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-1e135a751ea4220c3885e23190e08be56d405c93a4772c37bd3d2aa2ca97206f | Tsumugi has a chocolate bar in the form of a grid measuring $2 \times n$ for $n \in \mathbb{N}$. Yui and Ritsu take turns removing a piece of chocolate, where each piece must be a rectangle measuring $1 \times 2$ or $2 \times 3$ units and the sides of the rectangle must match the grid. The player who cannot pick up the... | [
"n \\not\\equiv 0 \\text{ or } 1 \\pmod{5}"
] | [
"Tsumugi has a chocolate bar in the form of a grid measuring $2 \\times n$ for $n \\in \\mathbb{N}$. Yui and Ritsu take turns removing a piece of chocolate, where each piece must be a rectangle measuring $1 \\times 2$ or $2 \\times 3$ units and the sides of the rectangle must match the grid. The player who cannot p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-e10f6143072bb450cdb4288f225ced0deab906f8a4314b60e48c85afb5677e93 | Tunna collects money from the king under the condition that the amount doubles each day from 1 taka. After 6 days, she collected a total of 43 taka. On how many days did she go to collect money? | [
"4"
] | [
"Tunna collects money from the king under the condition that the amount doubles each day from 1 taka. After 6 days, she collected a total of 43 taka. On how many days did she go to collect money?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20638e6d82cd5216b7954eced8c93e173978c3b0f2ed4420118b710568ecbc3e | Turbo the snail is given $4n$ positive real numbers $x_1, x_2, \ldots, x_{4n}$. He starts at a point in the plane and slides $x_1$ meters north, $x_2$ meters east, $x_3$ meters south, $x_4$ meters west, and so on, cycling between north, east, south, and west. After going through all $4n$ numbers, he finds himself back ... | [
"36"
] | [
"Turbo the snail is given $4n$ positive real numbers $x_1, x_2, \\ldots, x_{4n}$. He starts at a point in the plane and slides $x_1$ meters north, $x_2$ meters east, $x_3$ meters south, $x_4$ meters west, and so on, cycling between north, east, south, and west. After going through all $4n$ numbers, he finds himself... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0453419981cb1be0d4bf18b3cdc21127980b3d5be2bc6399707adbba9ac488d5 | Turbo the snail slides along $n$ circles drawn on a piece of paper such that any two circles intersect in two points, and no three circles pass through the same point. Initially moving in a clockwise direction on one of the circles, Turbo changes direction (clockwise to anticlockwise or vice versa) at each intersection... | [
"3"
] | [
"Turbo the snail slides along $n$ circles drawn on a piece of paper such that any two circles intersect in two points, and no three circles pass through the same point. Initially moving in a clockwise direction on one of the circles, Turbo changes direction (clockwise to anticlockwise or vice versa) at each interse... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-6a763314e566fc27532881a2eab5e80d2146de316e588c72b7b376d020515748 | Turgenev arrived in Kiev by train and visited the city. After having dinner on G square, he decided to return to the train station using only those paths he had stridden an odd number of times. Let the train station be vertex $A$ and G square be vertex $B$ in a graph representing the city's paths. If Turgenev can retur... | [
"3"
] | [
"Turgenev arrived in Kiev by train and visited the city. After having dinner on G square, he decided to return to the train station using only those paths he had stridden an odd number of times. Let the train station be vertex $A$ and G square be vertex $B$ in a graph representing the city's paths. If Turgenev can ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-4c550aa316606a714de1716bebcf1dc8789a6b908a662e1d593f0b9f17a63326 | Tushar the frog is trying to climb a set of stairs. He can jump up one stair at a time or two stairs at a time. How many ways can Tushar the Frog jump up a set of 12 stairs? | [
"233"
] | [
"Tushar the frog is trying to climb a set of stairs. He can jump up one stair at a time or two stairs at a time. How many ways can Tushar the Frog jump up a set of 12 stairs?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb97946d37cafddf8c44e2d7493193dd09b028396c86bd856c70217f4e92b2ae | Tushar wrote the numbers from 1 to 2019 side by side to form a large number: 123456...2019. How many digits appear an even number of times in this large number? | [
"9"
] | [
"Tushar wrote the numbers from 1 to 2019 side by side to form a large number: 123456...2019. How many digits appear an even number of times in this large number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2cef9aaf824e9961c0f7a997f5016a068d650dd5815e47d2ce51903273cc0c76 | Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that sat... | [
"5"
] | [
"Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination tha... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9a0780b8979966b0fd2d0d0dd229f36cf47d2477c62f858800c7384e96096010 | Twain is walking randomly on a number line. They start at \(0\), and flip a fair coin 10 times. Every time the coin lands heads, they increase their position by 1, and every time the coin lands tails, they decrease their position by 1. What is the probability that at some point the absolute value of their position is a... | [
"\\dfrac{175}{256}"
] | [
"Twain is walking randomly on a number line. They start at \\(0\\), and flip a fair coin 10 times. Every time the coin lands heads, they increase their position by 1, and every time the coin lands tails, they decrease their position by 1. What is the probability that at some point the absolute value of their positi... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-927727119b0e2edf3d3898380d88785f40a42da0fb10b0b2f0c9d92dc8b8ceae | Tweedledee and Tweedledum start at the points $(0.5, 0.5)$ and $(0.7, 0.7)$, respectively, on the Cartesian coordinate plane. Lines are drawn for every change of $\pm 0.1$ from a lattice point in the $x$ or $y$ coordinates. They conduct a random walk within the unit square defined by $\{(0, 0), (1, 0), (0, 1), (1, 1)\}... | [
"3"
] | [
"Tweedledee and Tweedledum start at the points $(0.5, 0.5)$ and $(0.7, 0.7)$, respectively, on the Cartesian coordinate plane. Lines are drawn for every change of $\\pm 0.1$ from a lattice point in the $x$ or $y$ coordinates. They conduct a random walk within the unit square defined by $\\{(0, 0), (1, 0), (0, 1), (... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-3f25044b0b27b03f7b7d0ebafba6a063ce823b72687a6b8be49e3045609f5a20 | Twelve 1-Euro-coins are laid flat on a table such that their midpoints form a regular 12-gon, and adjacent coins are tangent to each other. Prove that it is possible to place another seven such coins into the interior of the ring formed by the twelve coins. | [
"7"
] | [
"Twelve 1-Euro-coins are laid flat on a table such that their midpoints form a regular 12-gon, and adjacent coins are tangent to each other. Prove that it is possible to place another seven such coins into the interior of the ring formed by the twelve coins."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-5e7b684d06d3c5fc1eb1ae90e0c8f7a6479faa74948932966b950a10278b73b6 | Twelve candidates for mayor participate in a TV talk show. Each candidate, in turn, makes a statement about the number of lies told so far: the first says "One lie has been told," the second says "Now two lies have been told," and so on until the twelfth says "Now twelve lies have been told." It is known that at least ... | [
"12"
] | [
"Twelve candidates for mayor participate in a TV talk show. Each candidate, in turn, makes a statement about the number of lies told so far: the first says \"One lie has been told,\" the second says \"Now two lies have been told,\" and so on until the twelfth says \"Now twelve lies have been told.\" It is known tha... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-66be823dee1aca65ae181f169e6a2726225bc9c084f08e6df972e3c1dd408068 | Twelve cards are placed in a box, with 5 red cards (1 point each), 4 white cards (2 points each), and 3 blue cards (3 points each). Three cards are picked at random from the box. What is the probability that the total point value of the three cards is greater than or equal to 7, rounded to two decimal places? | [
