $\\angle AOB = 60^\\circ$. The heights of points $A$ and $B$ from the ground are $10$ cm and $5$ cm, resp... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7a03011b4eb0d4638e814706f7aea939a399d6cfa2d968bc147c00a83481385b | A glass with the form \( z = x^2 + y^2 \) and \( z \leq 1 \) is filled with water to a certain height \( h \). If the glass is tilted \( 45^\circ \), the water is just about to stream out. Determine \( h \). | [
"\\( h = \\frac{1}{4} \\)"
] | [
"A glass with the form \\( z = x^2 + y^2 \\) and \\( z \\leq 1 \\) is filled with water to a certain height \\( h \\). If the glass is tilted \\( 45^\\circ \\), the water is just about to stream out. Determine \\( h \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 7
} | 17 | 0 | 0 |
omr-ws-23b6e3b88ba702b74c7cdbe8112d1370892b648e69424380cdaa0b99f3483f7f | A goat is tethered to a circular hay silo of radius 10 ft by means of a rope 10π feet long. Find the area of the region the goat can reach. | [
"\\(\\frac{250}{3}\\pi^3\\)"
] | [
"A goat is tethered to a circular hay silo of radius 10 ft by means of a rope 10π feet long. Find the area of the region the goat can reach."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 3,
"tir": 3
} | 23 | 0 | 0 |
omr-ws-6b089925f79d5e4ef5d3935ef69d60bdf0b34e1d8473a313584d2f911babad90 | A goat is tethered to a corner of a building which is in the shape of a regular $n$-gon. The length of a side of the building and the length of the tether are each $r$. Find the area of the region over which the goat can graze as a function of $r$ and $n$. | [
"\\(\\left(\\frac{1}{2} + \\frac{1}{n}\\right) \\pi r^2\\)"
] | [
"A goat is tethered to a corner of a building which is in the shape of a regular $n$-gon. The length of a side of the building and the length of the tether are each $r$. Find the area of the region over which the goat can graze as a function of $r$ and $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 13
} | 27 | 0 | 0 |
omr-ws-18ccaec491da61eac0f4da773bb7fd54ef2a1caa6534e81d1ace57b4af00be6d | A goat is tethered to a corner of a building which is in the shape of a regular n-gon. The length of a side of the building and the length of the tether are each r. Find the area of the region over which the goat can graze as a function of n and r. | [
"\\(\\frac{n+2}{2n} \\cdot \\pi r^2\\)"
] | [
"A goat is tethered to a corner of a building which is in the shape of a regular n-gon. The length of a side of the building and the length of the tether are each r. Find the area of the region over which the goat can graze as a function of n and r."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-42e5d4ba171bbd0c6c51ff533b353679b70c135519a3810aee9be317b91e866f | A goat is tethered to a point on the circumference of a circular field of radius $r$. If the goat can eat half the grass inside the circle, how long is the rope? | [
"1.1587r"
] | [
"A goat is tethered to a point on the circumference of a circular field of radius $r$. If the goat can eat half the grass inside the circle, how long is the rope?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8e83722a07709f9c9297c17a5b3214e434b64b52c93f4e3ff862e3e6e66f2f65 | A goat is tethered to the outside of a $30$ ft. by $10$ ft. barn, $10$ ft. from the corner on one of the longer sides of the barn. If the rope is $30$ ft. long, what is the total area of the land the goat can graze, assuming the goat does not eat the rope? | [
"\\( 600\\pi \\)"
] | [
"A goat is tethered to the outside of a $30$ ft. by $10$ ft. barn, $10$ ft. from the corner on one of the longer sides of the barn. If the rope is $30$ ft. long, what is the total area of the land the goat can graze, assuming the goat does not eat the rope?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-8d59dda1532fe8f9c5c477afac14a55068d8a58a00dc7e4ca7ca2488ccfc0809 | A goat is tied by a rope of length \( r \) to one 25-foot side of a 15-by-25 foot barn, 10 feet from the nearest corner. Find a formula or set of formulas to describe the area that the goat can graze for any rope length \( r \) up to 40 feet. | [
"\\[\n\\begin{cases}\n\\frac{\\pi r^2}{2} & \\text{if } r < 10 \\\\\n\\frac{\\pi (3r^2 - 20r + 100)}{4} & \\text{if } 10 \\leq r < 15 \\\\\n\\frac{\\pi (4r^2 - 50r + 325)}{4} & \\text{if } 15 \\leq r < 25 \\\\\n\\frac{\\pi (5r^2 - 100r + 950)}{4} & \\text{if } 25 \\leq r < 30 \\\\\n\\frac{\\pi (6r^2 - 160r + 1850)}... | [
"A goat is tied by a rope of length \\( r \\) to one 25-foot side of a 15-by-25 foot barn, 10 feet from the nearest corner. Find a formula or set of formulas to describe the area that the goat can graze for any rope length \\( r \\) up to 40 feet."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 6
} | 12 | 0 | 0 |
omr-ws-3aecfce345e4960d19910e1bbba8d5d0417365578f57426beec8ff94328c66f6 | A goat is tied to a corner of a rectangular house with dimensions 10 m * 5 m * 10 m using a 15 m long rope. The goat cannot enter the house but can graze outside as far as the rope allows. Calculate the total area the goat can graze, using π = 22/7. | [
"629"
] | [
"A goat is tied to a corner of a rectangular house with dimensions 10 m * 5 m * 10 m using a 15 m long rope. The goat cannot enter the house but can graze outside as far as the rope allows. Calculate the total area the goat can graze, using π = 22/7."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 28
} | 46 | 0 | 0 |
omr-ws-136ffca19521b78dfb222357de58cfd892945bbec2321aebed51f4619a5d2805 | A goat is tied to a hexagon with sides of 4 feet using a leash of 6 feet. Determine the area the goat can graze. | [
"\\dfrac{76}{3}\\pi"
] | [
