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-a7c047d4786341ca4d9cf7b0a814e4bfe2b3839a42747c5298d75c671a5e8b1b | A token is placed on the corner cell of an $n \times n$ chess-board. Each of two players moves it in turn to an orthogonally neighboring cell. A player loses if they move the token to a cell that was already visited (including the starting cell). Determine who wins if the token starts on a corner cell for each $n$. | [
"\\text{First player wins if } n \\text{ is even; second player wins if } n \\text{ is odd}"
] | [
"A token is placed on the corner cell of an $n \\times n$ chess-board. Each of two players moves it in turn to an orthogonally neighboring cell. A player loses if they move the token to a cell that was already visited (including the starting cell). Determine who wins if the token starts on a corner cell for each $n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-3b14ffbf6875dfa6886e99310acfe696225f550a73246991f80b7e63ad0dc1d2 | A token starts at the point $(0,0)$ of an $xy$ -coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a po... | [
"391"
] | [
"A token starts at the point $(0,0)$ of an $xy$ -coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 20
} | 41 | 0 | 0 |
omr-ws-a6d64c3dea9a838547cbcbd1823d11640ca6e520706c36401895e86caedd050c | A token starts at the point $(0,0)$ of an $xy$-coordinate grid and makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes and is selected randomly from the four possible directions, independently of the other moves. What is the probability that the token ends at a poi... | [
"391"
] | [
"A token starts at the point $(0,0)$ of an $xy$-coordinate grid and makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes and is selected randomly from the four possible directions, independently of the other moves. What is the probability that the token ends at ... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 22
} | 41 | 0 | 0 |
omr-ws-211ebe10e2765db4c37940a3ba45ed32e36b011c0fdb0ac3c48d7400a6b98f5d | A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. What is the probability that the token ... | [
"\\(\\frac{135}{256}\\)"
] | [
"A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. What is the probability that the t... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-c273c882407c69070337a9c6abf157b7eee5ee57719164670e09aa60a8356c33 | A tortoise (T) and a hare (H) are racing in opposite directions around a circular track with a diameter of 100 yards. They start at the same spot, but H does not move until T has already gone $1/8$ of the distance. H meets T after going $1/6$ of the distance. How many times faster must H now go to finish the race compa... | [
"\\(\\frac{5}{8}\\)"
] | [
"A tortoise (T) and a hare (H) are racing in opposite directions around a circular track with a diameter of 100 yards. They start at the same spot, but H does not move until T has already gone $1/8$ of the distance. H meets T after going $1/6$ of the distance. How many times faster must H now go to finish the race ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93a4a349e353ced2f00fe5a89ebe6247d73bee582070b84829599003be33099f | A tortoise is given an 80-second head start in a race. When Achilles catches up to where the tortoise was when he began running, he finds that while he is now 40 meters ahead of the starting line, the tortoise is now 5 meters ahead of him. At this point, how long will it be, in seconds, before Achilles passes the torto... | [
"\\(\\frac{10}{7}\\)"
] | [
"A tortoise is given an 80-second head start in a race. When Achilles catches up to where the tortoise was when he began running, he finds that while he is now 40 meters ahead of the starting line, the tortoise is now 5 meters ahead of him. At this point, how long will it be, in seconds, before Achilles passes the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-c9fc3135ab558a4b1059cf281ade3c6e543ad706b41b62d833dab2567372ff85 | A tortoise starts at the bottom left corner of a \( 4 \times 4 \) grid of points. Each point is colored red or blue, and there are exactly 8 of each color. It can only move up or right, and can only move to blue points. If the bottom left corner is always blue, find the number of ways we can color the grid such that th... | [
"48"
] | [
"A tortoise starts at the bottom left corner of a \\( 4 \\times 4 \\) grid of points. Each point is colored red or blue, and there are exactly 8 of each color. It can only move up or right, and can only move to blue points. If the bottom left corner is always blue, find the number of ways we can color the grid such... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-0f5362019d0b7f2fd6d606cc7c8e371dd3ebc802950d74f8473d708ba3ebede3 | A torus $\mathcal T$ is formed by revolving a circle with radius $3$ around an axis in the plane that is $6$ units away from the center of the circle. When a sphere of radius $11$ is placed inside $\mathcal T$, it is internally tangent to $\mathcal T$ along a circle with radius $r_{i}$. When the sphere is placed outsid... | [
"127"
] | [
"A torus $\\mathcal T$ is formed by revolving a circle with radius $3$ around an axis in the plane that is $6$ units away from the center of the circle. When a sphere of radius $11$ is placed inside $\\mathcal T$, it is internally tangent to $\\mathcal T$ along a circle with radius $r_{i}$. When the sphere is place... | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-d282bcb6529057d40681f2e4c2534a9d24ed3e3517871864310c166ac91ec65b | A torus (donut) having inner radius $2$ and outer radius $4$ sits on a flat table. What is the radius of the largest spherical ball that can be placed on top of the center torus so that the ball still touches the horizontal plane? (If the $xy$-plane is the table, the torus is formed by revolving the circle in the $xz$-... | [
"\\frac{9}{4}"
] | [
"A torus (donut) having inner radius $2$ and outer radius $4$ sits on a flat table. What is the radius of the largest spherical ball that can be placed on top of the center torus so that the ball still touches the horizontal plane? (If the $xy$-plane is the table, the torus is formed by revolving the circle in th... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-75a8fd73eb8de97ad255bb9b9c4c3819c7c97686543fd280d375f3104f70ee01 | A tosses a coin \( n+1 \) times and B tosses a coin \( n \) times. What is the probability that A gets more heads than B? | [
"\\(\\frac{1}{2}\\)"
] | [
"A tosses a coin \\( n+1 \\) times and B tosses a coin \\( n \\) times. What is the probability that A gets more heads than B?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-a469e09ecf02e1bc6b28ea60166ae765596d171e26835577841e0e0afef025dc | A total fixed amount of $N$ thousand rupees is given to $A$, $B$, and $C$ every year, each being given an amount proportional to her age. In the first year, $A$ got half the total amount. When the sixth payment was made, $A$ got six-seventh of the amount that she had in the first year; $B$ got 1000 Rs less than that sh... | [
"35"
] | [
"A total fixed amount of $N$ thousand rupees is given to $A$, $B$, and $C$ every year, each being given an amount proportional to her age. In the first year, $A$ got half the total amount. When the sixth payment was made, $A$ got six-seventh of the amount that she had in the first year; $B$ got 1000 Rs less than th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-856992a06e4144b44d64c886f86a8bf0bf8bf186345c6adf5708d0e545f910e8 | A total of $2n$ people are lining up for Broadway show tickets where $n \ge 5$. Each person can only buy one ticket and each ticket costs $5$ dollars. Half of the people only have one $10$ dollar bill per person, and the other half have only one $5$ dollar bill per person. If the ticket booth has no money to start with... | [
"\\(\\frac{1}{(n+1)!n!}\\)"
] | [
"A total of $2n$ people are lining up for Broadway show tickets where $n \\ge 5$. Each person can only buy one ticket and each ticket costs $5$ dollars. Half of the people only have one $10$ dollar bill per person, and the other half have only one $5$ dollar bill per person. If the ticket booth has no money to star... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02067c9e41fe3bad14b1c6fb855f90e3a10159325c086b76683b256628162231 | A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$. | [
"17"
] | [
"A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2975261e0670b782195ce67dd3b0e6387f34481a42300ed46960f381be3ed414 | A total of $n$ balls, numbered 1 through $n$, are put into $n$ urns, also numbered 1 through $n$, such that ball $i$ is equally likely to go into any of the urns $1, 2, \ldots, i$. Find the expected number of urns that are empty. | [
"\\( n \\left(\\frac{n-1}{n}\\right)^n \\approx \\frac{n}{e} \\)"
] | [