"0.21"
] | [
"Twelve cards are placed in a box, with 5 red cards (1 point each), 4 white cards (2 points each), and 3 blue cards (3 points each). Three cards are picked at random from the box. What is the probability that the total point value of the three cards is greater than or equal to 7, rounded to two decimal places?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-24c957cf9d159529509f19197db5b4cff1e1cf4d8c4547f9ef7fd7f32cad5b38 | Twelve cards lie in a row. The cards are of three kinds: with both sides white, both sides black, or with a white and a black side. Initially, nine of the twelve cards have a black side up. The cards $1-6$ are turned, and subsequently four of the twelve cards have a black side up. Now cards $4-9$ are turned, and six ca... | [
"9 BW, 3 WW, 0 BB"
] | [
"Twelve cards lie in a row. The cards are of three kinds: with both sides white, both sides black, or with a white and a black side. Initially, nine of the twelve cards have a black side up. The cards $1-6$ are turned, and subsequently four of the twelve cards have a black side up. Now cards $4-9$ are turned, and s... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-2c88aa34c51eafd96a183fccafb78b06a1c40d9d1574b9b019a9bb8c036ad36d | Twelve children are to be separated into 4 groups of 3. Find the probability that Marcus, Maxim, and Meep are in the same group. | [
"\\dfrac{1}{55}"
] | [
"Twelve children are to be separated into 4 groups of 3. Find the probability that Marcus, Maxim, and Meep are in the same group."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ade3ff528979dfaa42d8dbd6c51d59b3476275f63ed28347150ebf580079f680 | Twelve children have the following amounts of candy: 123, 154, 245, 250, 305, 310, 363, 382, 483, 485, 497, and 605 pieces. A group of children is called "neat" if the total amount of candy they have can be divided into packets of twelve pieces with no leftovers. What is the smallest number of packets that can be made ... | [
"114"
] | [
"Twelve children have the following amounts of candy: 123, 154, 245, 250, 305, 310, 363, 382, 483, 485, 497, and 605 pieces. A group of children is called \"neat\" if the total amount of candy they have can be divided into packets of twelve pieces with no leftovers. What is the smallest number of packets that can b... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b41b17b6de4e8a4041a2fec37cd4ad0946b668331c72b63159202ffc02465e3d | Twelve congruent CDs are placed on a circle \( C \) of radius such that they cover the circle \( C \). Each CD is tangent to each other and none overlaps. Let \( r \) be the sum of the areas of the twelve CDs. Let \( R \) be the area of the circle \( C \). If \( \frac{R}{r} = \frac{a + b \sqrt{c}}{d} \) where \( c \) i... | [
"26"
] | [
"Twelve congruent CDs are placed on a circle \\( C \\) of radius such that they cover the circle \\( C \\). Each CD is tangent to each other and none overlaps. Let \\( r \\) be the sum of the areas of the twelve CDs. Let \\( R \\) be the area of the circle \\( C \\). If \\( \\frac{R}{r} = \\frac{a + b \\sqrt{c}}{d}... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-b3a1c285b16d069d9becab886e44396c296e304e53994e8f07819475a0fa1a53 | Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and each disk is tangent to its two neighbors. The sum of the areas of the twelve disks can be written in the form $\pi(a - b\sqrt{c})$, where $a$, $b$, and $c$ are positive integers... | [
"135"
] | [
"Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and each disk is tangent to its two neighbors. The sum of the areas of the twelve disks can be written in the form $\\pi(a - b\\sqrt{c})$, where $a$, $b$, and $c$ are positive i... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-fd124c30ea0248f5c050d8b99ee54c64441261e21dc051274e9c70186f1143ed | Twelve congruent disks are placed on a circle \( C \) of radius 1 in such a way that the twelve disks cover \( C \), no two of the disks overlap, and each disk is tangent to its two neighbors. The sum of the areas of the twelve disks can be written in the form \( \pi(a - b\sqrt{c}) \), where \( a, b, c \) are positive ... | [
"135"
] | [
"Twelve congruent disks are placed on a circle \\( C \\) of radius 1 in such a way that the twelve disks cover \\( C \\), no two of the disks overlap, and each disk is tangent to its two neighbors. The sum of the areas of the twelve disks can be written in the form \\( \\pi(a - b\\sqrt{c}) \\), where \\( a, b, c \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-1a66920d0a08d4e13d3b02a7fe4a97dcc23d93c5d5b48b907aeb767eb71c217f | Twelve congruent disks are placed on a circle of radius 1 such that the disks cover the circle, do not overlap, and each disk is tangent to its two neighbors. The sum of the areas of the twelve disks can be written in the form $\pi(a - b\sqrt{c})$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible ... | [
"12"
] | [
"Twelve congruent disks are placed on a circle of radius 1 such that the disks cover the circle, do not overlap, and each disk is tangent to its two neighbors. The sum of the areas of the twelve disks can be written in the form $\\pi(a - b\\sqrt{c})$, where $a$, $b$, and $c$ are positive integers and $c$ is not div... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c636d9810b81c5517132cc26255600fa9536d6e682e4020e8f86e506244a4abb | Twelve couples are seated around a circular table such that all men are seated side by side, and each woman is seated opposite her husband. In each step, a woman and a man next to her swap places. What is the least possible number of swaps until all couples are seated side by side? | [
"66"
] | [
"Twelve couples are seated around a circular table such that all men are seated side by side, and each woman is seated opposite her husband. In each step, a woman and a man next to her swap places. What is the least possible number of swaps until all couples are seated side by side?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 8,
"tir": 1
} | 44 | 0 | 0 |
omr-ws-d52a52987f48439a16fb43cfb80a49ee8dc08780ace25a5d72c89d58fec31562 | Twelve delegates attend a conference and are seated around a circular table. Each delegate brings three copies of a document and passes one copy to each of his immediate neighbors, keeping one copy for himself. A thief steals one document from each chair. What is the probability that the thief obtains a complete set of... | [
"\\(\\frac{4}{6561}\\)"
] | [
"Twelve delegates attend a conference and are seated around a circular table. Each delegate brings three copies of a document and passes one copy to each of his immediate neighbors, keeping one copy for himself. A thief steals one document from each chair. What is the probability that the thief obtains a complete s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 15
} | 16 | 0 | 0 |
omr-ws-c4862ebb296e17d203e29dc4206bda08f7588201df404d3ab5a394bb15f45fe6 | Twelve equal disks are arranged without overlapping so that each disk touches two others and together they cover every point inside a circle of radius 1. Find the total area of the disks. | [
"\\( 12 \\pi \\left( \\frac{1}{\\sqrt{6} + \\sqrt{2} + 1} \\right)^2 \\)"
] | [
"Twelve equal disks are arranged without overlapping so that each disk touches two others and together they cover every point inside a circle of radius 1. Find the total area of the disks."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-c8f30ccefb87ec819968562af07dca609f9ce5b779de5dbbccb9e8e0b5ec25cd | Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime? | [
"\\(\\left(\\frac{1}{6}\\right)^{10}\\)"
] | [
"Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 63
} | 95 | 0 | 0 |
omr-ws-86bcecd42faebee9bd093a8c09f3a0810557e2d61ae0de1a4f4722fdb28febc0 | Twelve keys, of which only one fits, are tried one after another until a door opens. Find the probability that the door will open a) on the 12th try b) on none of the first three tries. | [
"a) 1, b) \\(\\frac{3}{4}\\)"
] | [
