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-1f674e3e22af688165237b1a6ff231e125d80bc921d5b63893d886ca42c32acc | To the circle inscribed in an equilateral triangle, there is a tangent perpendicular to the side of the triangle. The segment of this tangent enclosed inside the triangle has length $9-3\sqrt{3}$ cm. Find the radius of the inscribed circle. | [
"3"
] | [
"To the circle inscribed in an equilateral triangle, there is a tangent perpendicular to the side of the triangle. The segment of this tangent enclosed inside the triangle has length $9-3\\sqrt{3}$ cm. Find the radius of the inscribed circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-b7d4cb31f1e94b779de97275d1873f9a713fbf345aed79ea2458dee6c5eaf03d | To the nearest degree, find the acute angle between the lines \(2x + y = 10\) and \(x + 2y = 10\). | [
"37"
] | [
"To the nearest degree, find the acute angle between the lines \\(2x + y = 10\\) and \\(x + 2y = 10\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b9b3a75a6c991fdba37d1ddd5f9e29bd4d4b1e0af4a0446982ca6dd1572c291 | To the nearest minute, what time is it the last time the hour and minute hands of an analog clock are in the same position before midnight? | [
"10:55"
] | [
"To the nearest minute, what time is it the last time the hour and minute hands of an analog clock are in the same position before midnight?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c76e4571e8d3103d3c14d351e5f0ac88341aa489a9965f5d70d7593ed67d1217 | To the nearest tenth, find the value of \( x \) that satisfies the equation \( 2^x = -2x + 11 \). | [
"2.6"
] | [
"To the nearest tenth, find the value of \\( x \\) that satisfies the equation \\( 2^x = -2x + 11 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5a029b3836ccab3be0759c216b1026e46cb8b13b2a8e9fb1e37cb59f0897bfa | To transmit a positive integer less than 1000, the Networked Number Node offers two options. Option 1. Pay $\$$d to send each digit d. Therefore, 987 would cost $\$$9 + $\$$8 + $\$$7 = $\$$24 to transmit. Option 2. Encode integer into binary (base 2) first, and then pay $\$$d to send each digit d. Therefore, 987 become... | [
"503"
] | [
"To transmit a positive integer less than 1000, the Networked Number Node offers two options.\n\nOption 1. Pay $\\$$d to send each digit d. Therefore, 987 would cost $\\$$9 + $\\$$8 + $\\$$7 = $\\$$24 to transmit.\n\nOption 2. Encode integer into binary (base 2) first, and then pay $\\$$d to send each digit d. Ther... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-2d0b0ed3580350afd11a7faf042cf96b6affe6ece571951dbc726d9443c78ab7 | To transport a factory's produce from point $N$ to a town $A$ a highway $NP$ is being built to connect the factory with a railway $A B$ which passes through town $A$. Carriage by road is twice as expensive as that by rail. To what point $P$ should the highway lead for the total transportation cost of the produce from p... | [
"$x(p)=\\min\\left(\\frac{100}{\\sqrt 3},a\\right)$, where $x (p)$ is the distance from the railway station to the point $P$"
] | [
"To transport a factory's produce from point $N$ to a town $A$ a highway $NP$ is being built to connect the factory with a railway $A B$ which passes through town $A$. Carriage by road is twice as expensive as that by rail. To what point $P$ should the highway lead for the total transportation cost of the produce f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3fbbef0e83c677e18b6a5dc576d5ffcf4e93a0fe45c1207377e88ede6f350dbf | To two externally tangent circles of radii $R$ and $r$ common exterior tangent lines are drawn. Find the area of the trapezoid bounded by the tangent lines and chords joining the points of tangency. | [
"$S=\\frac{8(Rr)^{3/2}}{R+r}$"
] | [
"To two externally tangent circles of radii $R$ and $r$ common exterior tangent lines are drawn. Find the area of the trapezoid bounded by the tangent lines and chords joining the points of tangency."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57b822e58a48cf726a7a4b3de06df1a2939686738472feb4aeecd73abebd94e2 | To two externally tangent circles of radii $R$ and $r$ common tangent lines are drawn: one interior and two exterior ones. Determine the length of the segment of the interior tangent line contained between the exterior tangents. | [
"$MN=2\\sqrt {Rr}$"
] | [
"To two externally tangent circles of radii $R$ and $r$ common tangent lines are drawn: one interior and two exterior ones. Determine the length of the segment of the interior tangent line contained between the exterior tangents."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b53f173bdcdb51d79ab90ec29bbfb54fbdffe8aebfa652260f8ba3411495f542 | To two externally tangent circles of radii \( R \) and \( r \), a secant is constructed so that the circles cut off three equal segments on it. Find the lengths of these segments. | [
"\\(\\sqrt{\\frac{14Rr - R^2 - r^2}{12}}\\)"
] | [
"To two externally tangent circles of radii \\( R \\) and \\( r \\), a secant is constructed so that the circles cut off three equal segments on it. Find the lengths of these segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 7
} | 39 | 0 | 0 |
omr-ws-efb7e0b483226f0acdc3bb31a2e51b7585e4d944e3657aaa8bb6eea64d183546 | To what number does the series \(\sum_{n=1}^{\infty}\frac{(-1)^{n+1} 2^{n+1}}{n!}\) converge? | [
"\\(2(1-e^{-2})\\)"
] | [
"To what number does the series \\(\\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1} 2^{n+1}}{n!}\\) converge?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1331e225346dc0c7f16655148b793ff3289866e812eda2288b513658b348a957 | To which famous ring is $\frac{\mathbb{Z}_{5}\left[X,Y\right]}{\left(Y-X^{2}, XY+Y+2\right)}$ isomorphic? | [
"\\(\\mathbb{F}_{125}\\)"
] | [
"To which famous ring is $\\frac{\\mathbb{Z}_{5}\\left[X,Y\\right]}{\\left(Y-X^{2}, XY+Y+2\\right)}$ isomorphic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-ac03db1c88de5d6fadf5e09d4794f6da245bcc5b92d7e46d71bef5212e8f0f64 | To which group of order 6 is the quotient group \( S_4/V_4 \) isomorphic? | [
"\\( S_3 \\)"
] | [
"To which group of order 6 is the quotient group \\( S_4/V_4 \\) isomorphic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-eadc4b0738e9a277f7d6b6d2d23439ea36e62d3b813095c5c0594d863ac16b8a | To which sequence does the exponential generating function $-\ln(1+x)$ belong? | [
"\\(\\frac{(-1)^n}{n}\\)"
] | [
"To which sequence does the exponential generating function $-\\ln(1+x)$ belong?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85ec3230bcf66f4373f1f3362ae9522e5869ea0c45da4ea4c31f33f2980e7624 | To which sequence does the exponential generating function $-\ln(1-x)$ belong? | [
"\\(H_n\\)"
] | [
"To which sequence does the exponential generating function $-\\ln(1-x)$ belong?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9fb9ee1890890a761c3aedf2c0c20f2749d1d3032fa8d93587e39d030bb5066 | Toady the frog counts from $1$ to $720$. If the number he says is relatively prime to his favorite number $x$, he hops forward $1$ unit. If the number is not relatively prime to $x$, he hops backward $1$ unit. At the end of his counting, he is $10$ units away from his original starting point. What is Toady's favorite n... | [
"134"
] | [
"Toady the frog counts from $1$ to $720$. If the number he says is relatively prime to his favorite number $x$, he hops forward $1$ unit. If the number is not relatively prime to $x$, he hops backward $1$ unit. At the end of his counting, he is $10$ units away from his original starting point. What is Toady's favor... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-b210546efe99867139056e86f1225bbea9362b8feb1ced027b23a207cb88711a | Today is Barbara's birthday, and Alberto wants to give her a gift by playing the following game. The numbers \( 0, 1, 2, \ldots, 1024 \) are written on a blackboard. First, Barbara erases \( 2^9 \) numbers, then Alberto erases \( 2^8 \) numbers, then Barbara \( 2^7 \), and so on, until there are only two numbers \( a \... | [
"32"
] | [
"Today is Barbara's birthday, and Alberto wants to give her a gift by playing the following game. The numbers \\( 0, 1, 2, \\ldots, 1024 \\) are written on a blackboard. First, Barbara erases \\( 2^9 \\) numbers, then Alberto erases \\( 2^8 \\) numbers, then Barbara \\( 2^7 \\), and so on, until there are only two ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8dee39369367eb868193c8f1ca5d11940da8561297527a44dd4dde8fd1241a0a | Today is February 26, 2014. What day and month will it be $12^{3456}$ days from now, taking into account the leap year rules? | [
"October 14"
] | [
"Today is February 26, 2014. What day and month will it be $12^{3456}$ days from now, taking into account the leap year rules?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-a2506cfabadcc338086e322c4dbf01c4c39ff939b9b23e461ce9e796397a91db | Today is Thursday. After $1998^{1998}$ days, which day of the week will it be? | [
"Friday"
] | [
"Today is Thursday. After $1998^{1998}$ days, which day of the week will it be?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-27465fca62341d09117e5a3fe4ff2a60a8ccfde03a46bea2bb68d5c7bec5ae23 | Today the age of Pedro is written followed by the age of Luisa, forming a four-digit number that is a perfect square. In 33 years, the same process will again form a four-digit perfect square. Find the current ages of Pedro and Luisa. | [
"Pedro is \\(11\\) years old and Luisa is \\(56\\) years old."
