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-9afcc0644debd3ac0490b39bc7c7ce6a400cea20131ae208d013e0f1da4921ef | A syndicate has 12 members, one of whom is the president. A board of 4 positions needs to be formed from these members, with the condition that the president cannot be one of the board members. How many ways can this board be formed? | [
"10890"
] | [
"A syndicate has 12 members, one of whom is the president. A board of 4 positions needs to be formed from these members, with the condition that the president cannot be one of the board members. How many ways can this board be formed?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6628165db1c26b23d95cf60db8491a451d261f032651a7c1c9c5bd20f273ccb9 | A system has two components: a main component and a back-up. The back-up starts working only after the main component fails. Let \( X \) be the lifetime in years of the first component and let \( Y \) be the lifetime in years of the second component. \( X \) and \( Y \) have the joint probability density function \( f(... | [
"\\dfrac{47}{192}"
] | [
"A system has two components: a main component and a back-up. The back-up starts working only after the main component fails. Let \\( X \\) be the lifetime in years of the first component and let \\( Y \\) be the lifetime in years of the second component. \\( X \\) and \\( Y \\) have the joint probability density f... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-da2e0ad62ca77fb38d49b3224fecb6a78bd4cc74609a5c376f217acfffbcb7b9 | A t-shirt vendor sells 30 t-shirts a day for $10 each. For each $1 increase in price, she sells 4 fewer shirts, and for each $1 decrease in price, she sells 4 more shirts. What price will maximize her revenue? | [
"10"
] | [
"A t-shirt vendor sells 30 t-shirts a day for $10 each. For each $1 increase in price, she sells 4 fewer shirts, and for each $1 decrease in price, she sells 4 more shirts. What price will maximize her revenue?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4bfffc2ce1403382557c819e393d5495c6943fe9a4e265df1bdca28a63ba384c | A table $2 \times 2024$ is filled with positive integers. The first row contains numbers from the set $\{1, \ldots, 2023\}$. For any two columns, the difference of numbers from the first row is divisible by the difference of numbers from the second row, and all numbers in the second row are pairwise different. Let $S$ ... | [
"2047276"
] | [
"A table $2 \\times 2024$ is filled with positive integers. The first row contains numbers from the set $\\{1, \\ldots, 2023\\}$. For any two columns, the difference of numbers from the first row is divisible by the difference of numbers from the second row, and all numbers in the second row are pairwise different.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e950300f31b596d06daabb64cbc1c64edd9c786420f45cc5099de7c56f2ae3f5 | A table \( 2008 \times 2008 \) is given. Two players alternately make moves. A move consists of choosing a white or black pawn and placing it on a chosen field. A player wins if after his move there is a sequence of five pawns of the same color in a line (horizontal, vertical, or diagonal). Determine if the first playe... | [
"Yes"
] | [
"A table \\( 2008 \\times 2008 \\) is given. Two players alternately make moves. A move consists of choosing a white or black pawn and placing it on a chosen field. A player wins if after his move there is a sequence of five pawns of the same color in a line (horizontal, vertical, or diagonal). Determine if the fir... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb8802d144f1ec60ed2f0cee7fdfd64d427462159e9440055979af7b72e842df | A table has $3,000,000$ rows and $100$ columns, with each row containing the numbers from $1$ to $100$ exactly once, and no two rows are the same. Above each column, the number of distinct entries in that column is written in red. Find the smallest possible sum of the red numbers. | [
"\\(10000\\)"
] | [
"A table has $3,000,000$ rows and $100$ columns, with each row containing the numbers from $1$ to $100$ exactly once, and no two rows are the same. Above each column, the number of distinct entries in that column is written in red. Find the smallest possible sum of the red numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-8e93c804efc99613a8cead746d9967b1f12f750d010a8936d51939f64ebfc78c | A table has 1000 cards numbered from 1 to 1000 in order. In each movement, the first card is moved just before the last card, and the new first card is moved to just after the last card. After 1000 movements, let $P$ be the position of the card initially numbered 1. Find the value of $P$. | [