"Twelve keys, of which only one fits, are tried one after another until a door opens. Find the probability that the door will open a) on the 12th try b) on none of the first three tries."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-bb3dc0e0ebcc4c1ebb4f611c079826fc97030d88f3ba23de39c8e5479b0ad7d9 | Twelve knights sit at a round table. Each knight hates the two knights sitting next to him but none of the others. A task group of five knights is to be selected such that no two knights who hate each other are included. In how many ways can this group be selected? | [
"36"
] | [
"Twelve knights sit at a round table. Each knight hates the two knights sitting next to him but none of the others. A task group of five knights is to be selected such that no two knights who hate each other are included. In how many ways can this group be selected?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e38aaa7f77951b512bbe2cd5910e6a694a7d1fe67fab75195691b4215b212a3 | Twelve labeled points are spaced equally on a circle. How many ways are there to color 3 of the points red, 3 green, 3 blue, and 3 white such that for every two different colors, there exists a straight line that separates all the points of those two colors? | [
"288"
] | [
"Twelve labeled points are spaced equally on a circle. How many ways are there to color 3 of the points red, 3 green, 3 blue, and 3 white such that for every two different colors, there exists a straight line that separates all the points of those two colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-78d95b6c5ef3179a44548eb5a1dfa0f641af82af4906a4816feff56a1bd70434 | Twelve musicians $M_1, \ldots, M_{12}$ gather at a week-long chamber music festival. Each day, there is one scheduled concert, and some of the musicians play while the others listen as members of the audience. For $i=1,2,\ldots,12$, let $t_i$ be the number of concerts in which musician $M_i$ plays, and let $t = t_1 + t... | [
"12"
] | [
"Twelve musicians $M_1, \\ldots, M_{12}$ gather at a week-long chamber music festival. Each day, there is one scheduled concert, and some of the musicians play while the others listen as members of the audience. For $i=1,2,\\ldots,12$, let $t_i$ be the number of concerts in which musician $M_i$ plays, and let $t = ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-adfecd66ebc80795badeaa6587048f9ff1766dda0e0f99d181993a30780b5d46 | Twelve musicians gather at a week-long chamber music festival. Each day, there is one scheduled concert and some of the musicians play while the others listen as members of the audience. For \( i=1,2,3,\ldots,12 \), let \( t_i \) be the number of concerts in which each musician plays, and let \( t = t_1 + t_2 + \ldots ... | [
"22"
] | [
"Twelve musicians gather at a week-long chamber music festival. Each day, there is one scheduled concert and some of the musicians play while the others listen as members of the audience. For \\( i=1,2,3,\\ldots,12 \\), let \\( t_i \\) be the number of concerts in which each musician plays, and let \\( t = t_1 + t_... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4ab70fb818091923748e3df73924ad063a540257fe5212c43bde72e0f4514ce | Twelve painters live in twelve houses, each one black or white, on a circular street. Each month, a different painter walks clockwise along the street, repainting the houses the opposite color, starting with his own, and stops as soon as he paints a white house black. After one year, calculate the total number of house... | [
"6"
] | [
"Twelve painters live in twelve houses, each one black or white, on a circular street. Each month, a different painter walks clockwise along the street, repainting the houses the opposite color, starting with his own, and stops as soon as he paints a white house black. After one year, calculate the total number of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2da1a979052dd3c610b4329237dab33601b4814f7db58d3c6eec66677c716899 | Twelve painters live in twelve houses, each one black or white, on a circular street. Each month, a different painter walks clockwise along the street, repainting the houses the opposite color, starting with his own, and stops as soon as he paints a white house black. After one year, calculate the total number of house... | [
"12"
] | [
"Twelve painters live in twelve houses, each one black or white, on a circular street. Each month, a different painter walks clockwise along the street, repainting the houses the opposite color, starting with his own, and stops as soon as he paints a white house black. After one year, calculate the total number of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-fe17c73e8112d5cf42b5d9c7a8472d6c505a2218b597817f3dd2eb7da87d43c0 | Twelve painters live in twelve houses, each one black or white, on a circular street. Each month, a different painter walks clockwise along the street, repainting the houses the opposite color, starting with his own, and stops as soon as he paints a white house black. After one year, calculate the total number of house... | [
"12"
] | [
"Twelve painters live in twelve houses, each one black or white, on a circular street. Each month, a different painter walks clockwise along the street, repainting the houses the opposite color, starting with his own, and stops as soon as he paints a white house black. After one year, calculate the total number of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d40dd409a24fbbe5e64008b30a0af014d1bb2ef74e5eef7b763feac1ad2d8c8a | Twelve people are playing hot potato in a circle. At each step, the person with the potato passes it right or left with equal probability. If Moe begins by passing the potato left to Larry, what is the probability that Curly, who is on Moe's immediate right, is the last player to touch the potato? | [
"\\(\\frac{2}{11}\\)"
] | [
"Twelve people are playing hot potato in a circle. At each step, the person with the potato passes it right or left with equal probability. If Moe begins by passing the potato left to Larry, what is the probability that Curly, who is on Moe's immediate right, is the last player to touch the potato?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-8c86e64e51ef49be72c2bf32bf6a9ef709dd9c3753b2acd7b20f11cc91f6a10d | Twelve people are seated around a circular table with 28 flower vases. Two people can see each other if and only if there is no flower vase aligned with them. Calculate the minimum number of pairs of people who are able to see each other. | [
"6"
] | [
"Twelve people are seated around a circular table with 28 flower vases. Two people can see each other if and only if there is no flower vase aligned with them. Calculate the minimum number of pairs of people who are able to see each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-db42783e22c0ae15f67ebce7c8b6ce63bc77332270bb904b3da520225f216537 | Twelve people are standing in a circle and play hot potato. At each step, the person with the potato passes it to the right or left with equal probability. If Moe begins by passing the potato left to Larry, what is the probability that Curly, who is on Moe’s immediate right, is the last player to touch the potato? | [
"\\(\\frac{2}{11}\\)"
] | [
"Twelve people are standing in a circle and play hot potato. At each step, the person with the potato passes it to the right or left with equal probability. If Moe begins by passing the potato left to Larry, what is the probability that Curly, who is on Moe’s immediate right, is the last player to touch the potato?... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a19eabc6cd2ea07235fa0fca61e47c1fe718708a926fafd707c06ea2f5364d71 | Twelve people go to a party. First, everybody with no friends at the party leaves. Then, at the $i$-th hour, everybody with exactly $i$ friends left at the party leaves. After the eleventh hour, what is the maximum number of people left? Note that friendship is mutual. | [
"0"
] | [