"A total of $n$ balls, numbered 1 through $n$, are put into $n$ urns, also numbered 1 through $n$, such that ball $i$ is equally likely to go into any of the urns $1, 2, \\ldots, i$. Find the expected number of urns that are empty."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d089e1426967337c4b6e90f39a4c82be685ef081e52ee3226bb26169c3f61d01 | A total of $n$ male psychologists attended a meeting and took their coats at random after the meeting. Each psychologist has a probability $p$ of losing their coat independently of the others. Find the probability that none of the psychologists had their own coat with them, and deduce that it is approximately $e^{-(1-p... | [
"\\( e^{-(1 - p)} \\)"
] | [
"A total of $n$ male psychologists attended a meeting and took their coats at random after the meeting. Each psychologist has a probability $p$ of losing their coat independently of the others. Find the probability that none of the psychologists had their own coat with them, and deduce that it is approximately $e^{... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f2875e59e790cf2f0488a50bd8dbfafe7eae8fd8c384b88de2e335f981327a57 | A total of $n$ points are chosen uniformly randomly on a sphere. What is the probability that the 3D object formed by the convex hull of these $n$ points contains the center of the sphere? | [
"\\(\\frac{n}{2^{n-1}}\\)"
] | [
"A total of $n$ points are chosen uniformly randomly on a sphere. What is the probability that the 3D object formed by the convex hull of these $n$ points contains the center of the sphere?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-38bec6839e32ca5e687a265d37e98701b76bc66b173dcdcda9188bf14d68bfd9 | A total of $n$ students sit around a round table. Each of $7$ women has a woman sitting beside her. Each of $12$ women has a man sitting beside her. For every $4$ men, $3$ of them each have a woman sitting on his left side. Find $n$. | [
"25"
] | [
"A total of $n$ students sit around a round table. Each of $7$ women has a woman sitting beside her. Each of $12$ women has a man sitting beside her. For every $4$ men, $3$ of them each have a woman sitting on his left side. Find $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-74d6b70ead4d79359cbd2367cb7bc8679bfc1b59e9363fd97f77598a8654f476 | A total of $n^2 + 2n + 29$ pieces of candy is divided equally among $15$ girls and $n$ boys. Find all possible values for $n$. | [
"1, 13, 17, 41, 97"
] | [
"A total of $n^2 + 2n + 29$ pieces of candy is divided equally among $15$ girls and $n$ boys. Find all possible values for $n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a0496100e9b6123ec229e8f3df2d1d8aa49c8ec6875aabde2c7748daa0edc6e | A total of 16 people, including Millinova and her two friends, are seated around a round table. The two birthday girls sit together and Millinova sits next to one of them. Caelestor, gfour84, and Rolloway sit together (but not necessarily in that order), and Caelestra, Caelestor's cousin, sits across the table from Cae... | [
"11"
] | [
"A total of 16 people, including Millinova and her two friends, are seated around a round table. The two birthday girls sit together and Millinova sits next to one of them. Caelestor, gfour84, and Rolloway sit together (but not necessarily in that order), and Caelestra, Caelestor's cousin, sits across the table fro... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-292a987cf63835530bf385f848a9acf83d017b3cb609cbd24d4ae99fa9ff3f9c | A total of 17 teams play in a single-elimination tournament. (A single-elimination tournament is one where once a team has lost, it is removed from the competition.) How many total games must be played before a winner can be declared, assuming there is no possibility of ties? | [
"16"
] | [
"A total of 17 teams play in a single-elimination tournament. (A single-elimination tournament is one where once a team has lost, it is removed from the competition.) How many total games must be played before a winner can be declared, assuming there is no possibility of ties?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-179b1673714f18c572c4d0d0a25830ac676428024b38741d7702ead46950a6a4 | A total of 180 marbles (gray, white, and black) are placed in a line. The first five are gray, followed by four white, followed by three black, followed by five gray, followed by four white, followed by three black,.... If this pattern continues, what is the color of the 158th marble in this line? | [
"\\text{gray}"
] | [
"A total of 180 marbles (gray, white, and black) are placed in a line. The first five are gray, followed by four white, followed by three black, followed by five gray, followed by four white, followed by three black,.... If this pattern continues, what is the color of the 158th marble in this line?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da0f871fcd924668dcebd4c8206a9ff2520063164dd597cc9bb47f417e690253 | A total of 20 cards numbered 1 to 20 are divided into groups such that each group contains at least two cards. For every two cards in the same group, the GCD of their numbers is either 1 or 2. What is the minimum number of groups that must be formed? | [
"6"
] | [
"A total of 20 cards numbered 1 to 20 are divided into groups such that each group contains at least two cards. For every two cards in the same group, the GCD of their numbers is either 1 or 2. What is the minimum number of groups that must be formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-b7e28386b81a15c11df2582aa8eaedd9ecb4cf7c7da8e857c43f3d50d1216a82 | A total of 2010 coins are distributed in 5 boxes with consecutive natural numbers of coins in each box. Martha can choose and take one of the boxes, but before that, she can perform the following transformation finitely many times: from a box with at least 4 coins, she can transfer one coin to each of the other boxes. ... | [
"2004"
] | [
"A total of 2010 coins are distributed in 5 boxes with consecutive natural numbers of coins in each box. Martha can choose and take one of the boxes, but before that, she can perform the following transformation finitely many times: from a box with at least 4 coins, she can transfer one coin to each of the other bo... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-15de8165eda37a82091c4c86400e860e4060c935d4186b70a4ae0cce2b805391 | A total of 2010 coins are distributed in 5 boxes with consecutive natural numbers of coins. Martha can perform the following transformation finitely many times: from a box with at least 4 coins, she can transfer one coin to each of the other boxes. What is the maximum number of coins that Martha can take away after per... | [
"2004"
] | [
"A total of 2010 coins are distributed in 5 boxes with consecutive natural numbers of coins. Martha can perform the following transformation finitely many times: from a box with at least 4 coins, she can transfer one coin to each of the other boxes. What is the maximum number of coins that Martha can take away afte... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-6547a73d9f78a385c7e08c293ab3c416affdf432226b913e878d5fa2b852f54f | A total of 2021 teams participated in an ARML Team Round. The table below shows the percentage of teams that answered each of the ten questions correctly, rounded to the nearest percent. A team's score was five times the number of correct answers. \begin{tabular}{|cccccccccc|} \hline T-1 & T-2 & T-3 & T-4 & T-5 & T-6 &... | [
"36"
] | [
"A total of 2021 teams participated in an ARML Team Round. The table below shows the percentage of teams that answered each of the ten questions correctly, rounded to the nearest percent. A team's score was five times the number of correct answers.\n\n\\begin{tabular}{|cccccccccc|}\n\\hline\nT-1 & T-2 & T-3 & T-4 &... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04c43b8cf4388f8e670c58e972ce88c381a59431b8caca97bdb02c697553a465 | A total of 57500 must be divided among 10 persons such that each subsequent person receives 500 less than the previous person. Find the amount of money received by the first and the last person on the list. | [
"First person: 8000, Last person: 3500"
] | [
"A total of 57500 must be divided among 10 persons such that each subsequent person receives 500 less than the previous person. Find the amount of money received by the first and the last person on the list."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 29
} | 31 | 0 | 0 |
omr-ws-67c4a888c7bdb1c6f0c2d4ede8ac57022802e8da049dc30f9e8ef210774eee6a | A total of \( n \) balls, numbered 1 through \( n \), are put into \( n \) urns, also numbered 1 through \( n \), such that for each \( i = 1, 2, \ldots, n \), ball \( i \) is equally likely to go into any of the urns \( 1, 2, \ldots, i \). The balls are placed independently. Define the indicator random variables \( I_... | [
"\\(\\frac{n-1}{2}\\)"
] | [