] | [
"Today the age of Pedro is written followed by the age of Luisa, forming a four-digit number that is a perfect square. In 33 years, the same process will again form a four-digit perfect square. Find the current ages of Pedro and Luisa."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 30
} | 31 | 0 | 0 |
omr-ws-fe5429ae98857046b45ce87fb34b197e3ea4f7a590cba960ca0dca675580ef9d | Todd and Allison are playing a game on a grid. At the beginning, an orange stone is placed in the center intersection. Todd moves the orange stone to a horizontally or vertically adjacent intersection that is not occupied by a blue stone and then places a blue stone in the orange stone’s previous spot. Allison places a... | [
"15"
] | [
"Todd and Allison are playing a game on a grid. At the beginning, an orange stone is placed in the center intersection. Todd moves the orange stone to a horizontally or vertically adjacent intersection that is not occupied by a blue stone and then places a blue stone in the orange stone’s previous spot. Allison pla... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-1ce3a64ed81aaeb858232dcec8f69d16badc09be1a60b31713345d01b61ef5ce | Together A, B, C, D, and E earned a total of $150, but in unequal amounts. In order to equalize their earnings exactly, B gave half her earnings to A. Next, C gave 1/3 of his earnings to B. Then D gave 1/4 of her earnings to C. Finally, E gave 1/6 of his earnings to D. How many dollars did A earn before equalization? | [
"11"
] | [
"Together A, B, C, D, and E earned a total of $150, but in unequal amounts. In order to equalize their earnings exactly, B gave half her earnings to A. Next, C gave 1/3 of his earnings to B. Then D gave 1/4 of her earnings to C. Finally, E gave 1/6 of his earnings to D. How many dollars did A earn before equalizati... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0af110f727810d9f2288035f8350734c0a8333e580e24671773e7681e3cea10b | Tom Sawyer can paint a fence by himself in 3 hours, and Huck Finn can paint the same fence by himself in 4 hours. They start painting together at 12:00 noon but get into a fight that lasts for 10 minutes during which no painting is done. After the fight, Huck leaves and Tom finishes painting alone by 2:25 pm. At what t... | [
"1:00"
] | [
"Tom Sawyer can paint a fence by himself in 3 hours, and Huck Finn can paint the same fence by himself in 4 hours. They start painting together at 12:00 noon but get into a fight that lasts for 10 minutes during which no painting is done. After the fight, Huck leaves and Tom finishes painting alone by 2:25 pm. At w... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 27
} | 50 | 0 | 0 |
omr-ws-02461f6669f305dbf30bee56ae96adb906efc46cc9098b6d1c6d8c685bb136d0 | Tom and Alice work independently to solve a certain problem. The probability that Tom solves it is 0.2, and the probability that Alice solves it is 0.55. If the problem is solved, what is the probability that Tom solves it? | [
"\\(\\frac{20}{89}\\)"
] | [
"Tom and Alice work independently to solve a certain problem. The probability that Tom solves it is 0.2, and the probability that Alice solves it is 0.55. If the problem is solved, what is the probability that Tom solves it?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dcfcffb05a88eef6af57f2f672f17f396d9ab4095217a46bfbbaec723224fbd | Tom and Bill live on the same street and need to deliver a package to each other. They start walking from their homes simultaneously and meet 600 yards from Bill's house. They continue walking and meet again 700 yards from Tom's house. How far apart are their houses? | [
"1100"
] | [
"Tom and Bill live on the same street and need to deliver a package to each other. They start walking from their homes simultaneously and meet 600 yards from Bill's house. They continue walking and meet again 700 yards from Tom's house. How far apart are their houses?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-95ef353444aa09e830a9a7083adbb400b9715fbd3224eaeb8cc2b5d38fe85df1 | Tom and Bill live on the same street. Each boy has a package to deliver to the other boy's house. The two boys start simultaneously from their own homes and meet 600 yards from Bill's house. The boys continue on their errand and they meet again 700 yards from Tom's house. How far apart do the boys live? | [
"1100"
] | [
"Tom and Bill live on the same street. Each boy has a package to deliver to the other boy's house. The two boys start simultaneously from their own homes and meet 600 yards from Bill's house. The boys continue on their errand and they meet again 700 yards from Tom's house. How far apart do the boys live?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 4
} | 57 | 0 | 0 |
omr-ws-404a8230522ae9b574666b6d0d0e8a09d43ebaee42ab5ce25e55b226c940d65b | Tom and Bill live on the same street. Each boy has a package to deliver to the other boy’s house. The two boys start simultaneously from their own homes and meet 600 yards from Bill's house. The boys continue on their errand and they meet again 700 yards from Tom's house. How far apart do the boys live? | [
"1100"
] | [
"Tom and Bill live on the same street. Each boy has a package to deliver to the other boy’s house. The two boys start simultaneously from their own homes and meet 600 yards from Bill's house. The boys continue on their errand and they meet again 700 yards from Tom's house. How far apart do the boys live?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 4
} | 25 | 0 | 0 |
omr-ws-6c1b9d88e246b349d8bac2dbea9e62be7c7d8466f9d16a95c80343e2b69ea4bc | Tom and George play the following game: First, Tom chooses three positive integers \( p, a, \) and \( b \) such that \( p \le a < b \). Then George chooses an integer \( n \) such that \( a \le n \le b \). They play with the set \( M = \{1, 2, \ldots, n\} \). The rules are: 1. The game is divided into turns, with Georg... | [
"10"
] | [
"Tom and George play the following game: First, Tom chooses three positive integers \\( p, a, \\) and \\( b \\) such that \\( p \\le a < b \\). Then George chooses an integer \\( n \\) such that \\( a \\le n \\le b \\). They play with the set \\( M = \\{1, 2, \\ldots, n\\} \\). The rules are:\n1. The game is divide... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb103eea4ed9283869faa213ec064db5be8932a6ff5fc892e02d3f85d0549ac5 | Tom and Huck paint a fence for four hours, after which Jim helps them and they finish two hours later. If Jim had not helped them, it would have taken them 5 more hours to paint the fence. How long would it take for Jim to paint the fence alone? | [
"6"
] | [