"Twelve people go to a party. First, everybody with no friends at the party leaves. Then, at the $i$-th hour, everybody with exactly $i$ friends left at the party leaves. After the eleventh hour, what is the maximum number of people left? Note that friendship is mutual."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-541e5ee5200a6f6ebf736398be7eb921b9e187f204a6313c0210fdf1c2a344f1 | Twelve people sit around a round table, labeled from 1 to 12 o'clock. A fair coin is given to the person at 6 o'clock. Each person with the coin flips it: if heads, the coin is passed counter-clockwise; if tails, the coin is passed clockwise. The game stops when everyone has touched the coin at least once. What is the ... | [
"\\(\\frac{1}{11}\\)"
] | [
"Twelve people sit around a round table, labeled from 1 to 12 o'clock. A fair coin is given to the person at 6 o'clock. Each person with the coin flips it: if heads, the coin is passed counter-clockwise; if tails, the coin is passed clockwise. The game stops when everyone has touched the coin at least once. What is... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 24
} | 43 | 0 | 0 |
omr-ws-265d1c0e129f4be68c402849588d8fe415efc026f7f112ab328ecd63eb09b4a7 | Twelve people used two different treatments (A and B) to help them get to sleep. Of the twelve, 2 preferred A, 8 preferred B, and 2 were undecided. Calculate the appropriate p-value for a sign test of the null hypothesis H0: "A and B are equally effective" against the alternative hypothesis H1: "A and B are not equally... | [
"7/64"
] | [
"Twelve people used two different treatments (A and B) to help them get to sleep. Of the twelve, 2 preferred A, 8 preferred B, and 2 were undecided. Calculate the appropriate p-value for a sign test of the null hypothesis H0: \"A and B are equally effective\" against the alternative hypothesis H1: \"A and B are not... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 23
} | 37 | 0 | 0 |
omr-ws-bf76ec67e2b7f2498ed414222876c0cc7ee99dd788dd9d312c5c92209223aac6 | Twelve people with different heights are to be placed in two parallel lines, each with 6 people. Determine the total number of possibilities if the people should be in ascending order from left to right in each line and no person should have in front of them a person with greater stature. | [
"132"
] | [
"Twelve people with different heights are to be placed in two parallel lines, each with 6 people. Determine the total number of possibilities if the people should be in ascending order from left to right in each line and no person should have in front of them a person with greater stature."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 3
} | 51 | 0 | 0 |
omr-ws-ac95d9359d2bfe6518c25fc148ea93938d81a931370cb64e54053c4b857cedf7 | Twelve people, three of whom are in the Mafia and one of whom is a police inspector, randomly sit around a circular table. What is the probability that the inspector ends up sitting next to at least one of the Mafia? | [
"\\dfrac{27}{55}"
] | [
"Twelve people, three of whom are in the Mafia and one of whom is a police inspector, randomly sit around a circular table. What is the probability that the inspector ends up sitting next to at least one of the Mafia?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb5a4e8a4c528b2f27b01d2bc44eb388c0006686ff574404d90f96ca7be7adaf | Twelve points \( A_1, A_2, A_3, \ldots, A_{12} \) are equally spaced around a circle of radius 6. What is the area of the region \( A_1A_6A_8 \)? | [
"\\( 9(2 + \\sqrt{3}) \\)"
] | [
"Twelve points \\( A_1, A_2, A_3, \\ldots, A_{12} \\) are equally spaced around a circle of radius 6. What is the area of the region \\( A_1A_6A_8 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7a3105bdc1d1b24af4f9f86812535ee9d0b8a5e69dedde654ead9827dbf32594 | Twelve points \( A_1, A_2, A_3, \ldots, A_{12} \) are equally spaced around a circle with area \( 4\pi \). What is the area of quadrilateral \( A_1A_2A_7A_{10} \)? | [
"6"
] | [
"Twelve points \\( A_1, A_2, A_3, \\ldots, A_{12} \\) are equally spaced around a circle with area \\( 4\\pi \\). What is the area of quadrilateral \\( A_1A_2A_7A_{10} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53fc2dce350e0eb7b93ed23ba5b1c9126e9e2d5a78b292d5d1aa95c17091738b | Twelve points are arranged along the edges of a 3 x 3 square as shown. How many triangles have all three of their vertices among these points? | [
"204"
] | [
"Twelve points are arranged along the edges of a 3 x 3 square as shown. How many triangles have all three of their vertices among these points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 16
} | 38 | 0 | 0 |
omr-ws-c9804ff4b7c5e3a310e12a8d4cd4c0a0a0e98b0096b2d289b60c81f06a9a843b | Twelve points are equally spaced on the circumference of a circle of radius 1. The distance between each pair of adjacent points is in the form \(\frac{a \pi}{b}\). Find \( a + b \). | [
"7"
] | [
"Twelve points are equally spaced on the circumference of a circle of radius 1. The distance between each pair of adjacent points is in the form \\(\\frac{a \\pi}{b}\\). Find \\( a + b \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-43c54f0554d8f1851324085f1d5770520a583c7b1b1333b18ad784c2de832fcf | Twelve points are located in space such that seven of them lie on a single plane (with no three collinear), and the remaining five are in general position (none of them lies on the planes determined by any other three points). How many distinct planes pass through at least three of the twelve points? | [
"116"
] | [
"Twelve points are located in space such that seven of them lie on a single plane (with no three collinear), and the remaining five are in general position (none of them lies on the planes determined by any other three points). How many distinct planes pass through at least three of the twelve points?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c4ada4b8f5c9acdf07068bd29241940b10d843e386694780e95817306e9dada2 | Twelve points are located on a clock with equal distance, numbered $1, 2, 3, \ldots, 12$ in clockwise order. Use four colors (red, yellow, blue, green) to color the points, with each color used for exactly three points. Construct $n$ convex quadrilaterals using these points as vertices such that: 1. The colors of the v... | [
"9"
] | [
"Twelve points are located on a clock with equal distance, numbered $1, 2, 3, \\ldots, 12$ in clockwise order. Use four colors (red, yellow, blue, green) to color the points, with each color used for exactly three points. Construct $n$ convex quadrilaterals using these points as vertices such that:\n1. The colors o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-b6d683a8af82b4137d519bc7cc72e6a2222fe6e94f435a4853c64ede65cc1303 | Twelve points, four of which are vertices, lie on the perimeter of a square. The distance between adjacent points is one unit. Some of the points have been connected by straight lines, and $B$ is the intersection of two of those lines. Find the ratio $AB:BC$ in its simplest form. | [
"1:2"
] | [
"Twelve points, four of which are vertices, lie on the perimeter of a square. The distance between adjacent points is one unit. Some of the points have been connected by straight lines, and $B$ is the intersection of two of those lines. Find the ratio $AB:BC$ in its simplest form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-b932a872ed14af94b8b2f220fd59cbbc551ec2a5f4878af2c8478c8e44bd9128 | Twelve positive integers are located around a circle such that each integer is equal to 2 plus the greatest common divisor of its two neighbors. Find the greatest possible sum of these twelve integers. | [
"56"
] | [
"Twelve positive integers are located around a circle such that each integer is equal to 2 plus the greatest common divisor of its two neighbors. Find the greatest possible sum of these twelve integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-e0ddc945840f201ba0d7748896283c0542787381c6c024657c280b16a50302d4 | Twelve ships will leave San Francisco, one per month, to travel around the horn of South America bound for New York. At the same time, twelve different ships will leave New York, one per month, along the same route bound for San Francisco. Each ship will take six months to reach its destination. Excluding meetings in t... | [
"72"
] | [