"A total of \\( n \\) balls, numbered 1 through \\( n \\), are put into \\( n \\) urns, also numbered 1 through \\( n \\), such that for each \\( i = 1, 2, \\ldots, n \\), ball \\( i \\) is equally likely to go into any of the urns \\( 1, 2, \\ldots, i \\). The balls are placed independently. Define the indicator r... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-8189f5fae0980438e14890375ceb8d9425caad85004ffdbbfe635338175a3892 | A total of \( n \) bar magnets are placed end to end in a line on a table, where the orientation of the south and north poles of each magnet is randomly chosen from the two possibilities with equal probability. Adjacent magnets with opposite poles facing each other join to form a block. Find the mean and variance of th... | [
"The mean and variance of the number of blocks of joined magnets are:\n\\[ \\text{Mean} = \\frac{n+1}{2} \\]\n\\[ \\text{Variance} = \\frac{n-1}{4} \\]"
] | [
"A total of \\( n \\) bar magnets are placed end to end in a line on a table, where the orientation of the south and north poles of each magnet is randomly chosen from the two possibilities with equal probability. Adjacent magnets with opposite poles facing each other join to form a block. Find the mean and varianc... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-a7818777d5d14a11ae9b97cb3813eeb1f2f6b8c0c8ef3a6edec799b429a0c2f8 | A total of \( n \) board members of an organization shall be divided into four commissions according to the following provisions: (i) each member belongs to exactly two commissions, and (ii) each two commissions has exactly one joint member. What is \( n \)? | [
"6"
] | [
"A total of \\( n \\) board members of an organization shall be divided into four commissions according to the following provisions: \n(i) each member belongs to exactly two commissions, and \n(ii) each two commissions has exactly one joint member. \nWhat is \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5b045f8850156a0009c3fcf4408a25f4cd230ef6c9f691c3cd3f643bf771efad | A total of fifteen delegates from Israel, Palestine, Russia, and the United States meet at a conference. Each country has sent a different number of delegates (with each country sending at least one delegate). Israel and Russia have sent a combined total of six delegates. Russia and the United States have sent a combin... | [
"Russia"
] | [
"A total of fifteen delegates from Israel, Palestine, Russia, and the United States meet at a conference. Each country has sent a different number of delegates (with each country sending at least one delegate). Israel and Russia have sent a combined total of six delegates. Russia and the United States have sent a c... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-f2e9f39aced5582c86d785965267888d9d2fcecb97ba6363377d49550c72b091 | A tour agency has signed up 100 people for a cruise on a ship with a maximum capacity of 150. A ticket costs $2000. For each $5 that this price is lowered, one new passenger signs up. Incidental costs to the agency are $500 per passenger, and there are fixed costs of $125,000 for the ship. By how much should the price ... | [
"250"
] | [
"A tour agency has signed up 100 people for a cruise on a ship with a maximum capacity of 150. A ticket costs $2000. For each $5 that this price is lowered, one new passenger signs up. Incidental costs to the agency are $500 per passenger, and there are fixed costs of $125,000 for the ship. By how much should the p... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f0efd4cb5fb081725e4f7cb46e8c2dfcf87b0b5199fe1125cbefa61a90a0bcd7 | A tourist bus is traveling along a straight highway. An ancient palace is located to one side of the highway and a road has been built from the main entrance at right angles to the highway. How far from the point where the roads intersect should the bus stop for the tourists to have the best view of the fa\c{c}ade of t... | [
"$\\left(\\sqrt {4b^2-3a^2}-b\\right)\\cdot 3^{-1/2}$"
] | [
"A tourist bus is traveling along a straight highway. An ancient palace is located to one side of the highway and a road has been built from the main entrance at right angles to the highway. How far from the point where the roads intersect should the bus stop for the tourists to have the best view of the fa\\c{c}ad... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a64fc68233528eed26a364c4aedbe2bc2957b0d225056c7df488ac51d19cb32 | A tourist fills two tanks of water using two hoses. 2.9 liters of water flow out per minute from the first hose, 8.7 liters from the second. At the moment when the smaller tank was half full, the tourist swapped the hoses, after which both tanks filled at the same time. What is the capacity of the larger tank if the ca... | [
"31.25 L"
] | [
"A tourist fills two tanks of water using two hoses. 2.9 liters of water flow out per minute from the first hose, 8.7 liters from the second. At the moment when the smaller tank was half full, the tourist swapped the hoses, after which both tanks filled at the same time. What is the capacity of the larger tank if t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1156ec0ccbc8533baa6139ac31921f40a0bdd18cb8ecd8f91c0af683c6ed4e27 | A tourist has wandered randomly through the streets of Moscow and decides to return to the train station using only the streets he has passed an odd number of times. Let $G$ be the graph representing the streets and intersections of Moscow, where each street is an edge and each intersection is a vertex. Suppose the tou... | [
"0"
] | [
"A tourist has wandered randomly through the streets of Moscow and decides to return to the train station using only the streets he has passed an odd number of times. Let $G$ be the graph representing the streets and intersections of Moscow, where each street is an edge and each intersection is a vertex. Suppose th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b31ca710463e4ce96dfc875f4ceb6f840234544d87b867b19e3a3c6f8ca814e1 | A tourist is on an island with 100 wizards, where exactly one is a knight and the rest are liars. The tourist can choose any two wizards $A$ and $B$ and ask $A$ to cast a spell on $B$ that will change $B$ from a knight to a liar or vice versa if $A$ is a knight, but will have no effect if $A$ is a liar. The tourist wan... | [
"1"
] | [
"A tourist is on an island with 100 wizards, where exactly one is a knight and the rest are liars. The tourist can choose any two wizards $A$ and $B$ and ask $A$ to cast a spell on $B$ that will change $B$ from a knight to a liar or vice versa if $A$ is a knight, but will have no effect if $A$ is a liar. The touris... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31e0afb43a35aad60c9c24fb768b52440a5e12706e61070de533867077d65e5b | A tourist on an island must open a series of doors, each colored with one of n colors, using n keys, each with a different color. Each key can open any door but is destroyed if it opens a door of the same color. Determine the minimum number of doors required to ensure that no tourist can get the treasure, regardless of... | [
"\\( n + 1 \\)"
] | [
"A tourist on an island must open a series of doors, each colored with one of n colors, using n keys, each with a different color. Each key can open any door but is destroyed if it opens a door of the same color. Determine the minimum number of doors required to ensure that no tourist can get the treasure, regardle... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a06a3aeb5c635dfdfb5f4dc777fc6b57c603e8ca28145061b5e031f90e865456 | A tourist sailed 90 km down the river in a boat and then walked 10 km. He walked 4 h less than he sailed. If the tourist walked as long as he sailed and sailed as long as he walked, the distances would be equal. How much time did he walk and how much time did he sail? | [
"6 h and 2 h"
] | [
"A tourist sailed 90 km down the river in a boat and then walked 10 km. He walked 4 h less than he sailed. If the tourist walked as long as he sailed and sailed as long as he walked, the distances would be equal. How much time did he walk and how much time did he sail?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac74dbfaae59b4fe8250fb506f87462e4038f85ace9f4c39b9ba771622570fa0 | A tourist stands on a straight road at point P. The tourist can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of points where the tourist can reach within an hour's walk. | [
"x^2 + y^2 \\leq 9 \\text{ or } |x| + \\sqrt{3} |y| \\leq 6"
] | [
"A tourist stands on a straight road at point P. The tourist can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of points where the tourist can reach within an hour's walk."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-7fcb348c1bbe80eea8e10e0d9ce5ab6c94542f131eadf153a6fb2d5feed45dad | A tourist takes a trip through a city in stages. Each stage consists of three segments of length 100 meters separated by right turns of 60°. Between the last segment of one stage and the first segment of the next stage, the tourist makes a left turn of 60°. At what distance will the tourist be from his initial position... | [
"200"
] | [