"1"
] | [
"A table has 1000 cards numbered from 1 to 1000 in order. In each movement, the first card is moved just before the last card, and the new first card is moved to just after the last card. After 1000 movements, let $P$ be the position of the card initially numbered 1. Find the value of $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-a721e6e653072ce44fb58a2a7de0f8886496b47a5616ad26f65a4fb1ac51acc3 | A table is covered by 15 pieces of paper. After removing 7 pieces, let $A$ be the area of the table covered by the remaining 8 pieces. If the total area of the table is 1, find the minimum possible value of $A$. | [
"0"
] | [
"A table is covered by 15 pieces of paper. After removing 7 pieces, let $A$ be the area of the table covered by the remaining 8 pieces. If the total area of the table is 1, find the minimum possible value of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-e81d1ab517ca52b702fa544eb0c81c6f394cdf998d17bd053067889add686737 | A table is drawn with $2011$ rows and $2012$ columns. In each cell of the table is written one of the numbers: $0,\, 1,\, 2$, so that the sum of the numbers in each row and each column is a multiple of $3$. Find the maximum possible number of $1$'s in the table. | [
"4043450"
] | [
"A table is drawn with $2011$ rows and $2012$ columns. In each cell of the table is written one of the numbers: $0,\\, 1,\\, 2$, so that the sum of the numbers in each row and each column is a multiple of $3$. Find the maximum possible number of $1$'s in the table."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 5,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-67a15eefd8a68bf93ff3d9f0e146f0375649d41e3ab961c9ba03139feab56a1b | A table of numbers has $21$ columns labelled $1, 2, 3, ..., 21$ and $33$ rows labelled $1, 2, 3, ..., 33$. Every element of the table is equal to $2$. All the rows whose label is not a multiple of $3$ are erased. All the columns whose label is not an even number are erased. What is the sum of the numbers that remain in... | [
"220"
] | [
"A table of numbers has $21$ columns labelled $1, 2, 3, ..., 21$ and $33$ rows labelled $1, 2, 3, ..., 33$. Every element of the table is equal to $2$. All the rows whose label is not a multiple of $3$ are erased. All the columns whose label is not an even number are erased. What is the sum of the numbers that rema... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-ef23fc6dbe6cf25d88a2d914f43cc55605ab7fdd3c887e00feac7d5b42fef888 | A table of size $n \times n$ will be filled with numbers $1$ or $-1$ such that the product of all the numbers in each row and the product of all the numbers in each column is $-1$. How many different ways can the table be filled? | [
"n!"
] | [
"A table of size $n \\times n$ will be filled with numbers $1$ or $-1$ such that the product of all the numbers in each row and the product of all the numbers in each column is $-1$. How many different ways can the table be filled?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba12b1b7e04e79d992d023fbfc26fe778fcf1d6b7b1b50550448525d0fa2c502 | A table of type \( (n_1, n_2, \ldots, n_m) \) with \( n_1 \ge n_2 \ge \ldots \ge n_m \) is defined such that there are \( n_1 \) squares in the first row, \( n_2 \) in the second, and so on, with a total of \( n = n_1 + n_2 + \ldots + n_m \) squares. The table is filled with numbers from 1 to \( n \) such that the numb... | [
"168"
] | [
"A table of type \\( (n_1, n_2, \\ldots, n_m) \\) with \\( n_1 \\ge n_2 \\ge \\ldots \\ge n_m \\) is defined such that there are \\( n_1 \\) squares in the first row, \\( n_2 \\) in the second, and so on, with a total of \\( n = n_1 + n_2 + \\ldots + n_m \\) squares. The table is filled with numbers from 1 to \\( n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d4adea42ca8bdf6e936f1d02ed964f3ae662bde481ac81ccf11f5ca8974e6dcb | A table with $m$ rows and $n$ columns is given. At any move, one chooses some empty cells such that any two of them lie in different rows and columns, places a white piece in those cells, and then places a black piece in the cells whose rows and columns contain the white pieces. The game ends when no more moves can be ... | [
"\\( m + n - 1 \\)"
] | [