"Tom and Huck paint a fence for four hours, after which Jim helps them and they finish two hours later. If Jim had not helped them, it would have taken them 5 more hours to paint the fence. How long would it take for Jim to paint the fence alone?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 4
} | 13 | 0 | 0 |
omr-ws-b20dacbac1178e2aea2462b1f11c3716259ca9dd8c80ec8d92bb96eeaf98d5eb | Tom and Jerry are dividing a chain of 101 sausages by taking turns biting the connections. When a connection is broken, the player may eat any single sausage that falls out. Tom takes the first bite. Each player tries to eat more sausages than the other. Who will succeed in eating more sausages? | [
"Tom"
] | [
"Tom and Jerry are dividing a chain of 101 sausages by taking turns biting the connections. When a connection is broken, the player may eat any single sausage that falls out. Tom takes the first bite. Each player tries to eat more sausages than the other. Who will succeed in eating more sausages?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-2164f32ecfaef493d5198e49051b9279cec7aed86eccf6ac413ce9a9d3674d28 | Tom and Jerry are dividing a chain of 7 sausages by taking turns biting the connections. When a connection is broken, the player may eat any single sausage that falls out. Tom takes the first bite. Each player tries to eat more sausages than the other. Who will succeed in eating more sausages? | [
"Tom"
] | [
"Tom and Jerry are dividing a chain of 7 sausages by taking turns biting the connections. When a connection is broken, the player may eat any single sausage that falls out. Tom takes the first bite. Each player tries to eat more sausages than the other. Who will succeed in eating more sausages?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6763b226fac7c370ff90d6216f8294601598ec0c66afcf3374e720bc78c6e231 | Tom and Jerry are playing a game on a \( 5 \times 5 \) grid. Tom starts at the bottom left corner and Jerry starts at the top right corner. In each move, Tom can move to any adjacent cell (up, down, left, right) and Jerry can move to any adjacent cell as well. Tom's goal is to land on the same cell as Jerry. Let \( A(5... | [
"8"
] | [
"Tom and Jerry are playing a game on a \\( 5 \\times 5 \\) grid. Tom starts at the bottom left corner and Jerry starts at the top right corner. In each move, Tom can move to any adjacent cell (up, down, left, right) and Jerry can move to any adjacent cell as well. Tom's goal is to land on the same cell as Jerry. Le... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-875bae8a5277fa96b825962d6e224e9622b75548112d9eb4ec2488b54487bee6 | Tom and Jerry play a game on a 25 x 25 chessboard by alternately placing pawns on empty squares, with Tom going first. A player wins if, after their move, some four pawns form the vertices of a rectangle with sides parallel to the sides of the board. Which player has a winning strategy? | [
"Tom has a winning strategy."
] | [
"Tom and Jerry play a game on a 25 x 25 chessboard by alternately placing pawns on empty squares, with Tom going first. A player wins if, after their move, some four pawns form the vertices of a rectangle with sides parallel to the sides of the board. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-9a930dfecbeb9916843279b2cf19f5bd968568b36db98ea62845efbaaae44de4 | Tom challenges Harry to a game. Tom first blindfolds Harry and places $4$ quarters on an index card, one on each corner of the card. It is Harry’s job to flip all the coins either face-up or face-down using the following rules: (a) Harry is allowed to flip as many coins as he wants during his turn. (b) A turn consists ... | [
"4"
] | [
"Tom challenges Harry to a game. Tom first blindfolds Harry and places $4$ quarters on an index card, one on each corner of the card. It is Harry’s job to flip all the coins either face-up or face-down using the following rules:\n(a) Harry is allowed to flip as many coins as he wants during his turn.\n(b) A turn co... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-18c678a062be489573ac799761067e4535b42863a90eebafa4fb0811c3b57337 | Tom colours 2006 vertices on a circle using 17 colours. Marry connects pairs of vertices with bowstrings such that the vertices are of the same colour and the bowstrings do not intersect except at the vertices. Determine the greatest number of bowstrings Marry can draw. | [
"233"
] | [
"Tom colours 2006 vertices on a circle using 17 colours. Marry connects pairs of vertices with bowstrings such that the vertices are of the same colour and the bowstrings do not intersect except at the vertices. Determine the greatest number of bowstrings Marry can draw."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e05dc2c5ce3598320a87e00df08c4f27510ba5f0da5efbdde01f097aa545fd84 | Tom comes up with a number whose sum of the digits is 2023. Sam knows only the sum of the digits of Tom's number. In each step, Sam chooses a natural number \( x \) and Tom responds with the sum of the digits of \( |a - x| \), where \( a \) is Tom's number. What is the minimum number of steps Sam needs to determine Tom... | [
"2023"
] | [
"Tom comes up with a number whose sum of the digits is 2023. Sam knows only the sum of the digits of Tom's number. In each step, Sam chooses a natural number \\( x \\) and Tom responds with the sum of the digits of \\( |a - x| \\), where \\( a \\) is Tom's number. What is the minimum number of steps Sam needs to de... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7fecb4482b9eed5ffd2426f20e2e0455cfd0a2e2ad5ec3f85533e6fba20a35b0 | Tom drives 4000 km during a trip and rotates the tires (four on the car and one spare) so that each tire is used for the same distance. How many kilometers does each tire cover? | [
"3200 km"
] | [
"Tom drives 4000 km during a trip and rotates the tires (four on the car and one spare) so that each tire is used for the same distance. How many kilometers does each tire cover?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-d180420d422cc21a3832a1e6d7c6c4049448b9f09408fb2fe08ed0ffd08f119f | Tom drives in a city with 11 two-way streets running north-south and 11 two-way streets running east-west. Starting one block east from the center, and only able to make left turns or go straight at intersections, how many different ways can he reach the northeast corner of the city without passing by the same spot twi... | [
"1"
] | [