"A tourist takes a trip through a city in stages. Each stage consists of three segments of length 100 meters separated by right turns of 60°. Between the last segment of one stage and the first segment of the next stage, the tourist makes a left turn of 60°. At what distance will the tourist be from his initial pos... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-8b2d0672bea7a0308eb93464d24cec05172f58298033ed8728f6f29838baaf75 | A tourist walks through Mendoza in stages. Each stage consists of 3 segments, each of length 100 m, with two $60^\circ$ turns to the right. Between the last segment of one stage and the first of the next, a $60^\circ$ left turn is made. How far will the tourist be from the starting point after having covered 1997 stage... | [
"200 m"
] | [
"A tourist walks through Mendoza in stages. Each stage consists of 3 segments, each of length 100 m, with two $60^\\circ$ turns to the right. Between the last segment of one stage and the first of the next, a $60^\\circ$ left turn is made. How far will the tourist be from the starting point after having covered 199... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 12
} | 34 | 0 | 0 |
omr-ws-152d0727f4425446b3be81c88fcd3a08dd6beca393aba498bebea993d9ec99ce | A tournament is held among \( n \) teams with the following rules: - Every team plays all others once at home and once away (i.e., a double round-robin schedule). - Each team may participate in several away games in a week (from Sunday to Saturday). - There is no away game arrangement for a team if it has a home game i... | [
"6"
] | [
"A tournament is held among \\( n \\) teams with the following rules:\n- Every team plays all others once at home and once away (i.e., a double round-robin schedule).\n- Each team may participate in several away games in a week (from Sunday to Saturday).\n- There is no away game arrangement for a team if it has a h... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-940661a72a62cfd4786c09690a9bcb6e2379b4e03d3db4443db795fb59270589 | A tournament sequence is a sequence \( \{t(i)\} \) of positive integers such that \( t(1) = 1 \) and \( t(n+1) \leq 2t(n) \). Suppose you randomly create a tournament sequence of length \( k \). In terms of \( k \), what is the probability that the terms of this tournament sequence are a geometric sequence where the ra... | [
"\\dfrac{1}{2^{\\frac{k(k-1)}{2}}}",
"\\dfrac{1}{2^{k(k-1)/2}}"
] | [
"A tournament sequence is a sequence \\( \\{t(i)\\} \\) of positive integers such that \\( t(1) = 1 \\) and \\( t(n+1) \\leq 2t(n) \\). Suppose you randomly create a tournament sequence of length \\( k \\). In terms of \\( k \\), what is the probability that the terms of this tournament sequence are a geometric seq... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-2fbfef5939233c36d5124dc5a69f5c497b1c405666fe0107951f37e31a1cab62 | A tower stands vertically in the interior of an equilateral triangle field with side length \( a \). If the angles of elevation of the top of the tower from the corners of the field are \( A \), \( B \), and \( C \), find the height of the tower. | [
"\\dfrac{a\\sqrt{3}}{\\cot A + \\cot B + \\cot C}"
] | [
"A tower stands vertically in the interior of an equilateral triangle field with side length \\( a \\). If the angles of elevation of the top of the tower from the corners of the field are \\( A \\), \\( B \\), and \\( C \\), find the height of the tower."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-313c0687ddc6323ab65cc1ebeeb5ae170a6d572b2d0ffef45ee6035e8edfd74b | A town contains 4 people who repair televisions. If 4 sets break down, what is the probability that exactly \( i \) of the repairers are called? Solve the problem for \( i = 1, 2, 3, 4 \). What assumptions are you making? | [
"\\( P(i=1) = \\frac{1}{64}, P(i=2) = \\frac{21}{64}, P(i=3) = \\frac{36}{64}, P(i=4) = \\frac{6}{64} \\)"
] | [
"A town contains 4 people who repair televisions. If 4 sets break down, what is the probability that exactly \\( i \\) of the repairers are called? Solve the problem for \\( i = 1, 2, 3, 4 \\). What assumptions are you making?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-01c64e1d897e92b3c3b2b859ae1ff34125e00f0140c8f2563b3ebcd48141e007 | A town has 1418 houses. The number of houses increases by five percent the first year and four percent the next. How many houses are in the town at the end of the second year? Round your answer to the nearest whole number. | [
"1548"
] | [
"A town has 1418 houses. The number of houses increases by five percent the first year and four percent the next. How many houses are in the town at the end of the second year? Round your answer to the nearest whole number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b8bfbbde7154e993c52b8fdd9afa2af658355189a56bcedf5639591b0fa4f1d | A town has several clubs. Given any two residents, there is exactly one club that both belong to. Given any two clubs, there is exactly one resident who belongs to both. Each club has at least 3 members, and at least one club has 17 members. How many residents are there? | [
"273"
] | [
"A town has several clubs. Given any two residents, there is exactly one club that both belong to. Given any two clubs, there is exactly one resident who belongs to both. Each club has at least 3 members, and at least one club has 17 members. How many residents are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 33,
"tir": 1
} | 34 | 0 | 0 |
omr-ws-b23d38f401ca51881a2fd24f6672a1863f5b8b7361a96f9b57aa6391203f1bb8 | A town has six parks. On a Saturday, six classmates, who are unaware of each other's decision, choose a park at random and go there at the same time. What is the probability that at least two of them go to the same park? | [
"\\dfrac{319}{324}"
] | [
"A town has six parks. On a Saturday, six classmates, who are unaware of each other's decision, choose a park at random and go there at the same time. What is the probability that at least two of them go to the same park?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c57f14cbb7be303aab9ca27d7203394a18dba35247f41247a8c098122ecff4e7 | A town's reservoir, when full, can supply 32000 gallons per day for 50 days or 37000 gallons per day for 40 days. What is the maximum daily usage in gallons that ensures the supply never runs out? | [
"12000"
] | [
"A town's reservoir, when full, can supply 32000 gallons per day for 50 days or 37000 gallons per day for 40 days. What is the maximum daily usage in gallons that ensures the supply never runs out?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 24
} | 72 | 0 | 0 |
omr-ws-0ff6313ddebc876b9a788c040f7b06ab528c2a1928ffe840136d021ab72e472a | A town-planner has built an isolated city with $2N$ roundabouts, each connecting exactly three roads. All roads are two-way, and each roundabout is oriented clockwise. Vlad starts driving from his house and always takes the first exit at each roundabout he encounters. His journey includes every road in the town in both... | [
"\\( N \\) must be odd."
] | [
"A town-planner has built an isolated city with $2N$ roundabouts, each connecting exactly three roads. All roads are two-way, and each roundabout is oriented clockwise. Vlad starts driving from his house and always takes the first exit at each roundabout he encounters. His journey includes every road in the town in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-519d3b9e6c219e15762539b7e16115d3fd25dca08831b9cb76c4ea6418ec16d3 | A toy car is released off a ramp of 130 cm length and 60 cm height. Assume no friction or air resistance. Determine the instantaneous speed of the car after it has rolled 30 cm. | [
"1.66 m/s"
] | [
"A toy car is released off a ramp of 130 cm length and 60 cm height. Assume no friction or air resistance. Determine the instantaneous speed of the car after it has rolled 30 cm."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2cad436a0404265049ca973617bc1107185abff761addc7d8a61f7b987eb1610 | A toy factory produces several kinds of clay toys, each painted in one of $k$ colours. The diversity of a colour is defined as the number of different types of toys painted in that colour. The painting protocol requires that each colour is used and the diversities of each two colours are different. A batch of clay Cheb... | [
"\\dfrac{k(k + 1)}{2}"
] | [
"A toy factory produces several kinds of clay toys, each painted in one of $k$ colours. The diversity of a colour is defined as the number of different types of toys painted in that colour. The painting protocol requires that each colour is used and the diversities of each two colours are different. A batch of clay... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-160269b63c42290baa98e8c5370c5f52a6f08f2d627a891d601fd763d01ccc89 | A toy manufacturer can produce 8 First Babies or 20 Real Babies in one hour. The company must produce at least twice as many First Babies as Real Babies and can spend no more than 48 hours per week making these dolls. The profit on each First Baby is $3.00 and the profit on each Real Baby is $7.50. Find the number and ... | [
"320 First Babies and 160 Real Babies to maximize profit.",
"\\( x = 320 \\) First Babies, \\( y = 160 \\) Real Babies."