"A table with $m$ rows and $n$ columns is given. At any move, one chooses some empty cells such that any two of them lie in different rows and columns, places a white piece in those cells, and then places a black piece in the cells whose rows and columns contain the white pieces. The game ends when no more moves ca... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95a76b71104c73294613c55ec7f49cfcb48c75104aaf1ecc06d7ede939b8591b | A table with \( m \) rows and \( n \) columns has all entries either \(-1\) or \(1\) such that the product of entries in each row and each column is \(-1\). If \( m \) and \( n \) have the same parity, find the number of such tables. | [
"\\(2^{(m-1)(n-1)}\\)"
] | [
"A table with \\( m \\) rows and \\( n \\) columns has all entries either \\(-1\\) or \\(1\\) such that the product of entries in each row and each column is \\(-1\\). If \\( m \\) and \\( n \\) have the same parity, find the number of such tables."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-2252491eecacd12e577b9dcee775ff9e55fda3987761ceda9c83c40155e513b1 | A table with an infinite number of fields contains numbers, with empty fields considered to contain zeros. In any square with sides along the grid lines, the absolute value of the sum of the numbers is equal to or less than 1. Determine the maximum value of the sum of all elements in a 5 x 1 rectangle. | [
"1"
] | [
"A table with an infinite number of fields contains numbers, with empty fields considered to contain zeros. In any square with sides along the grid lines, the absolute value of the sum of the numbers is equal to or less than 1. Determine the maximum value of the sum of all elements in a 5 x 1 rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 19,
"genselect": 2
} | 21 | 0 | 0 |
omr-ws-36faaa0663f0f5a8be7161e23f804e9fafbe320eedea1b210717d413bd828bd3 | A tailor cuts 120 square patches of side 1 from a 25x20 rectangular cloth. Determine the minimum radius of a circular patch that can always be cut from the remaining material, regardless of how the squares were cut. Express your answer as a decimal rounded to the nearest hundredth. | [
"0.91"
] | [
"A tailor cuts 120 square patches of side 1 from a 25x20 rectangular cloth. Determine the minimum radius of a circular patch that can always be cut from the remaining material, regardless of how the squares were cut. Express your answer as a decimal rounded to the nearest hundredth."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-c8cb721e7e7859c634b7393c82f82f99ac486b9ef5a29fbb97dd9f171dc91d59 | A tailor shop has an order for 810 suits, another shop has to make 900 soils in the same period of time. The first shop has completed its task 3 days before the target date, and the second, 6 days ahead of time. How many suits does each shop produce per day, if the second shop makes 4 suits per day more than the first? | [
"The first shop produces 20, and the second, 24 suits per day"
] | [
"A tailor shop has an order for 810 suits, another shop has to make 900 soils in the same period of time. The first shop has completed its task 3 days before the target date, and the second, 6 days ahead of time. How many suits does each shop produce per day, if the second shop makes 4 suits per day more than the f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aed22eb665d9bfa32c098b91b147567926c5a349f97140dc550b5b3034882ce8 | A talk show host has just bought 10 new jokes. Each night he tells some of the jokes. What is the largest number of nights on which you can tune in so that you never hear on one night at least all the jokes you heard on one of the other nights? | [
"252"
] | [
"A talk show host has just bought 10 new jokes. Each night he tells some of the jokes. What is the largest number of nights on which you can tune in so that you never hear on one night at least all the jokes you heard on one of the other nights?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-0740b037e29a0aa4f73e260723f1ee69935cc36435a6d9008d1b7309eb989a06 | A tangent $AB$ is drawn to a circle from a point $A$ which lies on the extension of the diameter $KL$ of the circle in the direction of the point $L$ ($B$ is the point of tangency). The tangent $AB$ makes an angle a with the diameter $KL$. Find the area of the figure formed by the sides of the angle and the arc $LB$ if... | [
"$\\frac{1}{2}R^2\\cot\\alpha -\\frac{1}{2}R^2\\left(\\frac{\\pi }{2}-\\alpha \\right)$"
] | [