"Tom drives in a city with 11 two-way streets running north-south and 11 two-way streets running east-west. Starting one block east from the center, and only able to make left turns or go straight at intersections, how many different ways can he reach the northeast corner of the city without passing by the same spo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-9c5475d87bd7bb07bc000b55ce1696b2978c477abec7334f7591bac5428de0af | Tom has 100 pairs of books, with 67 pairs written in Chinese and 33 pairs written in English. He arranges all books in a row in all possible ways and calculates how many times books written in different languages are adjacent to each other for each arrangement. Let \( S \) be the arithmetic mean of these numbers. Deter... | [
"88.4"
] | [
"Tom has 100 pairs of books, with 67 pairs written in Chinese and 33 pairs written in English. He arranges all books in a row in all possible ways and calculates how many times books written in different languages are adjacent to each other for each arrangement. Let \\( S \\) be the arithmetic mean of these numbers... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 5,
"tir": 17
} | 46 | 0 | 0 |
omr-ws-7f150333bc52cbad2833021a7505a190ad8a2ea7ca3b519d0581136de453b136 | Tom has 13 weight pieces, 12 of which weigh the same and one fake piece that weighs more. He has two balances: one that correctly shows which pan is heavier or if they are equal, and another that gives the correct result when the weights differ but a random result when they are equal. Tom does not know which balance is... | [
"4"
] | [
"Tom has 13 weight pieces, 12 of which weigh the same and one fake piece that weighs more. He has two balances: one that correctly shows which pan is heavier or if they are equal, and another that gives the correct result when the weights differ but a random result when they are equal. Tom does not know which balan... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-78a20859477a508c564807b4b03ae3081dd0af89bf476348a4dee3d1bab5a8fe | Tom has 3 blue and 4 red balls in a bag. Dick takes 2 balls out of the bag and keeps them. Harry then takes out one of the balls left in the bag. If the probability of Harry taking out a blue ball is \( \frac{x}{210} \), what is the value of \( x \)? | [
"90"
] | [
"Tom has 3 blue and 4 red balls in a bag. Dick takes 2 balls out of the bag and keeps them. Harry then takes out one of the balls left in the bag. If the probability of Harry taking out a blue ball is \\( \\frac{x}{210} \\), what is the value of \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-b1afabe3d82d4562c4ff483a44276b5e22d10ffde9dfe02245dfa0c6eba61120 | Tom has a problematic keyboard where each key hit has a $10\%$ chance of being interpreted as two hits. This applies to all letter keys and the backspace key. For example, hitting the $\texttt{p}$ key outputs $\texttt{p}$ $90\%$ of the time and $\texttt{pp}$ $10\%$ of the time. Hitting the backspace key removes one let... | [
"\\(\\frac{130}{9}\\)"
] | [
"Tom has a problematic keyboard where each key hit has a $10\\%$ chance of being interpreted as two hits. This applies to all letter keys and the backspace key. For example, hitting the $\\texttt{p}$ key outputs $\\texttt{p}$ $90\\%$ of the time and $\\texttt{pp}$ $10\\%$ of the time. Hitting the backspace key remo... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-50ac660147e3ea86a5c7f1f6bb7926a0a00754252ce87ac67dffb0a9026724e3 | Tom has a red marble, a green marble, a blue marble, and three identical yellow marbles. How many different groups of two marbles can Tom choose? | [
"7"
] | [
"Tom has a red marble, a green marble, a blue marble, and three identical yellow marbles. How many different groups of two marbles can Tom choose?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-690e8759a62d1b947966f77e01c6152104b376838bdc44a65b9c7c2eb20ef609 | Tom has a scientific calculator with only the keys 1, 2, 3, +, and - working. He presses a sequence of 5 random keystrokes, and the calculator evaluates the expression to yield a result $E$. Find the expected value of $E$. (Note: Negative numbers are permitted, excess operators are parsed as signs, and trailing operato... | [
"0"
] | [
"Tom has a scientific calculator with only the keys 1, 2, 3, +, and - working. He presses a sequence of 5 random keystrokes, and the calculator evaluates the expression to yield a result $E$. Find the expected value of $E$. (Note: Negative numbers are permitted, excess operators are parsed as signs, and trailing op... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-3008dfa34df93f379260deca05832241512e84b85676791f515fa1583d0c7144 | Tom has a triangle $\triangle ABC$ such that $AB = BC = 40$. What is the sum of the minimum and maximum possible integer value of $AC$? | [
"79"
] | [
"Tom has a triangle $\\triangle ABC$ such that $AB = BC = 40$. What is the sum of the minimum and maximum possible integer value of $AC$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c20766eaae9b61ff4436fa19db20d458de7f55ab9ad8f258a9f791bc9bdb303 | Tom has selected a number at random between 2 and 10, and Kay has selected a number between 6 and 12 with the condition that the likelihood that her number is between 9 and 12 is twice the likelihood that her number is between 6 and 9. Find the probability that the sum of the two numbers is greater than 13. | [
"\\(\\frac{115}{144}\\)"
] | [
"Tom has selected a number at random between 2 and 10, and Kay has selected a number between 6 and 12 with the condition that the likelihood that her number is between 9 and 12 is twice the likelihood that her number is between 6 and 9. Find the probability that the sum of the two numbers is greater than 13."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7502853b6750ae76e6b2819d5063dd29f9714c4122bbc7da2a8293670959240e | Tom has twelve slips of paper with the numbers 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4, and 4.5. He wants to distribute these slips into five cups labeled $A$, $B$, $C$, $D$, and $E$ such that the sum of the numbers in each cup is an integer and the sums are consecutive and increasing from $A$ to $E$. Given that a slip w... | [
"D"
] | [
"Tom has twelve slips of paper with the numbers 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4, and 4.5. He wants to distribute these slips into five cups labeled $A$, $B$, $C$, $D$, and $E$ such that the sum of the numbers in each cup is an integer and the sums are consecutive and increasing from $A$ to $E$. Given that a s... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e92f07557cad36a529a59975be79b22d3660cd0ad4bea79654105bda8c40cfa | Tom has two sets \( A = \{1, 2, 3, 4, 5, 6, 7, 8\} \) and \( B = \{1, 2, 3, 4, 5, 6, 7, 8\} \). Jerry has two sets \( C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) and \( D = \{2, 3, 4, 5\} \). Both randomly pick a number from each of their respective sets and sum them. Who has a higher probability of obtaining a higher sum... | [
"Tom has a higher probability of winning."