] | [
"A toy manufacturer can produce 8 First Babies or 20 Real Babies in one hour. The company must produce at least twice as many First Babies as Real Babies and can spend no more than 48 hours per week making these dolls. The profit on each First Baby is $3.00 and the profit on each Real Baby is $7.50. Find the number... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 24,
"tir": 63
} | 87 | 0 | 0 |
omr-ws-0ba97a50b22279c38489e0a213d91972ab7670ff7b57a43538101619eda06831 | A toy store is selling small bags with a mini-figure inside, where each bag contains 1 of 8 different mini-figures. The bags are not marked, so you do not know which mini-figure you will get. If the probability of getting a particular mini-figure is the same for all 8 mini-figures, then how many bags, on average, must ... | [
"\\( 8 \\left(1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{7} + \\frac{1}{8}\\right) \\) or approximately 21.742856."
] | [
"A toy store is selling small bags with a mini-figure inside, where each bag contains 1 of 8 different mini-figures. The bags are not marked, so you do not know which mini-figure you will get. If the probability of getting a particular mini-figure is the same for all 8 mini-figures, then how many bags, on average, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-7f147a9087dfc9ce5c73451e5465d5d788818b37a8c2ffdd9ac0a9ade33de092 | A toy store manager purchased a total of 43 items including blocks (at $8.50 each), dolls (at $12.20 each), trucks (at $10.40 each), and puzzles (at $6.40 each), for a total of $410.00. The amount spent on trucks and puzzles exceeded the amount spent on blocks and dolls by $30. The number of blocks and trucks purchased... | [
"156"
] | [
"A toy store manager purchased a total of 43 items including blocks (at $8.50 each), dolls (at $12.20 each), trucks (at $10.40 each), and puzzles (at $6.40 each), for a total of $410.00. The amount spent on trucks and puzzles exceeded the amount spent on blocks and dolls by $30. The number of blocks and trucks purc... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a24ef013f4832aa847621d6383c522a45719414a1fb9903829dd5185e63a68fc | A toy store manager received a large order of Mr. Slinkums just in time for the holidays. The manager places $20\%$ of them on the shelves, leaving the other 120 Mr. Slinkums in storage. How many Mr. Slinkums were in this order? | [
"150"
] | [
"A toy store manager received a large order of Mr. Slinkums just in time for the holidays. The manager places $20\\%$ of them on the shelves, leaving the other 120 Mr. Slinkums in storage. How many Mr. Slinkums were in this order?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa67328b3dd428c593f80afad848b53512e8fa66cb206271e849c2797ad76525 | A toymaker has $k$ dice, each with 6 blank sides. The toymaker must draw one of the digits $0, 1, 2, \ldots, 9$ on each side of each die. Determine the largest integer $n$ (in terms of $k$) such that for any positive integer $r \leq n$, it is possible to choose some of the $k$ dice and form the decimal representation o... | [
"\\( 10^k - 1 \\)"
] | [
"A toymaker has $k$ dice, each with 6 blank sides. The toymaker must draw one of the digits $0, 1, 2, \\ldots, 9$ on each side of each die. Determine the largest integer $n$ (in terms of $k$) such that for any positive integer $r \\leq n$, it is possible to choose some of the $k$ dice and form the decimal represent... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dca4f257bfdf5e272690b5472335030f4eba340cc255ea768f3ca5cd42c781d | A track and field playing area is in the shape of a rectangle with semicircles at each end. The inside perimeter of the track is 1800 meters. What should the dimensions of the rectangle be so that the area of the rectangle is maximized? Solve without using calculus. | [
"\\( d = \\frac{900}{\\pi} \\) meters, \\( l = 450 \\) meters."
] | [
"A track and field playing area is in the shape of a rectangle with semicircles at each end. The inside perimeter of the track is 1800 meters. What should the dimensions of the rectangle be so that the area of the rectangle is maximized? Solve without using calculus."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 30
} | 41 | 0 | 0 |
omr-ws-6c7ccf9fab7b4292b973cdc39d954c174507caed3ac3acf467dd178902a4e6f4 | A track field is to be built in the shape of a region bounded between two parallel lines and two semi-circles. If the perimeter of the field is exactly 4 km, find a design to maximize the rectangular area within it. What are the side lengths of the rectangular area of your design, and what is the maximum rectangular ar... | [
"\\( x = 1 \\) km, \\( y = \\frac{2}{\\pi} \\) km, and the maximum rectangular area is \\( \\frac{2}{\\pi} \\) square kilometers."
] | [
"A track field is to be built in the shape of a region bounded between two parallel lines and two semi-circles. If the perimeter of the field is exactly 4 km, find a design to maximize the rectangular area within it. What are the side lengths of the rectangular area of your design, and what is the maximum rectangul... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 7,
"tir": 23
} | 31 | 0 | 0 |
omr-ws-5948a432fd61fbc3b304a92b3ba44023930c3ec49206a7d5ad9a15df6b00381c | A tractor moves from point $A$ to point $B$. The radius of the front wheel of the tractor is smaller than that of the rear wheel. On the way from $A$ to $B$ the front wheel made 200 rotations more than the rear wheel. If the circumference of the front wheel was 5/4 times longer, then on the way from $A$ to $B$ it would... | [
"The length of the circumference of the front wheel is 2 m and that of the rear wheel is 3 m"
] | [
"A tractor moves from point $A$ to point $B$. The radius of the front wheel of the tractor is smaller than that of the rear wheel. On the way from $A$ to $B$ the front wheel made 200 rotations more than the rear wheel. If the circumference of the front wheel was 5/4 times longer, then on the way from $A$ to $B$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b90cc53c8b04061e38f10a3cb7be862f86cdf3c58b609eb5b337cc00df1508f1 | A trader buys a chair for Rs.600 and sells it for Rs.765 at a credit of 4 months. Reckoning money worth 6% per annum, what is his profit percentage? | [
"25%"
] | [
"A trader buys a chair for Rs.600 and sells it for Rs.765 at a credit of 4 months. Reckoning money worth 6% per annum, what is his profit percentage?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-6f5ab983d5675ecc1e1a1e3d57772510c5dcfdf992a0732b5f1aa65ab096c4cf | A trader owns horses of 3 races, with exactly \( b_j \) horses of each race (for \( j = 1, 2, 3 \)). He wants to distribute these horses to his 3 sons such that each son values his horses more than the other sons' horses. The value each son \( i \) places on a horse of race \( j \) is \( a_{ij} \) golds, with \( a_{ii}... | [
"1"
] | [
"A trader owns horses of 3 races, with exactly \\( b_j \\) horses of each race (for \\( j = 1, 2, 3 \\)). He wants to distribute these horses to his 3 sons such that each son values his horses more than the other sons' horses. The value each son \\( i \\) places on a horse of race \\( j \\) is \\( a_{ij} \\) golds,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-ff0514a3d077898f0fd9a7b79c8f4c4966d89eab7f287f38f50fa61c23fac5d2 | A traffic cone has a slant height of 20 inches and a base radius of 5 inches. The cone is supported by a square base with each side measuring 12 inches. Assuming the cone has no thickness, what is the exterior surface area of the cone and the square base? Answer to the nearest square inch. | [