"A tangent $AB$ is drawn to a circle from a point $A$ which lies on the extension of the diameter $KL$ of the circle in the direction of the point $L$ ($B$ is the point of tangency). The tangent $AB$ makes an angle a with the diameter $KL$. Find the area of the figure formed by the sides of the angle and the arc $L... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-242b09e87155799e6474013f6907422ac148aa6571cdd542605cb5797c3ba710 | A tangent $\overline{MA}$ is drawn from a point $M$ to circle $O$. $\overline{MC}$ is a secant line intersecting $O$ at $B$. Given that $\angle AMC$'s bisector intersects $\overline{AC}$ at $E$ and $\overline{AB}$ at $D$, compute $\frac{DB}{AB} + \frac{EC}{AC}$. | [
"1"
] | [
"A tangent $\\overline{MA}$ is drawn from a point $M$ to circle $O$. $\\overline{MC}$ is a secant line intersecting $O$ at $B$. Given that $\\angle AMC$'s bisector intersects $\\overline{AC}$ at $E$ and $\\overline{AB}$ at $D$, compute $\\frac{DB}{AB} + \\frac{EC}{AC}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-ba7a68adefdc15fe35015955d254fcf3dfd0f7475db4fbf0eb3a6aefaab969dc | A tangent TP is drawn to a fixed circle from a point T on a fixed line passing through the centre of the circle. A point Q is taken on the normal at P such that PQ = PT. Find the locus of Q. | [
"\\( r(\\theta) = 1 + |\\tan(\\theta)| \\)"
] | [
"A tangent TP is drawn to a fixed circle from a point T on a fixed line passing through the centre of the circle. A point Q is taken on the normal at P such that PQ = PT. Find the locus of Q."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fcd381dad969cf910a22de484e96375248b4412e5a3a8d6e0a3f3ef051331ade | A tangent at any point on a curve meets the coordinate axes at points \( A \) and \( B \). The area of triangle \( \Delta OAB \) is 2 square units. Determine the equation of the curve. | [
"\\( y = \\frac{1}{x} \\)"
] | [
"A tangent at any point on a curve meets the coordinate axes at points \\( A \\) and \\( B \\). The area of triangle \\( \\Delta OAB \\) is 2 square units. Determine the equation of the curve."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-8df8ff7d0657ac570a7a92f823fe62055a870e0d48aebefb781c73430cfa4715 | A tangent is drawn at any point \( P(t) \) on the parabola \( y^2 = 8x \). From this tangent, a point \( Q(\alpha, \beta) \) is taken, from which tangents \( QA \) and \( QB \) are drawn to the circle \( x^2 + y^2 = 4 \). Find the locus of the point of concurrency of the chord of contact \( AB \) of the circle \( x^2 +... | [
"\\( y^2 = -2x \\)"
] | [
"A tangent is drawn at any point \\( P(t) \\) on the parabola \\( y^2 = 8x \\). From this tangent, a point \\( Q(\\alpha, \\beta) \\) is taken, from which tangents \\( QA \\) and \\( QB \\) are drawn to the circle \\( x^2 + y^2 = 4 \\). Find the locus of the point of concurrency of the chord of contact \\( AB \\) o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-01f31e745836facc89f21eb1b69ffe8f73e6402ea5a17b8fe177af99a68b863c | A tangent is drawn from $O(0,0)$ to the circle defined by the equation $(x-5)^2+(y-7)^2=25$. What is the distance from the origin to the point of contact of the tangent? | [
"7"
] | [
"A tangent is drawn from $O(0,0)$ to the circle defined by the equation $(x-5)^2+(y-7)^2=25$. What is the distance from the origin to the point of contact of the tangent?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d21d703d36b1b7111ad6ab5e79dda036e3db4b79a1947ddda73fbd07e207e920 | A tangent line is drawn to a unit circle \( O \), and a point \( B \) 2 units away from the point of tangency \( A \) is chosen on the tangent line. Construct the other tangent to \( O \) going through \( B \) and extend this to meet line \( AO \) at point \( P \). Draw the line parallel to \( AB \) and tangent to circ... | [
"\\(\\frac{1}{6}\\)"
] | [
"A tangent line is drawn to a unit circle \\( O \\), and a point \\( B \\) 2 units away from the point of tangency \\( A \\) is chosen on the tangent line. Construct the other tangent to \\( O \\) going through \\( B \\) and extend this to meet line \\( AO \\) at point \\( P \\). Draw the line parallel to \\( AB \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f6fcfe6a91d5f98abbf2956d23216d8155fd50f27594e4afa4059c2b91e7951d | A tank containing 1000 L of salt water with an initial concentration of 0.1 kg/L is continuously mixed. Pure water is added at a rate of 5 L/sec, and the solution exits at the same rate. Write a conservation of mass principle to obtain the concentration \( C(t) \) and solve it to find the time \( t \) when \( C(t) = 0.... | [