] | [
"Tom has two sets \\( A = \\{1, 2, 3, 4, 5, 6, 7, 8\\} \\) and \\( B = \\{1, 2, 3, 4, 5, 6, 7, 8\\} \\). Jerry has two sets \\( C = \\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\} \\) and \\( D = \\{2, 3, 4, 5\\} \\). Both randomly pick a number from each of their respective sets and sum them. Who has a higher probability of o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-867e16627f0fcce67492af96e06535983f4ef0fe9e214d519b23cfffc00b5458 | Tom is twice as old as Harry was when Harry was one-third as old as Tom will be when Tom is as old as Harry is now. If the difference in their ages is 666, how old is Harry? | [
"1998"
] | [
"Tom is twice as old as Harry was when Harry was one-third as old as Tom will be when Tom is as old as Harry is now. If the difference in their ages is 666, how old is Harry?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5e83d98b0a65fe4ff1091d6ab37b45a1013d29a052acf36d844e023382e7bf2 | Tom is twice the age that Ben was when Ben was 75% of Tom’s age, which was 10 years ago. How many years older is Tom than Ben? | [
"5"
] | [
"Tom is twice the age that Ben was when Ben was 75% of Tom’s age, which was 10 years ago. How many years older is Tom than Ben?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a83457a2622ebeab8a5489e5fcfc9dcf02c4a595a6d3cfb10f3c903568ef9669 | Tom is walking across a bridge on a train track and is 5/12 of the way across when he notices a train approaching at 90 km/h. If he can just escape death by running in either direction, how fast can Tom run? | [
"15 km/h"
] | [
"Tom is walking across a bridge on a train track and is 5/12 of the way across when he notices a train approaching at 90 km/h. If he can just escape death by running in either direction, how fast can Tom run?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-f00f9ca7fd2eef333fa848b6e7bc0c6627d865931afafba2fa58e6f5fdc8452e | Tom made two rectangles of $2 \times 6$ and $7 \times 8$ from several rectangular tiles measuring $1 \times 3$ and $1 \times 4$, but Jerry snatched and hid one tile from each of the rectangles. Will Tom be able to make a $5 \times 12$ rectangle from the remaining tiles? | [
"\\text{No}"
] | [
"Tom made two rectangles of $2 \\times 6$ and $7 \\times 8$ from several rectangular tiles measuring $1 \\times 3$ and $1 \\times 4$, but Jerry snatched and hid one tile from each of the rectangles. Will Tom be able to make a $5 \\times 12$ rectangle from the remaining tiles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-1659902da5deeeafb84232ccec609d200ccb94fec546468f7f8044b795ac8f9e | Tom painted a round fence consisting of $2n \ge 6$ sections in such a way that every section is painted in one of four colors. Then he repeats the following while it is possible: he chooses three neighboring sections of distinct colors and repaints them into the fourth color. For which $n$ can Tom repaint the fence in ... | [
"\\( n \\ge 3 \\)"
] | [
"Tom painted a round fence consisting of $2n \\ge 6$ sections in such a way that every section is painted in one of four colors. Then he repeats the following while it is possible: he chooses three neighboring sections of distinct colors and repaints them into the fourth color. For which $n$ can Tom repaint the fen... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b42abe6175f3ddde9bb3590880cb2a6d8e26d81f9d803b8e99d3071763f84a95 | Tom painted a round fence consisting of $2n \ge 6$ sections, with each section painted in one of four colors. He repeats the following process while possible: he chooses three neighboring sections of distinct colors and repaints them into the fourth color. For which values of $n$ can Tom repaint the fence in such a way... | [
"n \\not\\equiv 0 \\pmod{3}"
] | [
"Tom painted a round fence consisting of $2n \\ge 6$ sections, with each section painted in one of four colors. He repeats the following process while possible: he chooses three neighboring sections of distinct colors and repaints them into the fourth color. For which values of $n$ can Tom repaint the fence in such... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-8587a0a0116652d517dd9b3a8e41b7d508d82ec1e017dec1605c68a094939efb | Tom painted a round fence consisting of 2005 sections, with each section painted in one of four colors. He repeats the following process while possible: he chooses three neighboring sections of distinct colors and repaints them into the fourth color. Let $N$ be the maximum number of times Tom can perform this operation... | [
"2"
] | [
"Tom painted a round fence consisting of 2005 sections, with each section painted in one of four colors. He repeats the following process while possible: he chooses three neighboring sections of distinct colors and repaints them into the fourth color. Let $N$ be the maximum number of times Tom can perform this oper... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-9b8bb462c7b2ec1e5ab73915c64b133d70c1dd4c9fe2a587e101ba539d07c822 | Tom picks a polynomial \( p \) with nonnegative integer coefficients. Sally asks for \( p(1) \) and \( p(p(1) + 1) \). Given \( p(1) = 10 \) and \( p(11) = 46610 \), determine the polynomial \( p \). | [
"\\( p(x) = 3x^4 + 2x^3 + 2x + 3 \\)"
] | [
"Tom picks a polynomial \\( p \\) with nonnegative integer coefficients. Sally asks for \\( p(1) \\) and \\( p(p(1) + 1) \\). Given \\( p(1) = 10 \\) and \\( p(11) = 46610 \\), determine the polynomial \\( p \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-c456a83de5ced9f935975de1ce50e7ee89921c2fbf46e8bd7a144983a698ebb5 | Tom randomly selects two distinct numbers from the set {1, 2, 3, 4, 5} and Steve randomly selects a number from the set {1, 2, 3, ..., 9, 10}. What is the probability that Steve's number is larger than the sum of the two numbers selected by Tom? | [
"\\(\\frac{2}{5}\\)"
] | [
"Tom randomly selects two distinct numbers from the set {1, 2, 3, 4, 5} and Steve randomly selects a number from the set {1, 2, 3, ..., 9, 10}. What is the probability that Steve's number is larger than the sum of the two numbers selected by Tom?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-e655e0c4351b75f1f8e47e34373b5f43921fd3e0282e8b5ea3c198c54580aca7 | Tom starts with a positive integer \( a \). At each step, he adds one of the numbers 67, 78, or 89 randomly to \( a \). Let \( N \) be the smallest number of steps after which Tom is guaranteed to have a number whose last two digits are either 67, 78, or 89. Find \( N \). | [
"34"
] | [
"Tom starts with a positive integer \\( a \\). At each step, he adds one of the numbers 67, 78, or 89 randomly to \\( a \\). Let \\( N \\) be the smallest number of steps after which Tom is guaranteed to have a number whose last two digits are either 67, 78, or 89. Find \\( N \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-7fcb4b0fcbbc138c9463f318cf056a9ecb8e0fb6bdff1d7b47ee76610cb3e079 | Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes a triathlon consisting of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run in four and a quarter hours. How many minutes does he spend bicycling? | [
"150"
] | [
"Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes a triathlon consisting of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run in four and a quarter hours. How many minutes does he spend bicycling?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-af1f5864e500eb0985eecab517b20dc4ee983a1587c2df3ee193e6a7eca31e5a | Tom tosses 5 coins and Jack tosses 6 coins at the same time. If the person who gets more heads wins, what is the probability that Jack wins? | [
"\\(\\frac{1}{2}\\)"
] | [
"Tom tosses 5 coins and Jack tosses 6 coins at the same time. If the person who gets more heads wins, what is the probability that Jack wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-82f66b729d3c0b613c56997923127fcb9ed5c64ab1ab9792c247c6d372146eb9 | Tom uniformly and randomly picks an integer \( N \in \{3, 6, 9, 12, 15\} \). Then, he randomly fills in the digits of a six-digit integer using the digits \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \) without repetition. Tom wins if and only if the six-digit integer is divisible by \( N \). The probability Tom wins can be written in... | [
"73 + 560 = 633"
] | [