"380"
] | [
"A traffic cone has a slant height of 20 inches and a base radius of 5 inches. The cone is supported by a square base with each side measuring 12 inches. Assuming the cone has no thickness, what is the exterior surface area of the cone and the square base? Answer to the nearest square inch."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 15
} | 34 | 0 | 0 |
omr-ws-5d31d38fe950ba2a439b12108e469cd32d369756a53854e1f8ea4cec5a41d1a7 | A traffic helicopter pilot finds that with a tailwind her 120km trip away from the airport takes 30 minutes. On her return trip to the airport, into the wind it takes her 10 minutes longer. What is the speed of the helicopter? What is the speed of the wind? | [
"\\( v_h = 210 \\) km/h, \\( v_w = 30 \\) km/h"
] | [
"A traffic helicopter pilot finds that with a tailwind her 120km trip away from the airport takes 30 minutes. On her return trip to the airport, into the wind it takes her 10 minutes longer. What is the speed of the helicopter? What is the speed of the wind?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-fbfefc0b40116d039c0837ac05b28fe0eaf690b2ce789be9f06693d38f3e15a8 | A traffic light has a 2-minute cycle: 30 seconds green light, 90 seconds red light. Let \( X \) be the random variable representing the waiting time \( 0 \leq X \leq 1.5 \) minutes. The cumulative distribution function (CDF) of \( X \) is given by: \[ P(X \leq t) = \begin{cases} 0 & \text{if } t < 0, \\ \frac{1}{4} + \... | [
"\\dfrac{3}{4}"
] | [
"A traffic light has a 2-minute cycle: 30 seconds green light, 90 seconds red light. Let \\( X \\) be the random variable representing the waiting time \\( 0 \\leq X \\leq 1.5 \\) minutes. The cumulative distribution function (CDF) of \\( X \\) is given by:\n\\[ P(X \\leq t) = \\begin{cases} \n0 & \\text{if } t < 0... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d94c37fe52a8d1fbcf5d2254568e21edcc076447762977e4ab95ac517275714a | A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? | [
"\\frac{1}{7}"
] | [
"A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d3f82e1aa36b23e86a7e6cb11790d1aa6ef1e782ab805fe045f3e18dd3b041d9 | A traffic light runs through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? | [
"\\(\\frac{1}{7}\\)"
] | [
"A traffic light runs through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 24
} | 38 | 0 | 0 |
omr-ws-70394ff634f47d67084cd749ca16956132f52f08e5169cf0f78a89fa0db40541 | A train and a Super Hero start at the beginning of a 60-mile long tunnel. The train travels at 30 miles per hour, and the Super Hero flies at 100 miles per hour. How long will it be before the Super Hero is crashed by the train? | [
"\\(\\frac{12}{13}\\) hours"
] | [
"A train and a Super Hero start at the beginning of a 60-mile long tunnel. The train travels at 30 miles per hour, and the Super Hero flies at 100 miles per hour. How long will it be before the Super Hero is crashed by the train?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 20
} | 37 | 0 | 0 |
omr-ws-63bb4e029999bfcb09eea30f57c96c7f5778aaac5436d281b2fd017e99eb524f | A train breaks down 1 hour after starting its journey. The engineer takes 30 minutes to make repairs, and the train continues at half its original speed, arriving 2 hours late. If the delay had occurred 100 km further along the journey, the train would have been only 1 hour late. What is the total distance of the journ... | [
"250km"
] | [
"A train breaks down 1 hour after starting its journey. The engineer takes 30 minutes to make repairs, and the train continues at half its original speed, arriving 2 hours late. If the delay had occurred 100 km further along the journey, the train would have been only 1 hour late. What is the total distance of the ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0addfdc22430f6d36e8c67c71193cee40cdc67cfecd15de1e2ca969b415bf18c | A train breaks down an hour after starting its journey. The engineer takes half an hour to complete repairs (while at rest), but the train can continue at only half its usual speed and arrives at its destination two hours late. Had the delay occurred 100 km further on the journey, the train would only have been an hour... | [
"250 kilometers"
] | [
"A train breaks down an hour after starting its journey. The engineer takes half an hour to complete repairs (while at rest), but the train can continue at only half its usual speed and arrives at its destination two hours late. Had the delay occurred 100 km further on the journey, the train would only have been an... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-3d4277965dd4da7627627c2e8add14803006ccd4f5ff3825f3fd382d5e6f316d | A train consists of 2010 wagons, each containing gold coins of the same weight. The weights of the coins in different wagons are distinct and given by \( m_1 < m_2 < \ldots < m_{2010} \). Each wagon is labeled with one of these weights. The controller has a pair of scales and can use an arbitrary number of coins from a... | [
"1"
] | [
"A train consists of 2010 wagons, each containing gold coins of the same weight. The weights of the coins in different wagons are distinct and given by \\( m_1 < m_2 < \\ldots < m_{2010} \\). Each wagon is labeled with one of these weights. The controller has a pair of scales and can use an arbitrary number of coin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-3990b87672827b9aecf2951d6db65525536befda2e8f2e19454f20625e77fe28 | A train consists of \( n \) carriages, each of which may have a defect with probability \( p \). Each carriage is inspected independently by two inspectors, with the first inspector detecting defects with probability \( P1 \) and the second with probability \( P2 \). If none of the carriages is found to have a defect, ... | [
"0.0253"
] | [
"A train consists of \\( n \\) carriages, each of which may have a defect with probability \\( p \\). Each carriage is inspected independently by two inspectors, with the first inspector detecting defects with probability \\( P1 \\) and the second with probability \\( P2 \\). If none of the carriages is found to ha... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-e8a2d91c05846a17b67b22dea030d4228bae2c512f372346530808c1088051aa | A train departs from the train station heading North at 50 km/h. After 2 hours, another train departs from the same station heading East at 60 km/h. Find the rate of change of the distance between the two trains 1.5 hours after the departure of the second train. | [
"71.9 km/hr"
] | [
"A train departs from the train station heading North at 50 km/h. After 2 hours, another train departs from the same station heading East at 60 km/h. Find the rate of change of the distance between the two trains 1.5 hours after the departure of the second train."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-6840a2de7b8320bcedec0fb17d43f0761753a97b2397e1c4c4484ef7063ce277 | A train descends 5 cm in the first second and takes 30 seconds to reach the foot of the mountain. What is the total length of the descent in centimeters? | [
"4500"
] | [
"A train descends 5 cm in the first second and takes 30 seconds to reach the foot of the mountain. What is the total length of the descent in centimeters?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7ed3299214223c63b8e366e58636ec0d9dd605843a8f365aa0e7b87351c91e4d | A train going from Delhi to Jaipur stops at 7 intermediate stations. If 5 persons enter the train during the journey with five different tickets of the same class, how many different sets of tickets could they have had? | [