"A goat is tied to a hexagon with sides of 4 feet using a leash of 6 feet. Determine the area the goat can graze."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fcca484d6a87ff6e6382c6145bf2ec6ce815c11970dd87be2aa4e95918465fc1 | A goat is tied to a rectangular barn with dimensions 18 feet by 12 feet using a leash of 24 feet. Determine the area the goat can graze. | [
"477\\pi"
] | [
"A goat is tied to a rectangular barn with dimensions 18 feet by 12 feet using a leash of 24 feet. Determine the area the goat can graze."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3024c825af0ef8b208274a4b598cc6b3493d8161f9f6ee42c935d3ef4678f9c2 | A goat is tied to an external corner of a rectangular shed measuring 4m by 6m. If the goat's rope is 8 m long, what is the total area, in square meters, in which the goat can graze? | [
"\\( 53\\pi \\)"
] | [
"A goat is tied to an external corner of a rectangular shed measuring 4m by 6m. If the goat's rope is 8 m long, what is the total area, in square meters, in which the goat can graze?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-a3d67761db1633582f11a60518af94982772b156edea1aff5b772d2cb6e3d9d5 | A goat is tied to the corner of a rectangular barn with dimensions 40 feet by 80 feet. The rope is 50 feet long. What is the maximum area that the goat can graze in square feet? | [
"\\(1900\\pi\\)"
] | [
"A goat is tied to the corner of a rectangular barn with dimensions 40 feet by 80 feet. The rope is 50 feet long. What is the maximum area that the goat can graze in square feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 19
} | 37 | 0 | 0 |
omr-ws-3260669881d18ecad3b28254c61aa96fa1906d3550a45ea59eae853407059bae | A goat is tied to the fence of a circular field with a radius of 40 feet. If the goat's string is \( n \) feet long and the goat can graze exactly half the area of the field, what is the value of \( n \)? | [
"46.3"
] | [
"A goat is tied to the fence of a circular field with a radius of 40 feet. If the goat's string is \\( n \\) feet long and the goat can graze exactly half the area of the field, what is the value of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-82782d36da409785d16d71c4109e42d96df4d4235d64c65719ddfc74154436f4 | A goat lives in a triangle with area $A$ and perimeter $P$. The goat can go up to a distance $d$ away from the triangle. What is the area of the land that the goat can walk within? | [
"\\( A + dP + \\pi d^2 \\)"
] | [
"A goat lives in a triangle with area $A$ and perimeter $P$. The goat can go up to a distance $d$ away from the triangle. What is the area of the land that the goat can walk within?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-3e432872495a69fc38664d24ce99c2aaa0ed028ba29d47190694861f286147c4 | A gold ball is hit off the ground at an angle of 30 degrees and travels 400 feet. How long was it in the air? Use the equations $x=v_0 \cdot T \cdot \cos{\theta}$ and $y=-16T^2+v_0\cdot T\cdot \sin{\theta} + h_0$. | [
"\\(\\frac{5}{3^{1/4}}\\)"
] | [
"A gold ball is hit off the ground at an angle of 30 degrees and travels 400 feet. How long was it in the air? Use the equations $x=v_0 \\cdot T \\cdot \\cos{\\theta}$ and $y=-16T^2+v_0\\cdot T\\cdot \\sin{\\theta} + h_0$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-4248f796af7dcdd4f667314902526273270f9147cf18e10eac616c3cf8014e6e | A gold bar with dimensions $2 \times 3 \times 4$ is melted and recast into three spheres of equal volume. What is the radius of each sphere? | [
"\\(\\sqrt[3]{\\frac{6}{\\pi}}\\)"
] | [
"A gold bar with dimensions $2 \\times 3 \\times 4$ is melted and recast into three spheres of equal volume. What is the radius of each sphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 20
} | 68 | 0 | 0 |
omr-ws-ecf636bafefb72d4236f0883036ad75ba4045dce966dc7d09cc1b5939206320a | A gold shipment consists of \( N > 1 \) bars containing a total of \( 2^{100} \) gold atoms. The number of atoms in each bar is a power of \( 2 \). Determine the number of ways to load the bars onto two ships such that each ship carries the same amount of gold. Express your answer as the sum of the exponents of all the... | [
"99"
] | [
"A gold shipment consists of \\( N > 1 \\) bars containing a total of \\( 2^{100} \\) gold atoms. The number of atoms in each bar is a power of \\( 2 \\). Determine the number of ways to load the bars onto two ships such that each ship carries the same amount of gold. Express your answer as the sum of the exponents... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-013e5df46c54a560be652552f8410395db3a630495abb92e65d2c490cefc8dfb | A good word is a word consisting of two letters $A$ and $B$ such that there is never a letter $B$ between any two $A$'s. Find the number of good words with length 8. | [
"37"
] | [
"A good word is a word consisting of two letters $A$ and $B$ such that there is never a letter $B$ between any two $A$'s. Find the number of good words with length 8."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b365746d66d6ddfab1a1280e18475c4c6704e351f0d4d6fc1027221681cef831 | A goods train left Moscow at \( x \) hours \( y \) minutes and arrived in Saratov at \( y \) hours \( z \) minutes. The journey took \( z \) hours \( x \) minutes. Find all possible values of \( x \). | [
"\\( x = 6 \\) or \\( x = 12 \\)"
] | [
"A goods train left Moscow at \\( x \\) hours \\( y \\) minutes and arrived in Saratov at \\( y \\) hours \\( z \\) minutes. The journey took \\( z \\) hours \\( x \\) minutes. Find all possible values of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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