"Twelve ships will leave San Francisco, one per month, to travel around the horn of South America bound for New York. At the same time, twelve different ships will leave New York, one per month, along the same route bound for San Francisco. Each ship will take six months to reach its destination. Excluding meetings... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e5e925e304bde5728374dda13b8a70b1d96322fa9f7db99f0860a484ff7ea49 | Twelve small circles, each with radius \( r \), are arranged on a plane such that each circle externally touches its adjacent circles, and all twelve circles externally touch a larger circle with radius \( R \). If the ratio \( \frac{r}{R} \) can be expressed as \(\left[\sqrt{m-1} + \sqrt{n} - 1\right]\) where \( m \) ... | [
"28"
] | [
"Twelve small circles, each with radius \\( r \\), are arranged on a plane such that each circle externally touches its adjacent circles, and all twelve circles externally touch a larger circle with radius \\( R \\). If the ratio \\( \\frac{r}{R} \\) can be expressed as \\(\\left[\\sqrt{m-1} + \\sqrt{n} - 1\\right]... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6c874881ee539989f3b1526368c3d0103984e187f9c47c11f4962ecc581cb692 | Twelve squares are arranged in a circular order and four neighboring squares are occupied by four counters of different color: red, yellow, green and blue. Any counter can be moved from the square it occupies across any four squares to the fifth one (provided that the latter is not occupied) in any of the two possible ... | [
"Change the order of the arrangement of the squares so that it becomes possible to move from any square to the neighboring ones"
] | [
"Twelve squares are arranged in a circular order and four neighboring squares are occupied by four counters of different color: red, yellow, green and blue. Any counter can be moved from the square it occupies across any four squares to the fifth one (provided that the latter is not occupied) in any of the two poss... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c578efe53e7e57340648f98e93ad00ed2fc0351eb6fb0cdf393f35b4e08929d | Twelve students in Mrs. Stephenson's class have brown eyes. Twenty students in the class have a lunch box. Of Mrs. Stephenson's 30 students, what is the least possible number of students who have brown eyes and a lunch box? | [
"2"
] | [
"Twelve students in Mrs. Stephenson's class have brown eyes. Twenty students in the class have a lunch box. Of Mrs. Stephenson's 30 students, what is the least possible number of students who have brown eyes and a lunch box?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-123232a8d0d5acdfe1ff2062747790bfbe97e75bf5535a1583da64a349e8c448 | Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$. | [
"4"
] | [
"Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-0e783396e217b20080466092dec38ba91d7352069086cfa0fd65e070ad896587 | Twelve tables are set up in a row for a Millenium party. You want to put 2000 cupcakes on the tables so that the numbers of cupcakes on adjacent tables always differ by one. Find a way to do this. | [
"\\( 166, 167, 166, 167, 166, 167, 166, 167, 166, 167, 166, 168 \\)"
] | [
"Twelve tables are set up in a row for a Millenium party. You want to put 2000 cupcakes on the tables so that the numbers of cupcakes on adjacent tables always differ by one. Find a way to do this."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80364e71685356efed69ad72247df4ebbda929409f438cfb5ec8bc804917f2a5 | Twenty ants crawled along the edges of the twenty vertices of a regular dodecahedron made of wire. Each ant can crawl on its edges as it pleases, with the only restriction being that it can never exceed $1$ edge per minute. After time $t$ after the beginning of the movement, it turned out that by this time each pair of... | [
"3"
] | [
"Twenty ants crawled along the edges of the twenty vertices of a regular dodecahedron made of wire. Each ant can crawl on its edges as it pleases, with the only restriction being that it can never exceed $1$ edge per minute. After time $t$ after the beginning of the movement, it turned out that by this time each pa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-2ef6398e95eacbc2c240eab7359de39e8c713ecf04eefa1f1025365470a1bb51 | Twenty ants live on the faces of an icosahedron, one ant on each face, where each face has a side length of 1. Each ant moves in a counterclockwise direction along the edges of the icosahedron at a speed of at least 1. If two ants meet, they meet at a vertex. If five ants meet at the same time at a vertex, it is called... | [
"2"
] | [
"Twenty ants live on the faces of an icosahedron, one ant on each face, where each face has a side length of 1. Each ant moves in a counterclockwise direction along the edges of the icosahedron at a speed of at least 1. If two ants meet, they meet at a vertex. If five ants meet at the same time at a vertex, it is c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df6e2f6daadbe057641b3ecc67da852ff8b17b3b7647add6e665536441b26819 | Twenty balls are placed in an urn: five are red, five are green, five are yellow, and five are blue. Three balls are drawn at random without replacement. What is the probability that exactly one of the balls drawn is red? | [
"\\(\\frac{35}{76}\\)"
] | [
"Twenty balls are placed in an urn: five are red, five are green, five are yellow, and five are blue. Three balls are drawn at random without replacement. What is the probability that exactly one of the balls drawn is red?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-236955eb86e0da696d4a0b32800cc8c4a9cf64fb35ddacbc4f394ae686713063 | Twenty balls are placed in an urn: five are red, five are green, five are yellow, and five are blue. Three balls are drawn at random without replacement. What is the probability that the number of blue balls drawn is strictly greater than the number of yellow balls drawn? | [
"\\(\\frac{10}{57}\\)"
] | [
"Twenty balls are placed in an urn: five are red, five are green, five are yellow, and five are blue. Three balls are drawn at random without replacement. What is the probability that the number of blue balls drawn is strictly greater than the number of yellow balls drawn?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51fb5de6d0bbb109c7e3fec0159f5285dfda5d195ac3ee3e4e397d9851cf051d | Twenty balls are placed in an urn: five are red, five are green, five are yellow, and five are blue. Three balls are drawn at random without replacement. What is the probability that the three balls drawn have different colors? | [
"\\(\\frac{25}{57}\\)"
] | [
"Twenty balls are placed in an urn: five are red, five are green, five are yellow, and five are blue. Three balls are drawn at random without replacement. What is the probability that the three balls drawn have different colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-7bb878debfb06f8dc5703c276421fb53ffc9570b6ad7af332a5f512bd84a6e48 | Twenty boxes with weights 10, 20, 30, ..., 200 pounds are given. One hand is needed to lift a box for every 10 pounds it weighs. Determine the number of people needed to lift all the boxes simultaneously, given that no person can help lift more than one box at a time. | [
"110"
] | [
"Twenty boxes with weights 10, 20, 30, ..., 200 pounds are given. One hand is needed to lift a box for every 10 pounds it weighs. Determine the number of people needed to lift all the boxes simultaneously, given that no person can help lift more than one box at a time."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-e0c312d40d2352b4a3987b9d7f5e28b095bcc90cad1db8f0f47adbdc6a625891 | Twenty chairs are set up in a row for a contest. Only five participants attend, but none will sit next to any other. In how many ways can the participants be seated? | [
"$5!{16 \\choose 5}$"
] | [
"Twenty chairs are set up in a row for a contest. Only five participants attend, but none will sit next to any other. In how many ways can the participants be seated?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-047a93c9d62d21ead361d6b55d4b3a6e0e3cb7296e920069ae70e35fa1538fac | Twenty chairs are set up in a row for the Princeton garlic-eating contest. Only five eaters attend the competition, but none will sit next to any other. In how many ways can the eaters be seated? | [