"376,992"
] | [
"A train going from Delhi to Jaipur stops at 7 intermediate stations. If 5 persons enter the train during the journey with five different tickets of the same class, how many different sets of tickets could they have had?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 24
} | 42 | 0 | 0 |
omr-ws-85153af9bedee4c43ed6955857700d9b6799df34b8768fef7f5d52643786371e | A train going from London to Cambridge stops at 12 intermediate stops. If 75 people enter the train from London with 75 different tickets of the same class, how many different sets of tickets they may be holding? | [
"\\(\\binom{91}{75}\\)"
] | [
"A train going from London to Cambridge stops at 12 intermediate stops. If 75 people enter the train from London with 75 different tickets of the same class, how many different sets of tickets they may be holding?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 20
} | 38 | 0 | 0 |
omr-ws-00ca9c86bc066c4b0fd5c892f8c1975d97e2d07eaf05bd787e3012ce4afe48a1 | A train has 12 boxcars linked together, packed with 4 cars of lions, 4 cars of tigers, and 4 cars of bears. The bears cannot be in the first 4 cars, the tigers cannot be in the next 4 cars, and the lions cannot be in the last 4 cars. Find the number of ways the cars can be packed. | [
"346"
] | [
"A train has 12 boxcars linked together, packed with 4 cars of lions, 4 cars of tigers, and 4 cars of bears. The bears cannot be in the first 4 cars, the tigers cannot be in the next 4 cars, and the lions cannot be in the last 4 cars. Find the number of ways the cars can be packed."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-fa63173fd6bf694c0c30cfe98ac3a3934756df3408a2eaaf5b93cb211882c3b4 | A train has 5 passengers and stops at exactly 4 random non-consecutive stations out of a total of 14 stations. If the passengers have fixed destinations which are also non-consecutive, find the probability that exactly 2 of them reach their correct destination. | [
"\\(\\frac{131}{5005}\\)"
] | [
"A train has 5 passengers and stops at exactly 4 random non-consecutive stations out of a total of 14 stations. If the passengers have fixed destinations which are also non-consecutive, find the probability that exactly 2 of them reach their correct destination."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d11fc8f87bb87e0fe73ba9a75ca8d6990a82d92d7f67d4cf488003fb81b65d6 | A train has to cover 840 km in a specified time. At the half-distance point it was held up for half an hour and so, in the remaining section of the route, it increased its speed by 2 km/h. How much time did the train spend en route? | [
"21 hours"
] | [
"A train has to cover 840 km in a specified time. At the half-distance point it was held up for half an hour and so, in the remaining section of the route, it increased its speed by 2 km/h. How much time did the train spend en route?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88de823a197d7d5b778d5001363a9ade8dca68d471cddde5e1d2b6cdf9572cfe | A train is approaching a 1 kilometer long tunnel at a constant 40 km/hr. If Roger, who is inside the tunnel, runs towards either end at a constant 10 km/hr, he will reach that end at the exact same time as the train. How far from the center of the tunnel is Roger? | [
"\\dfrac{1}{8}"
] | [
"A train is approaching a 1 kilometer long tunnel at a constant 40 km/hr. If Roger, who is inside the tunnel, runs towards either end at a constant 10 km/hr, he will reach that end at the exact same time as the train. How far from the center of the tunnel is Roger?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-504b060b487aac46340a41ae2f44a836713ac5a81ffbcceabcf8f4b81d91cef0 | A train leaves Newcastle for Sydney at 7:45 am while another train leaves from Sydney for Newcastle at 8:15 am. The first train arrives in Sydney 40 minutes after the trains pass, while the train to Newcastle arrives 1 hour and 40 minutes after they pass. Assuming that the trains traveled at constant speeds (not necess... | [
"9:05 am"
] | [
"A train leaves Newcastle for Sydney at 7:45 am while another train leaves from Sydney for Newcastle at 8:15 am. The first train arrives in Sydney 40 minutes after the trains pass, while the train to Newcastle arrives 1 hour and 40 minutes after they pass. Assuming that the trains traveled at constant speeds (not n... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-19ae19580d05d68a27595e5d827da197af2255d69415a144936defa3b36cd1f0 | A train leaves Newcastle for Sydney at 7:45 am, and another train leaves Sydney for Newcastle at 8:15 am. The first train arrives in Sydney 40 minutes after they pass each other, while the second train arrives in Newcastle 1 hour and 40 minutes after they pass. Assuming both trains travel at constant speeds, at what ti... | [
"9:05 am"
] | [
"A train leaves Newcastle for Sydney at 7:45 am, and another train leaves Sydney for Newcastle at 8:15 am. The first train arrives in Sydney 40 minutes after they pass each other, while the second train arrives in Newcastle 1 hour and 40 minutes after they pass. Assuming both trains travel at constant speeds, at wh... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-44c650c7a0f8c4750c11314bfc60ce7b82781d5a02beeaac5345eaeaadd33f23 | A train leaves Pune at 7:30 am and reaches Mumbai at 11:30 am. Another train leaves Mumbai at 9:30 am and reaches Pune at 1:00 pm. Assuming that the two trains travel at constant speeds, at what time do the two trains cross each other? | [
"10:26 am"
] | [
"A train leaves Pune at 7:30 am and reaches Mumbai at 11:30 am. Another train leaves Mumbai at 9:30 am and reaches Pune at 1:00 pm. Assuming that the two trains travel at constant speeds, at what time do the two trains cross each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-4edfeb0f08f52437432611a836529255ffe7b87a28d38e2ab6c48db8557dea32 | A train leaves from Newcastle for Sydney at 7:45 am while another train leaves from Sydney for Newcastle at 8:15 am. The first train arrives in Sydney 40 minutes after the trains pass, while the train to Newcastle arrives 1 hour 40 minutes after they pass. Assuming that the trains travel at constant speeds (not necessa... | [
"9:05"
] | [
"A train leaves from Newcastle for Sydney at 7:45 am while another train leaves from Sydney for Newcastle at 8:15 am. The first train arrives in Sydney 40 minutes after the trains pass, while the train to Newcastle arrives 1 hour 40 minutes after they pass. Assuming that the trains travel at constant speeds (not ne... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-519b1cce03e02c7bf1abd1eda409ea34b4f6bd7d68548ab3bbdb778e19709b91 | A train leaves from station A for station B at 7:45 AM while another train leaves from station B for station A at 8:15 AM. The first train arrives at station B 40 minutes after the trains pass, while the other arrives at station A 1 hour and 40 minutes after they pass. Assuming the trains travel at constant speeds, wha... | [
"9:05 AM"
] | [
"A train leaves from station A for station B at 7:45 AM while another train leaves from station B for station A at 8:15 AM. The first train arrives at station B 40 minutes after the trains pass, while the other arrives at station A 1 hour and 40 minutes after they pass. Assuming the trains travel at constant speeds... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-5a2081518537a2f5ee19bcaff1b61856e4116d9c467bbf21c6d3e374b06fa2d2 | A train leaving L at 3:10 P.M. reaches W at 5:00 P.M. Another train leaving W at 3:30 P.M. reaches L at 5:50 P.M. At what time do they pass each other? | [
"4:20.4 P.M."