"Tom uniformly and randomly picks an integer \\( N \\in \\{3, 6, 9, 12, 15\\} \\). Then, he randomly fills in the digits of a six-digit integer using the digits \\( 1, 2, 3, 4, 5, 6, 7, 8, 9 \\) without repetition. Tom wins if and only if the six-digit integer is divisible by \\( N \\). The probability Tom wins can... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf68b77f05c8822c8835b543ffea6faa648d40851c790effb7cd1f01e3fa7de6 | Tom writes \( n \ge 3 \) distinct positive real numbers on the board. Jerry writes all the \( n(n - 1)(n - 2) \) quadratic polynomials whose coefficients are three distinct numbers written by Tom. Among the polynomials that Jerry writes, at most how many of them have two distinct real roots? | [
"\\dfrac{n(n-1)(n-2)}{3}"
] | [
"Tom writes \\( n \\ge 3 \\) distinct positive real numbers on the board. Jerry writes all the \\( n(n - 1)(n - 2) \\) quadratic polynomials whose coefficients are three distinct numbers written by Tom. Among the polynomials that Jerry writes, at most how many of them have two distinct real roots?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-5db83a63b3e9db6ac7d158ad63f104a9f1dff45f6e1edb7e3a82e4f0d64347f0 | Tom writes down the number $1$. For any number $n$ already written down, Tom writes down the numbers $5n$ and $5n+1$, provided that they are less than $1000$. Jerry computes all sums of two different numbers Tom has written down. How many different sums does Jerry get? | [
"206"
] | [
"Tom writes down the number $1$. For any number $n$ already written down, Tom writes down the numbers $5n$ and $5n+1$, provided that they are less than $1000$. Jerry computes all sums of two different numbers Tom has written down. How many different sums does Jerry get?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-3105c9698bddc3a076f2fbf9865e59c72b285f230ff1f21406e1c20e382a0283 | Tom writes down the number 1. For any number \( n \) already written down, Tom writes down the numbers \( 5n \) and \( 5n + 1 \), provided that they are less than 1000. Jerry computes all sums of two different numbers Tom has written down. How many different sums does Jerry get? | [
"206"
] | [
"Tom writes down the number 1. For any number \\( n \\) already written down, Tom writes down the numbers \\( 5n \\) and \\( 5n + 1 \\), provided that they are less than 1000. Jerry computes all sums of two different numbers Tom has written down. How many different sums does Jerry get?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 24
} | 29 | 0 | 0 |
omr-ws-26253d402cb3096357d1c039621702f67fa0c2142661ff68b2bf008482adb742 | Tom writes down the number 1. For any number \( n \) already written down, Tom writes down the numbers \( 5n \) and \( 5n+1 \), provided that they are less than 1000. Jerry computes all sums of two different numbers Tom has written down. How many different sums does Jerry get? | [
"211"
] | [
"Tom writes down the number 1. For any number \\( n \\) already written down, Tom writes down the numbers \\( 5n \\) and \\( 5n+1 \\), provided that they are less than 1000. Jerry computes all sums of two different numbers Tom has written down. How many different sums does Jerry get?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-2ab6ac974b2bd15bae7f181941e272880d373d75319a86386515b1a679c06af7 | Tom wrote down several distinct positive integers, not exceeding 100. Their product was not divisible by 18. At most how many numbers could he have written? | [
"68"
] | [
"Tom wrote down several distinct positive integers, not exceeding 100. Their product was not divisible by 18. At most how many numbers could he have written?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-187f56b5814da77fcaae3bd5269874bfd7f31fca4b33a4f97f278a37055e1f7a | Tom wrote the following eight distinct positive single-digit integers: \[ \textbf{ \Large 1 \ ? \ 6 \ 4 \ ? \ 3 \ ? \ 5 \normalsize } \] The eight numbers are split into four pairs, and the sum of each pair is identical. Three numbers are missing. Find the product of the three missing numbers. | [
"56"
] | [
"Tom wrote the following eight distinct positive single-digit integers:\n\\[\n \\textbf{ \\Large 1 \\ ? \\ 6 \\ 4 \\ ? \\ 3 \\ ? \\ 5 \\normalsize }\n\\]\nThe eight numbers are split into four pairs, and the sum of each pair is identical. Three numbers are missing. Find the product of the three missing numbers."... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fec7d07219ce9a2cde3f85c9ea76bd7ddccc7f068aae2614520b900282e55157 | Tom's backyard has 2 trees 40 feet apart. Tom wants to plant new trees that are 30 feet from each grown tree. In how many different locations could Tom plant the new trees? | [
"2"
] | [
"Tom's backyard has 2 trees 40 feet apart. Tom wants to plant new trees that are 30 feet from each grown tree. In how many different locations could Tom plant the new trees?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af304bee14b19d30b9afd6e9db2d8c43a584a7c086a70cc611abf174c175d06d | Tom's bricks are all parallelepipeds with integer dimensions not exceeding 7. For each brick, Tom calculates the volume and divides it by the square of the largest dimension. He then sums all these results. What is the irreducible fraction obtained from this sum? | [
"\\(\\frac{947}{1}\\)"
] | [
"Tom's bricks are all parallelepipeds with integer dimensions not exceeding 7. For each brick, Tom calculates the volume and divides it by the square of the largest dimension. He then sums all these results. What is the irreducible fraction obtained from this sum?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1b9ff481f85a8ea15b4a4bb630b5762083521534263b72423e28fef379c31b8 | Tom's graduating class has 288 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 10 rows and at least 15 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$? | [
"58"
] | [
"Tom's graduating class has 288 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 10 rows and at least 15 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea5d3e7ca226d9363f2e899dc744489c4e77fb2322025dcab421abeec78d9cd6 | Tom, Dick, and Harry play a game of dice. Tom wins if he rolls a 1, Dick wins if he rolls a 2 or 3, and Harry wins if he rolls a 4, 5, or 6. They take turns rolling the dice in the order Tom, Dick, Harry, and continue until one of them wins. The probability that Tom wins is in the form \(\frac{a}{b}\), where \(a\) and ... | [
"16"
] | [
"Tom, Dick, and Harry play a game of dice. Tom wins if he rolls a 1, Dick wins if he rolls a 2 or 3, and Harry wins if he rolls a 4, 5, or 6. They take turns rolling the dice in the order Tom, Dick, Harry, and continue until one of them wins. The probability that Tom wins is in the form \\(\\frac{a}{b}\\), where \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-c0b9da2bbf5cc462a7fe3e95b5a0259727acc079deb944f9f2a933ad087dcb88 | Tomek draws $n$ distinct equilateral triangles on a plane and colours each triangle with a different colour. He then draws a line $\ell$ such that each triangle has a side parallel to $\ell$. For an arbitrary point $P$ on $\ell$, if there exists a point $A$ with colour $c$ on some triangle such that $AP \perp \ell$ and... | [
"2n - 1"
] | [
"Tomek draws $n$ distinct equilateral triangles on a plane and colours each triangle with a different colour. He then draws a line $\\ell$ such that each triangle has a side parallel to $\\ell$. For an arbitrary point $P$ on $\\ell$, if there exists a point $A$ with colour $c$ on some triangle such that $AP \\perp ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-e3f1211facae43db65f2bd2e82fe117f4dcd773f544528b57754d14703bca6f0 | Tommy Theta has a bag with 11 marbles. One is black and the rest are white. Every time he picks a marble, the bag is reset and one white marble is exchanged for a black marble. What is the probability that Tommy draws a black marble on or before the third draw? | [
"\\(\\frac{611}{1331}\\)"
] | [
"Tommy Theta has a bag with 11 marbles. One is black and the rest are white. Every time he picks a marble, the bag is reset and one white marble is exchanged for a black marble. What is the probability that Tommy draws a black marble on or before the third draw?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-fdcb32d2103518c138d7a406264cdec8d9767f608835473323b21eede65a8db0 | Tommy is bidding on a cell phone from Jim, where Jim's asking price is uniformly distributed between 0 and 100. If Tommy buys the phone, he can resell it for 80% more than his bid. What should Tommy bid to maximize his expected return? | [
"100"
] | [
"Tommy is bidding on a cell phone from Jim, where Jim's asking price is uniformly distributed between 0 and 100. If Tommy buys the phone, he can resell it for 80% more than his bid. What should Tommy bid to maximize his expected return?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-baf8a497fdf2325c93024c153bf05d649e3df25fe80ade27e157b22d63a5530b | Tommy the Ant is tied to the corner \( O \) of a unit cube with a string of length \( \sqrt{2} \). Tommy can only walk along the surface of the cube. What fraction of the surface of the cube can Tommy walk to? | [