"524160",
"524160.",
"\\( \\binom{16}{5} \\times 5! \\)"
] | [
"Twenty chairs are set up in a row for the Princeton garlic-eating contest. Only five eaters attend the competition, but none will sit next to any other. In how many ways can the eaters be seated?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 16,
"tir": 81
} | 138 | 0 | 0 |
omr-ws-98d864e9b3eb0f5ac35fe2642d3117bb67346ac8e910a9dd1ce6b6a6296b95d6 | Twenty children are queueing for ice cream that is sold at R5 per cone. Ten of the children have a R5 coin, and the others have a R10 bill. At the beginning, the ice cream man doesn't have any change. How many possible arrangements of the twenty kids in a queue are there so that the ice cream man will never run out of ... | [
"16796"
] | [
"Twenty children are queueing for ice cream that is sold at R5 per cone. Ten of the children have a R5 coin, and the others have a R10 bill. At the beginning, the ice cream man doesn't have any change. How many possible arrangements of the twenty kids in a queue are there so that the ice cream man will never run ou... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-09b41f157e9fb4fc6be1b9b52bee8c01f1ed63b6a93cbabd37db02036dd135f1 | Twenty congruent circles are arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p} - q\right),$ where $p$ and $q$ are positive ... | [
"154"
] | [
"Twenty congruent circles are arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\\frac{1}{2}\\left(\\sqrt{p} - q\\right),$ where $p$ and $q$ are ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-e52a9bdd55b88806a1c4d4e2cce514f17ab376ca10751461e888cf0484648a3f | Twenty cubical blocks are arranged in layers as follows: the bottom layer has 10 blocks, the second layer has 6 blocks, the third layer has 3 blocks, and the top layer has 1 block. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2, 3, and 4 is assigned the number which is th... | [
"114"
] | [
"Twenty cubical blocks are arranged in layers as follows: the bottom layer has 10 blocks, the second layer has 6 blocks, the third layer has 3 blocks, and the top layer has 1 block. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2, 3, and 4 is assigned the number which ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 3,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-759195a0aae04d720b25b596aee7cce5041832d0cecc93b9abdb6598d135c8fd | Twenty dice numbered 1-20 are rolled independently. Any dice that rolls a prime number is taken out. The remaining dice are rolled again. What is the expected value of the sum of the numbers on the remaining dice after the second roll? | [
"126"
] | [
"Twenty dice numbered 1-20 are rolled independently. Any dice that rolls a prime number is taken out. The remaining dice are rolled again. What is the expected value of the sum of the numbers on the remaining dice after the second roll?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-66ca73530e9d0bcb21c9b62e1ff09b69d68dc70929f823b8c4cb0afc9b2cc3b6 | Twenty different books are to be put on 5 shelves, each of which holds at least 20 books. How many different arrangements are there if the order on the shelves matters? | [
"\\( 20! \\cdot 21^4 \\)"
] | [
"Twenty different books are to be put on 5 shelves, each of which holds at least 20 books. How many different arrangements are there if the order on the shelves matters?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-b1fe5bc61a5ef8093c3a0183676831e92a1ad3fdc10e3ecb8e3b10f1322c1c5d | Twenty different books are to be put on 5 shelves, each of which holds at least 20 books. How many different arrangements are there if we only care about the number of books on the shelves (and not which books are where)? | [
"\\(\\binom{24}{20}\\)"
] | [
"Twenty different books are to be put on 5 shelves, each of which holds at least 20 books. How many different arrangements are there if we only care about the number of books on the shelves (and not which books are where)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 20,
"tir": 22
} | 42 | 0 | 0 |
omr-ws-7b153bafd54a08d1c6b5bdadef9a19102a5c003302b9bdc8a6967bb173c60562 | Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points. | [
"072",
"72"
] | [
"Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points."
] | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 6,
"tir": 40
} | 72 | 0 | 0 |
omr-ws-5761c5f42b19b9585c99f788b70d2448fe66c02c2dd989c44e98f9efc260ddfa | Twenty equal circles are packed in a honeycomb fashion within a rectangle. The outer rows each contain 7 circles, and the middle row contains 6 circles. The outer circles touch the sides of the rectangle. Find the ratio of the long side of the rectangle to the short side. | [
"\\dfrac{7(\\sqrt{3} - 1)}{2}"
] | [
"Twenty equal circles are packed in a honeycomb fashion within a rectangle. The outer rows each contain 7 circles, and the middle row contains 6 circles. The outer circles touch the sides of the rectangle. Find the ratio of the long side of the rectangle to the short side."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3901e13a78cb849c3a06250bf173ef9b516bf98274d4ad2fd307bb7261064041 | Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in l... | [
"057",
"57"
] | [
"Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 24
} | 56 | 0 | 0 |
omr-ws-674added107ec9c059ea0466e5a0a11110659b6dfeee88d306dde363a00f6423 | Twenty five points are evenly spaced around a circle. Among the $300$ chords connecting two of these points, how many ways can two distinct chords be chosen and extended into lines, such that the two lines intersect and that the angle formed by their intersection is an integer? | [
"2530"
] | [
"Twenty five points are evenly spaced around a circle. Among the $300$ chords connecting two of these points, how many ways can two distinct chords be chosen and extended into lines, such that the two lines intersect and that the angle formed by their intersection is an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-d25dd9ae0fa4b7a14b5516ea52916c69183b7a54c46a1ad311a3c444a599a427 | Twenty girls are sitting around a table and are playing a game with \( n \) cards. Initially, one girl holds all the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends if and only if each girl is holding at most one card. L... | [
"171"
] | [
"Twenty girls are sitting around a table and are playing a game with \\( n \\) cards. Initially, one girl holds all the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends if and only if each girl is holding at most one ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-ec415984a728bd43cce4e17f400d364582c9d49c5cc5ebb294e1ec4bb038430a | Twenty gremlins and fifteen imps are at the Annual Mischief Convention. The imps have had a lot of in-fighting lately and refuse to shake hands with each other, but they readily shake hands with all of the gremlins. Meanwhile, all the gremlins are quite friendly and shake hands with all of the other gremlins as well as... | [
"490"
] | [
"Twenty gremlins and fifteen imps are at the Annual Mischief Convention. The imps have had a lot of in-fighting lately and refuse to shake hands with each other, but they readily shake hands with all of the gremlins. Meanwhile, all the gremlins are quite friendly and shake hands with all of the other gremlins as we... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d7b3ba2ff70bfcb806deab741d365c334610d8c8dcac627ed98c362f5c3e25b | Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After each of the first $10$ customers has made their purchase, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exactly ... | [
"10 kg"
] | [
"Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After each of the first $10$ customers has made their purchase, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-969711e52fbba8873949f93bf19878c8a299bd5367d9b9baf8a6c0cc0837aa91 | Twenty lamps are placed on the vertices of a regular $20$-gon. The lights turn on and off according to the following rule: if, in second $k$ the two lights adjacent to a light $L$ are either both on or both off, $L$ is on in second $k+1$; otherwise, $L$ is off in second $k+1$. Let $N$ denote the number of possible orig... | [