] | [
"A train leaving L at 3:10 P.M. reaches W at 5:00 P.M. Another train leaving W at 3:30 P.M. reaches L at 5:50 P.M. At what time do they pass each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 20
} | 28 | 0 | 0 |
omr-ws-4f1b2cdaa1cfe4d4968f53c43557d9f5b8871db2bbef645cf43d642c2b8884a7 | A train left a station $A$ for $B$ at 13:00. At 19:00 the train was brought to a halt by a snow drift. Two hours later the railway line was cleared and to make up for the lost time the train proceeded at a speed exceeding the original speed by 20\% and arrived at $B$ only one hour later. The next day a train going from... | [
"600 km"
] | [
"A train left a station $A$ for $B$ at 13:00. At 19:00 the train was brought to a halt by a snow drift. Two hours later the railway line was cleared and to make up for the lost time the train proceeded at a speed exceeding the original speed by 20\\% and arrived at $B$ only one hour later. The next day a train goin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a370c70bd8d2fbef5fbf496caff26e376d5a1a073ece41f26a5158da65d1f736 | A train left station $A$ for $C$ via $B$. The speed of the train in the section from $A$ to $B$ was as required, but it fell off by 25\% in the section between $B$ and $C$. On the return trip, the required speed was maintained between $C$ and $B$, but decreased 25\% between $B$ and $A$. How long did it take for the tra... | [
"10 hours"
] | [
"A train left station $A$ for $C$ via $B$. The speed of the train in the section from $A$ to $B$ was as required, but it fell off by 25\\% in the section between $B$ and $C$. On the return trip, the required speed was maintained between $C$ and $B$, but decreased 25\\% between $B$ and $A$. How long did it take for ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e47cf98c9e3e8d584494abe5158bf921462565420ef766a54c63780a24fa0337 | A train starting at noon travels north at 40 miles per hour. Another train, starting from the same point at 2 PM, travels east at 50 miles per hour. How fast are the trains separating at 3 PM? | [
"64 mph"
] | [
"A train starting at noon travels north at 40 miles per hour. Another train, starting from the same point at 2 PM, travels east at 50 miles per hour. How fast are the trains separating at 3 PM?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41f4fc5c64aebd6ffc24025a7c68b15211a74a2a6690ed0e024fbe97c110ca28 | A train stops at station $\alpha$ for each $0 \leqslant \alpha \leqslant \omega_1$. It leaves station $0$ empty. When it arrives at station $\alpha$ $(0 < \alpha < \omega_1)$, one person gets off (unless it is empty), and then a countable number of people get on. What are the possible numbers of people on the train as ... | [
"0"
] | [
"A train stops at station $\\alpha$ for each $0 \\leqslant \\alpha \\leqslant \\omega_1$. It leaves station $0$ empty. When it arrives at station $\\alpha$ $(0 < \\alpha < \\omega_1)$, one person gets off (unless it is empty), and then a countable number of people get on. What are the possible numbers of people on ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-8b1dc1e79f4efe9647630d4d75b45d0a224dc20f9c15b79fad5968bb0fe48d33 | A train traveling at a steady speed crossed a bridge which was 200 meters long in 1 minute. The whole train passed a person standing on the bridge in 12 seconds. How long was the train? | [
"50m"
] | [
"A train traveling at a steady speed crossed a bridge which was 200 meters long in 1 minute. The whole train passed a person standing on the bridge in 12 seconds. How long was the train?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-3d7b681b97ca0e8407946620ba98c922074660cff6550fd7d1430d577df4ac25 | A train traveling from Aytown to Beetown meets with an accident after 1 hour. It is stopped for $\frac{1}{2}$ hour, after which it proceeds at four-fifths of its usual rate, arriving at Beetown 2 hours late. If the train had covered 80 miles more before the accident, it would have been just 1 hour late. What is the usu... | [
"20 mph"
] | [
"A train traveling from Aytown to Beetown meets with an accident after 1 hour. It is stopped for $\\frac{1}{2}$ hour, after which it proceeds at four-fifths of its usual rate, arriving at Beetown 2 hours late. If the train had covered 80 miles more before the accident, it would have been just 1 hour late. What is t... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-7fa4f9c4afa304bc6390263552f77b9a5abf267bd65e4fcbba6e9a179f4074d4 | A train traveling from Aytown to Beetown meets with an accident after 1 hour. It is stopped for 0.5 hours, after which it proceeds at four-fifths of its usual rate, arriving at Beetown 2 hours late. If the train had covered 80 miles more before the accident, it would have been just 1 hour late. What is the usual rate o... | [
"20 mph"
] | [
"A train traveling from Aytown to Beetown meets with an accident after 1 hour. It is stopped for 0.5 hours, after which it proceeds at four-fifths of its usual rate, arriving at Beetown 2 hours late. If the train had covered 80 miles more before the accident, it would have been just 1 hour late. What is the usual r... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-86c25b15d93e12578de6f40e9266e2ea4d2f975550f8f6bc5194dee96987ce45 | A train traveling from Aytown to Beetown meets with an accident after 1 hour. The train is stopped for 30 minutes, after which it proceeds at four-fifths of its usual rate, arriving at Beetown 2 hours late. If the train had covered 80 miles more before the accident, it would have been just one hour late. What is the us... | [
"20",
"20 mph"
] | [
"A train traveling from Aytown to Beetown meets with an accident after 1 hour. The train is stopped for 30 minutes, after which it proceeds at four-fifths of its usual rate, arriving at Beetown 2 hours late. If the train had covered 80 miles more before the accident, it would have been just one hour late. What is t... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 80,
"genselect": 3,
"tir": 3
} | 86 | 0 | 0 |
omr-ws-72cefc727f6c767d59ecbace9973ef3f31edd555dc3a4b8834a305bb0f72ae0b | A train travels $\frac{3}{5}$ of its total distance at $60$ km/h, then stops for $24$ minutes. To arrive on time, it continues the remaining distance at $80$ km/h. Find the total distance from the beginning to the point where the train stopped. | [
"144"
] | [
"A train travels $\\frac{3}{5}$ of its total distance at $60$ km/h, then stops for $24$ minutes. To arrive on time, it continues the remaining distance at $80$ km/h. Find the total distance from the beginning to the point where the train stopped."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-10d2350b5d853abcc2c2ccad1d3457e6bdb904625345faf1c95022f63f749623 | A train travels for 1.5 hours at a constant rate, then suffers a breakdown and takes 0.5 hours to repair. After the repair, it continues at 4/5 of its original speed and arrives at its destination 3 hours late. Find the ratio of the length of the entire trip, in miles, to the starting rate, in mph. | [
"\\(\\frac{11}{2}\\)"
] | [
"A train travels for 1.5 hours at a constant rate, then suffers a breakdown and takes 0.5 hours to repair. After the repair, it continues at 4/5 of its original speed and arrives at its destination 3 hours late. Find the ratio of the length of the entire trip, in miles, to the starting rate, in mph."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1973ebf88dc23a860e59a6503491defa3aea9bcc7bdad6656c1bc3b9e4fa4f51 | A train travels from Berkeley to San Francisco under a tunnel of length 10 kilometers, and then returns to Berkeley using a bridge of length 7 kilometers. If the train travels at 30 km/hr underwater and 60 km/hr above water, what is the train’s average speed in km/hr on the round trip? | [
"42.5"
] | [
"A train travels from Berkeley to San Francisco under a tunnel of length 10 kilometers, and then returns to Berkeley using a bridge of length 7 kilometers. If the train travels at 30 km/hr underwater and 60 km/hr above water, what is the train’s average speed in km/hr on the round trip?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9cc97ad9886990daf6f8f56b8d0bcb2fd7507280c983dbce237f1dd6cd585e22 | A train was moving at a constant speed when it suddenly stopped for 12 minutes. After resuming, the driver increased the speed by 20 km/h and made up for the lost time over the remaining 99 km. What was the train's speed before it stopped? | [
"90 km/h"
] | [
"A train was moving at a constant speed when it suddenly stopped for 12 minutes. After resuming, the driver increased the speed by 20 km/h and made up for the lost time over the remaining 99 km. What was the train's speed before it stopped?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-830aa661514ec3ba6392c671ab026be57dc7f5297f7b34cc0c5f828b34d7cc49 | A train will come into view from a restaurant sometime between 3:00 and 6:00 PM and will be visible for 20 minutes. A man arrives at the restaurant sometime between 2:00 and 5:00 PM and stays for 40 minutes. What is the probability that the man sees the train while he is in the restaurant? | [
"\\(\\frac{5}{54}\\)"
] | [
"A train will come into view from a restaurant sometime between 3:00 and 6:00 PM and will be visible for 20 minutes. A man arrives at the restaurant sometime between 2:00 and 5:00 PM and stays for 40 minutes. What is the probability that the man sees the train while he is in the restaurant?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40badbbe5533394657f3fa4ad5c4d4886448db0dc364d2b22860bd20e986c5b6 | A train, \( x \) meters long, traveling at a constant speed, takes 20 seconds from the time it first enters a 300-meter-long tunnel until it completely emerges from the tunnel. One of the stationary ceiling lights in the tunnel is directly above the train for 10 seconds. Find \( x \). | [
"300"
] | [
"A train, \\( x \\) meters long, traveling at a constant speed, takes 20 seconds from the time it first enters a 300-meter-long tunnel until it completely emerges from the tunnel. One of the stationary ceiling lights in the tunnel is directly above the train for 10 seconds. Find \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d3a098a5e63b6832cf2aec85627793f715b3c6c1a4dc9a333351fa839434068d | A trainee swordsman Chris takes a six-round training. He can choose to train his fighting skill or running skill in each round, but he cannot choose the same kind of training in three consecutive rounds. In odd-numbered rounds (Round 1, Round 3, and Round 5), if Chris chooses to train his fighting skill, he gets 2 figh... | [
"15"
] | [
"A trainee swordsman Chris takes a six-round training. He can choose to train his fighting skill or running skill in each round, but he cannot choose the same kind of training in three consecutive rounds. In odd-numbered rounds (Round 1, Round 3, and Round 5), if Chris chooses to train his fighting skill, he gets 2... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 13
} | 21 | 0 | 0 |
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