"\\(\\frac{1}{4}\\)"
] | [
"Tommy the Ant is tied to the corner \\( O \\) of a unit cube with a string of length \\( \\sqrt{2} \\). Tommy can only walk along the surface of the cube. What fraction of the surface of the cube can Tommy walk to?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b64e2c6a0ead117b10b8a7f09a7b2e46807a93c4428e1de094205d3555e5579 | TongTong is doodling in class and draws a $3 \times 3$ grid. She then decides to color some (that is, at least one) of the squares blue, such that no two $1 \times 1$ squares that share an edge or a corner are both colored blue. In how many ways may TongTong color some of the squares blue? TongTong cannot rotate or ref... | [
"34\n\nTherefore, the number of ways TongTong can color some of the squares blue under the given conditions is:"
] | [
"TongTong is doodling in class and draws a $3 \\times 3$ grid. She then decides to color some (that is, at least one) of the squares blue, such that no two $1 \\times 1$ squares that share an edge or a corner are both colored blue. In how many ways may TongTong color some of the squares blue? TongTong cannot rotate... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-f27abddb0777eba1c04c25c11592fa0b458ad97b0f718543a41a789f180f639c | Tony Stark has two children. He says one of his children, who is male, was born on February 29. What is the probability that both children are male? | [
"\\(\\frac{13}{27}\\)"
] | [
"Tony Stark has two children. He says one of his children, who is male, was born on February 29. What is the probability that both children are male?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5993d6eab79b89f447fef2bf93017f63813f35aa2e05ec5adaacbc2f56f1e2b1 | Tony Stark has two children. He says one of his children, who is male, was born on Wednesday. What is the probability that both children are male? | [
"\\(\\frac{13}{27}\\)"
] | [
"Tony Stark has two children. He says one of his children, who is male, was born on Wednesday. What is the probability that both children are male?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e720efc54afa1512ee6516480af6683a9ad21efed131e7045e40192b6e811360 | Tony and Maria are training for a race by running up and down a 700 m long ski slope. They each run up the slope at different constant speeds. Coming down the slope, each runs at double their uphill speed. Maria reaches the top first and immediately starts running back down, meeting Tony 70 m from the top. When Maria r... | [
"300 meters"
] | [
"Tony and Maria are training for a race by running up and down a 700 m long ski slope. They each run up the slope at different constant speeds. Coming down the slope, each runs at double their uphill speed. Maria reaches the top first and immediately starts running back down, meeting Tony 70 m from the top. When Ma... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 9
} | 31 | 0 | 0 |
omr-ws-27b386332c745790baceb41ef92a58640571565dcb9f1ec8b20a99d03281f5ee | Tony can buy a bundle of 11 bears for $3 or a bundle of 15 bears for $4. How many amounts of bears are impossible to buy? | [
"70"
] | [
"Tony can buy a bundle of 11 bears for $3 or a bundle of 15 bears for $4. How many amounts of bears are impossible to buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-304ef8dce03f8fc00fc7cd43e7c52e8c05d3bf51bf11ec4cc7bdcc60ffbaa35b | Tony defines the polynomial \( p(x) = x^3 - 35x^2 + bx - 1 \) with roots \( r, s, \) and \( t \). He notices that \( r+s+t, r^2+s^2+t^2, \) and \( r^3+s^3+t^3 \) are all one less than a multiple of \( k \), and \( r^4+s^4+t^4 \) is five less than a multiple of \( k \), for a positive integer \( k \). Compute the produc... | [
"1"
] | [
"Tony defines the polynomial \\( p(x) = x^3 - 35x^2 + bx - 1 \\) with roots \\( r, s, \\) and \\( t \\). He notices that \\( r+s+t, r^2+s^2+t^2, \\) and \\( r^3+s^3+t^3 \\) are all one less than a multiple of \\( k \\), and \\( r^4+s^4+t^4 \\) is five less than a multiple of \\( k \\), for a positive integer \\( k ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-8d087608b1347171484e4230bba519677137bc45ae49c53bf5431b191aa77296 | Tony has 5 distinct shirts and 5 distinct hats. He can wear up to 5 shirts and any number of hats, but he must wear at least one shirt and one hat. How many different outfits can he wear? | [
"961"
] | [
"Tony has 5 distinct shirts and 5 distinct hats. He can wear up to 5 shirts and any number of hats, but he must wear at least one shirt and one hat. How many different outfits can he wear?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d80651a4d040a2e6fe9a2ffcaeb68b0feeb76c64935a7f02ec57f3802bd441f | Tony has \(k\) stones on square 0 in a row of \(n+1\) squares labelled from 0 to \(n\). On each move, he may choose a stone and advance the stone up to \(m\) squares where \(m\) is the number of stones on the same square (including itself) or behind it. Tony's goal is to get all stones to square \(n\). Calculate the mi... | [
"16"
] | [
"Tony has \\(k\\) stones on square 0 in a row of \\(n+1\\) squares labelled from 0 to \\(n\\). On each move, he may choose a stone and advance the stone up to \\(m\\) squares where \\(m\\) is the number of stones on the same square (including itself) or behind it. Tony's goal is to get all stones to square \\(n\\).... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-b50b90d57a0151c33ba5224a9ab486d659fcc380d6b0fdc73e3b8a4eab3b35f6 | Tony has a board with 2 rows and 4 columns. Tony will use 8 numbers from 1 to 8 to fill in this board, each number in exactly one entry. Let array $(a_1, \ldots, a_4)$ be the first row of the board and array $(b_1, \ldots, b_4)$ be the second row of the board. Let $F = \sum_{i=1}^{4} |a_i - b_i|$, calculate the average... | [
"12"
] | [
"Tony has a board with 2 rows and 4 columns. Tony will use 8 numbers from 1 to 8 to fill in this board, each number in exactly one entry. Let array $(a_1, \\ldots, a_4)$ be the first row of the board and array $(b_1, \\ldots, b_4)$ be the second row of the board. Let $F = \\sum_{i=1}^{4} |a_i - b_i|$, calculate the... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-d9a8e76310e762d8c051514c885dcd6b702bc796032d06c4e266abcbd72aea82 | Tony has a laser located at (0,0) in the plane, initially pointing in the positive direction of the x-axis. He rotates the laser counterclockwise and labels the first point with integer coordinates in each direction, where both coordinates have an absolute value of at most 5. After completing a full rotation, he has la... | [
"58"
] | [
"Tony has a laser located at (0,0) in the plane, initially pointing in the positive direction of the x-axis. He rotates the laser counterclockwise and labels the first point with integer coordinates in each direction, where both coordinates have an absolute value of at most 5. After completing a full rotation, he h... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5da96747f5bfdd6a41eedf8500e7d7dd6a9a4425e39250d7361bb145b0260b76 | Tony has two boxes, $A$ and $B$. Box $A$ contains 5 blue sticks, and box $B$ contains 4 blue sticks and 1 green stick. Tony randomly selects one of the boxes and then randomly draws one stick from the chosen box. He repeats this process until one of the boxes is empty. Calculate the probability that Tony does not draw ... | [
"47"
] | [
"Tony has two boxes, $A$ and $B$. Box $A$ contains 5 blue sticks, and box $B$ contains 4 blue sticks and 1 green stick. Tony randomly selects one of the boxes and then randomly draws one stick from the chosen box. He repeats this process until one of the boxes is empty. Calculate the probability that Tony does not ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab536df2130293e1316d5e7e81b5501998339c38b82a6c585bfceb3d112bc41b | Tony has two candles, both 30 cm tall. The height of the thicker candle decreases by 3/4 of the amount the thinner candle loses each minute. After 30 minutes, the height of the thinner candle is 2/3 of the thicker candle. What is the height of the thicker candle 15 minutes after it has been lit? | [
"22.5"
] | [