"7"
] | [
"Twenty lamps are placed on the vertices of a regular $20$-gon. The lights turn on and off according to the following rule: if, in second $k$ the two lights adjacent to a light $L$ are either both on or both off, $L$ is on in second $k+1$; otherwise, $L$ is off in second $k+1$. Let $N$ denote the number of possible... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-70c1cbe234c244fcd9d7e564f7d6972219a89f45401308776b4dbf4ebb751b57 | Twenty letters $a, b, c, d, e, f, g, h, i, j, A, B, C, D, E, F, G, H, I, J$ are written on separate slips of paper. The ten slips containing capital letters are shuffled and placed in a circle in random order. Next, the ten slips containing lowercase letters are shuffled and placed randomly in the spaces between the sl... | [
"\\dfrac{1919}{14175}"
] | [
"Twenty letters $a, b, c, d, e, f, g, h, i, j, A, B, C, D, E, F, G, H, I, J$ are written on separate slips of paper. The ten slips containing capital letters are shuffled and placed in a circle in random order. Next, the ten slips containing lowercase letters are shuffled and placed randomly in the spaces between t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-60d9651b64a7fd757bd95776f6256927ee4058d5a3f596a7b66da5133f98ccc5 | Twenty married couples are at a party. Every man shakes hands with everyone except himself and his spouse. Half of the women refuse to shake hands with any other women. The other 10 women all shake hands with each other (but not themselves). How many handshakes are there at the party? | [
"925"
] | [
"Twenty married couples are at a party. Every man shakes hands with everyone except himself and his spouse. Half of the women refuse to shake hands with any other women. The other 10 women all shake hands with each other (but not themselves). How many handshakes are there at the party?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b41ae98690968d503c5e84aeed5e504b750b8c70577ae9f24b1efeb5bc56a53 | Twenty married couples are at a party. Every man shakes hands with everyone except himself and his spouse. Half of the women refuse to shake hands with any other women. The other 10 women all shake hands with each other (but not with themselves). How many handshakes are there at the party? | [
"615"
] | [
"Twenty married couples are at a party. Every man shakes hands with everyone except himself and his spouse. Half of the women refuse to shake hands with any other women. The other 10 women all shake hands with each other (but not with themselves). How many handshakes are there at the party?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 80
} | 80 | 0 | 0 |
omr-ws-a7306225a5f0e8e4d4a1dec5d5eeebec6ce6e868864880804c986d19a12bdd70 | Twenty of King Arthur's knights are seated at a round table. Three of them are chosen at random to be sent off to slay a dragon. Find the probability that at least two of the three chosen knights had been sitting next to each other. | [
"$\\frac{17}{57}$"
] | [
"Twenty of King Arthur's knights are seated at a round table. Three of them are chosen at random to be sent off to slay a dragon. Find the probability that at least two of the three chosen knights had been sitting next to each other."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-93629f06f962a2b68d271ad323851e0d7171078342acd440d740be8322ede0c5 | Twenty pegs are arranged in a $2 \times 10$ rectangular grid. You have 10 small rubber bands that can stretch to enclose two adjacent pegs (including diagonally adjacent pegs). In how many ways can you stretch the rubber bands over ten pairs of pegs to enclose all twenty pegs? | [
"683"
] | [
"Twenty pegs are arranged in a $2 \\times 10$ rectangular grid. You have 10 small rubber bands that can stretch to enclose two adjacent pegs (including diagonally adjacent pegs). In how many ways can you stretch the rubber bands over ten pairs of pegs to enclose all twenty pegs?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 5
} | 27 | 0 | 0 |
omr-ws-4d9b5284817d96929a4c8668e8ba7ee683864bd6abf6f4a34149ca99f66e0b65 | Twenty points are marked on the circumference of a circle. Two players take turns connecting two of the 20 points with a segment, following these rules: - A segment can only appear once during the game. - No two segments can intersect, except at the endpoints. - The player who has no valid move left loses the game. Ass... | [
"first"
] | [
"Twenty points are marked on the circumference of a circle. Two players take turns connecting two of the 20 points with a segment, following these rules:\n- A segment can only appear once during the game.\n- No two segments can intersect, except at the endpoints.\n- The player who has no valid move left loses the g... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 26
} | 30 | 0 | 0 |
omr-ws-d2e7ec45b1d48e912baa57435ecebda6caf2688588a3b18ca7c406196770dba5 | Twenty points, which form a regular 20-gon, are chosen on a circle. These points are split into ten pairs, and the points in each pair are connected by a chord. Calculate the number of distinct possible lengths of the chords formed. | [
"10"
] | [
"Twenty points, which form a regular 20-gon, are chosen on a circle. These points are split into ten pairs, and the points in each pair are connected by a chord. Calculate the number of distinct possible lengths of the chords formed."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f107a324a7a8a40db178425032b0ae168465f3caf98c23027f6832a2c39e953 | Twenty random cards are placed in a row all face down. A move consists of turning a face down card face up, and turning over the card that is immediately to its right. Let $N$ be the maximum number of moves required to ensure all cards are face up, regardless of the initial configuration. Find $N$. | [
"20"
] | [
"Twenty random cards are placed in a row all face down. A move consists of turning a face down card face up, and turning over the card that is immediately to its right. Let $N$ be the maximum number of moves required to ensure all cards are face up, regardless of the initial configuration. Find $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-f0a1d2fcc879934e3b5628463f25a9cb2f109a081ffae584ac3f14bae9af0120 | Twenty random cards are placed in a row, all face down. A move consists of turning a face down card up, and also turning up the card immediately to the right. If the process is repeated until no more moves can be made, what is the maximum possible number of cards that can be face up at the end of the process? | [
"20"
] | [
"Twenty random cards are placed in a row, all face down. A move consists of turning a face down card up, and also turning up the card immediately to the right. If the process is repeated until no more moves can be made, what is the maximum possible number of cards that can be face up at the end of the process?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4abbe67624812b25a0b59a1304ca0bc674032dd5707424ed07f7870102ce873a | Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a $3\times 3 \times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c},$ where $p,q,$ and $r$ are distinct ... | [
"074"
] | [
"Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a $3\\times 3 \\times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\\frac{p^a}{q^br^c},$ where $p,q,$ and $r$ a... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 5,
"tir": 1
} | 32 | 0 | 0 |
omr-ws-e550be7756c9ceb2690a576ccf2d126d586f8e01c5537dce2487627cbf0f0b96 | Twenty straight trominoes are placed end to end to form a row of 60 unit squares. One square within each tromino is randomly chosen to be colored black (so there are 20 black squares). The other two are colored white. Let \( N \) be the number of unordered pairs of adjacent squares with opposing colors. Find the probab... | [
"13"
] | [