"Tony has two candles, both 30 cm tall. The height of the thicker candle decreases by 3/4 of the amount the thinner candle loses each minute. After 30 minutes, the height of the thinner candle is 2/3 of the thicker candle. What is the height of the thicker candle 15 minutes after it has been lit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-85a5c453e680556ffd83226c030755698a5ebd618960e22d6decc39c26544b85 | Tony is a Ping Pong player who can feel whether a ping pong ball is good or not. The Ping Pong club orders balls from either PPB company (with an $80\%$ chance of being good) or MIO company (with a $50\%$ chance of being good). The club orders from each company with equal probability. Tony tries the first ball and find... | [
"\\dfrac{89}{130}"
] | [
"Tony is a Ping Pong player who can feel whether a ping pong ball is good or not. The Ping Pong club orders balls from either PPB company (with an $80\\%$ chance of being good) or MIO company (with a $50\\%$ chance of being good). The club orders from each company with equal probability. Tony tries the first ball a... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-764a8b85e5ef91a9c3a923612d34d755f8a424a3d4974c8ecaea5db5698c84b1 | Tony is moving on a straight line for 6 minutes. Several observers watch him such that he is never unattended for any second. Each observer watches Tony continuously for exactly one minute during which Tony covers exactly one meter. What is the sum of the minimal and maximal possible distances Tony can walk during the ... | [
"14"
] | [
"Tony is moving on a straight line for 6 minutes. Several observers watch him such that he is never unattended for any second. Each observer watches Tony continuously for exactly one minute during which Tony covers exactly one meter. What is the sum of the minimal and maximal possible distances Tony can walk during... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2f6ff23678d8c4ea0ae09be3b2791f8c6e013ad533305f95bd1afc9dc6854b2 | Tony is moving on a straight line for 6 minutes. Several observers watch him such that he is never unattended for any second. Each observer watches Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony can wa... | [
"12"
] | [
"Tony is moving on a straight line for 6 minutes. Several observers watch him such that he is never unattended for any second. Each observer watches Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-ebcb2298a06e266f2ae27440848a26b8a5de1a6de267f1ed0d9a56cd024b5f88 | Tony lists all the factors of $105$ in order. Bryan randomly selects one of these factors and writes down all the numbers from $1$ to that factor. Mike then chooses an integer $x$ uniformly at random from Bryan's list, and Carl selects a number $y$ uniformly at random from $1$ to $x$. If the expected value of $y$ can b... | [
"31"
] | [
"Tony lists all the factors of $105$ in order. Bryan randomly selects one of these factors and writes down all the numbers from $1$ to that factor. Mike then chooses an integer $x$ uniformly at random from Bryan's list, and Carl selects a number $y$ uniformly at random from $1$ to $x$. If the expected value of $y$ ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-b5591dded2e490f85255da01eaa5c4b7f48fb3f2623038efc8360b1dcbb705bb | Tony places numbers on a grid starting from the origin. For each lattice point, he places the minimum nonnegative number that is not located directly to the left or below. What number does Tony place at the coordinate \((123, 89)\)? | [
"0"
] | [
"Tony places numbers on a grid starting from the origin. For each lattice point, he places the minimum nonnegative number that is not located directly to the left or below. What number does Tony place at the coordinate \\((123, 89)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1846085e6cc8d2e2d000abe7cbad28a38c6028690daa1a8cb0f2df9cf9ab7530 | Tony the Turkey is jumping on a trampoline. On his first bounce, he reaches a height of 1 foot. On his second bounce, he reaches a height of 2 feet. On his third bounce, he reaches a height of 6 feet. He continues alternately doubling/tripling the height of his bounces (so 1, 2, 6, 12, 36, 72, etc.). If Tim bounces 100... | [
"\\(3 \\times \\frac{6^{500} - 1}{5}\\)"
] | [
"Tony the Turkey is jumping on a trampoline. On his first bounce, he reaches a height of 1 foot. On his second bounce, he reaches a height of 2 feet. On his third bounce, he reaches a height of 6 feet. He continues alternately doubling/tripling the height of his bounces (so 1, 2, 6, 12, 36, 72, etc.). If Tim bounce... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-3c1b71e787928a8df7e93b35dfee09f8e90d2d9b73a2e5202a0b098652f40449 | Tony the mouse starts in the top left corner of a $3 \times 3$ grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center? | [
"\\(\\frac{1}{7}\\)"
] | [
"Tony the mouse starts in the top left corner of a $3 \\times 3$ grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-1cc3d2d53d5c71bdc34bc241fee8a7c1c61bc556399a6da8d1d4ae877cf32875 | Tony the mouse starts in the top left corner of a \(3 \times 3\) grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center? | [
"\\dfrac{1}{7}"
] | [
"Tony the mouse starts in the top left corner of a \\(3 \\times 3\\) grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d2cd9bbf7442fb122f07511f429fe62e0259e7599a14123bf2a9b09079d216cb | Tony the mouse starts in the top left corner of a \(3 \times 3\) grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability that he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center? | [
"\\(\\frac{1}{7}\\)"
] | [
"Tony the mouse starts in the top left corner of a \\(3 \\times 3\\) grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability that he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-f03b9eef05edb023276fa2f5f17a0ae005427386400f646c205bb2a500f31085 | Tony will paint 16 right, cylindrical columns. The top face and bottom face of each column will be covered, so those parts will not be painted. Each column is 18 feet tall and has a diameter of 10 feet. One gallon of paint will cover 350 square feet. If paint is sold only in full gallons, how many gallons of paint must... | [
"26"
] | [
"Tony will paint 16 right, cylindrical columns. The top face and bottom face of each column will be covered, so those parts will not be painted. Each column is 18 feet tall and has a diameter of 10 feet. One gallon of paint will cover 350 square feet. If paint is sold only in full gallons, how many gallons of pa... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e262c34130c458d445eeabf0fa8fcbe9b560af889074050cefff8e6696d2f085 | Tony will raise twenty dollars, plus two dollars for each lap he walks. Sandy will raise thirty dollars, plus 1.50 dollars for each lap she walks. After how many laps will they have raised the same amount of money? | [
"20"
] | [
"Tony will raise twenty dollars, plus two dollars for each lap he walks. Sandy will raise thirty dollars, plus 1.50 dollars for each lap she walks. After how many laps will they have raised the same amount of money?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3583dc59f480af10131da9b25f1f2900275885aa86b1507be1ea97bfea218ba6 | Tony works 2 hours a day and is paid $0.50 per hour for each full year of his age. During a six-month period, Tony worked 50 days and earned $630. How old was Tony at the end of the six-month period? | [
"13"
] | [
"Tony works 2 hours a day and is paid $0.50 per hour for each full year of his age. During a six-month period, Tony worked 50 days and earned $630. How old was Tony at the end of the six-month period?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-b957e5fea3c9c14ee6fbae3b3124cf7b668f6527c71c5cdee98ce71ad9f21016 | Tony's frisbee has a diameter of 24 inches, while Steven's frisbee has a radius of 12 inches. Tony claims that the surface area of his frisbee is $n$ times the surface area of Steven's frisbee. What is $n$? | [
"1"
] | [
"Tony's frisbee has a diameter of 24 inches, while Steven's frisbee has a radius of 12 inches. Tony claims that the surface area of his frisbee is $n$ times the surface area of Steven's frisbee. What is $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-18117043533027641f6a9da10755148226d9c9a5367c0edd6c2e40b9f7bcf911 | Toothless the dragon is at $(0,0)$ and wants to get to $(6,6)$. There is a pond full of fish at $(2,5)$. If Toothless can only fly east one unit or north one unit at a time, how many paths can he take that will allow him to catch some fish before returning home? | [
"105"
] | [
"Toothless the dragon is at $(0,0)$ and wants to get to $(6,6)$. There is a pond full of fish at $(2,5)$. If Toothless can only fly east one unit or north one unit at a time, how many paths can he take that will allow him to catch some fish before returning home?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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