"Twenty straight trominoes are placed end to end to form a row of 60 unit squares. One square within each tromino is randomly chosen to be colored black (so there are 20 black squares). The other two are colored white. Let \\( N \\) be the number of unordered pairs of adjacent squares with opposing colors. Find the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-e568acc5e6f45a119dda4f79f230ef38407ebf5c8d8680fbefadb35a630e1c18 | Twenty students sit around a round table. One student starts with 19 cards, and the others have no cards. In each step, a student with at least two cards gives one card to each of their two neighbors. After a series of moves, the game ends when no student has two or more cards. Determine the maximum number of moves tha... | [
"95"
] | [
"Twenty students sit around a round table. One student starts with 19 cards, and the others have no cards. In each step, a student with at least two cards gives one card to each of their two neighbors. After a series of moves, the game ends when no student has two or more cards. Determine the maximum number of move... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ff21552170936ab6b9753f23f8776442e94001d1cf85a99cc2d74f87967b7251 | Twenty students take turns walking down a hall with 20 lockers numbered 1 to 20. The first student opens all the lockers. The second student closes all the lockers numbered 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. The third student toggles the state (opens if closed, closes if open) of lockers numbered 3, 6, 9, 12, 15, 18, ... | [
"4"
] | [
"Twenty students take turns walking down a hall with 20 lockers numbered 1 to 20. The first student opens all the lockers. The second student closes all the lockers numbered 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. The third student toggles the state (opens if closed, closes if open) of lockers numbered 3, 6, 9, 12, 15,... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-483cc5acb3dbd57d76cc635bf764096e20a1ec460c3fb58d686d8d594caa8d73 | Twenty switches in an office computer network are to be connected so that each switch has a direct connection to exactly three other switches. How many connections will be necessary? | [
"30"
] | [
"Twenty switches in an office computer network are to be connected so that each switch has a direct connection to exactly three other switches. How many connections will be necessary?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-180a270ff64e9b9eb1ae3bdeb991f5e96af7ff34473e43e5b7defe8f7ca72cdc | Twenty tiles are numbered 1 through 20 and are placed into box $A$. Twenty other tiles numbered 11 through 30 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than 15 and the tile from box $B$ is either even or greater than 25? Express your an... | [
"\\frac{21}{50}"
] | [
"Twenty tiles are numbered 1 through 20 and are placed into box $A$. Twenty other tiles numbered 11 through 30 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than 15 and the tile from box $B$ is either even or greater than 25? Express yo... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94ed53da52ff8a960fae28efdfa97ed8597a4319c9ebe4d5e16d6cc745821d6a | Twenty-eight points are equally spaced around the circumference of a circle. What is the total number of triangles whose three vertices are from those twenty-eight points and the size of one of the angles is twice the size of another? | [
"420",
"700"
] | [
"Twenty-eight points are equally spaced around the circumference of a circle. What is the total number of triangles whose three vertices are from those twenty-eight points and the size of one of the angles is twice the size of another?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-36bc838a864908b29aec84fe4e3bfb6fec97397cec569ff86e0902bde9a93dac | Twenty-five boys and girls are seated at a round table. Let $N$ be the number of students who have two girls as their neighbors. Find the minimum possible value of $N$. | [
"0"
] | [
"Twenty-five boys and girls are seated at a round table. Let $N$ be the number of students who have two girls as their neighbors. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-3a7b8c910c6d0ebb07db4a5154e98802660331a141f8187b0c0e185b8e14e810 | Twenty-five boys and twenty-five girls are seated around a table. Let $N$ be the number of people who have two girls as neighbors. Find the minimum possible value of $N$. | [
"1"
] | [
"Twenty-five boys and twenty-five girls are seated around a table. Let $N$ be the number of people who have two girls as neighbors. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-86d82bf33d32d9fad04300842662485f9a4f7d270abc47e9dfe1a7cb85010aca | Twenty-five different positive integers add to 2008. What is the largest value that the least of them can have? | [
"68"
] | [
"Twenty-five different positive integers add to 2008. What is the largest value that the least of them can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 28
} | 60 | 0 | 0 |
omr-ws-49e384c1bb71d4b53237432cefba964ea5964d154c6c7bf37330c1a69dc448d2 | Twenty-five hobbits split a 5x5 plot into 1x1 plots. No hobbit wants to be an enemy to a neighbor (even touching corners). It is known that no hobbit has more than three enemies. Find the minimum number of rearrangements needed so that each hobbit is surrounded only by friends, assuming initially each hobbit has exactl... | [
"12"
] | [
"Twenty-five hobbits split a 5x5 plot into 1x1 plots. No hobbit wants to be an enemy to a neighbor (even touching corners). It is known that no hobbit has more than three enemies. Find the minimum number of rearrangements needed so that each hobbit is surrounded only by friends, assuming initially each hobbit has e... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-d678ef8aa08a563104a1fd4e236cf5c8f1f233b3b04cf45f313fdd94a9eb3328 | Twenty-five men sit around a circular table. Every hour, there is a vote, and each must respond "yes" or "no". Each man behaves as follows: if his response on the $n$-th vote is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)$-th vote; otherwis... | [
"12"
] | [
"Twenty-five men sit around a circular table. Every hour, there is a vote, and each must respond \"yes\" or \"no\". Each man behaves as follows: if his response on the $n$-th vote is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)$-th vote;... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-375b58d3f9a82f093f751969e4af6f7160a5f3829f92735cf4717b3971fbe3e1 | Twenty-five numbers are chosen from the set $\{1, 2, \ldots, 50\}$ and twenty-five numbers are chosen from the set $\{51, 52, \ldots, 100\}$. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers. | [
"2525"
] | [
"Twenty-five numbers are chosen from the set $\\{1, 2, \\ldots, 50\\}$ and twenty-five numbers are chosen from the set $\\{51, 52, \\ldots, 100\\}$. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fed992a3afb6115e8e9653152821b6dbdad07d931306bb4373a7952efe897806 | Twenty-five of King Arthur's knights are seated at a round table. Three knights are chosen at random and sent off to slay a dragon. What is the probability that at least two of the three chosen knights were sitting next to each other? If this probability is written as a fraction in lowest terms, what is the sum of the ... | [
"57"
] | [
"Twenty-five of King Arthur's knights are seated at a round table. Three knights are chosen at random and sent off to slay a dragon. What is the probability that at least two of the three chosen knights were sitting next to each other? If this probability is written as a fraction in lowest terms, what is the sum of... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 21
} | 53 | 0 | 0 |
omr-ws-b0176131dd9af08b824da358398615dc9f4294a396d224ed9cd070e0c9c422ea | Twenty-five of King Arthur's knights are seated at a round table. Three of them are chosen at random and sent off to slay a dragon. Let \( P \) be the probability that at least two of the three chosen knights were sitting next to each other. If \( P \) is written as a fraction in lowest terms, what is the sum of the nu... | [
"57"
] | [
"Twenty-five of King Arthur's knights are seated at a round table. Three of them are chosen at random and sent off to slay a dragon. Let \\( P \\) be the probability that at least two of the three chosen knights were sitting next to each other. If \\( P \\) is written as a fraction in lowest terms, what is the sum ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
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