"\\( 200 \\ln(10) \\)"
] | [
"A tank containing 1000 L of salt water with an initial concentration of 0.1 kg/L is continuously mixed. Pure water is added at a rate of 5 L/sec, and the solution exits at the same rate. Write a conservation of mass principle to obtain the concentration \\( C(t) \\) and solve it to find the time \\( t \\) when \\(... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-412bc8975f9952dcb41a6609c4bc77c7f37f69e5a8e3fb4e6193e2f72581026c | A tank contains 1000 L of salt water with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min, and the solution drains from the bottom at the same rate, keeping the volume constant. How many minutes does it take for the total amount of salt in the solution to be reduced by half its original amount... | [
"69.3147"
] | [
"A tank contains 1000 L of salt water with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min, and the solution drains from the bottom at the same rate, keeping the volume constant. How many minutes does it take for the total amount of salt in the solution to be reduced by half its original a... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 24
} | 45 | 0 | 0 |
omr-ws-7410de703d15b0811616c54abc54fbf74df10ee05ec8d1bc52a7f8a7f1d222a1 | A tank contains 125 gallons of heating oil at time \( t = 0 \). During the time interval \( 0 \leq t \leq 12 \) hours, heating oil is pumped into the tank at the rate \( H(t) = 5 + 2\sin\left(\frac{\pi t}{6}\right) \) gallons per hour and removed at the rate \( R(t) = 3 + \cos\left(\frac{\pi t}{6}\right) \) gallons per... | [
"a) 60, b) Rising, c) 149, d) \\( t = 6 \\)"
] | [
"A tank contains 125 gallons of heating oil at time \\( t = 0 \\). During the time interval \\( 0 \\leq t \\leq 12 \\) hours, heating oil is pumped into the tank at the rate \\( H(t) = 5 + 2\\sin\\left(\\frac{\\pi t}{6}\\right) \\) gallons per hour and removed at the rate \\( R(t) = 3 + \\cos\\left(\\frac{\\pi t}{6... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e9ab83e6685c95f460db79f3de3be2205cd0e4d588f5aa6b7b08d91b1389cfc | A tank contains 40 liters of a solution with 2 grams of a substance per liter. A liquid solution containing 3 grams of the substance per liter flows into the tank at a rate of 4 liters per minute, and the well-stirred mixture flows out at the same rate. Find the amount of the substance in the tank after 15 minutes. | [
"\\( 111.076 \\) grams"
] | [
"A tank contains 40 liters of a solution with 2 grams of a substance per liter. A liquid solution containing 3 grams of the substance per liter flows into the tank at a rate of 4 liters per minute, and the well-stirred mixture flows out at the same rate. Find the amount of the substance in the tank after 15 minutes... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 1,
"tir": 26
} | 40 | 0 | 0 |
omr-ws-ade7b81d817f3891c92e201b581ca03497537063aa1873716c3ff7cd5d6391c7 | A tank full of water is in the form of a rectangular parallelepiped 4m deep, 15m wide, and 30m long. Find the work required to pump the water in the tank up to a level 50 cm above the top of the tank. | [
"\\( 44100000 \\) Joules."
] | [
"A tank full of water is in the form of a rectangular parallelepiped 4m deep, 15m wide, and 30m long. Find the work required to pump the water in the tank up to a level 50 cm above the top of the tank."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 27
} | 40 | 0 | 0 |
omr-ws-a9c0fb7ede0572816e6dc64a6fe2270bfce51dd61ed0799484c27ae0a20f77b6 | A tank has dimensions of 10 cm by 15 cm by 18 cm and contains water at a depth of 3 cm. Eight cubes, each with a side length of 5 cm, are placed into the tank. Find the maximum height the water can rise. | [
"6.5"
] | [
"A tank has dimensions of 10 cm by 15 cm by 18 cm and contains water at a depth of 3 cm. Eight cubes, each with a side length of 5 cm, are placed into the tank. Find the maximum height the water can rise."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-80929c714fbdc0a4f29451ce98e6105851de701c49f166921bbc1858a689af17 | A tank has the shape of a regular hexagonal prism with a base side length of $1$ m and a height of $10$ m. The tank is partially filled with $9$ m$^3$ of water, and the plane of the free surface of the water cuts all lateral edges. One of the lateral edges has $2$ m under water. Determine the length of the part of the ... | [
"2"
] | [
"A tank has the shape of a regular hexagonal prism with a base side length of $1$ m and a height of $10$ m. The tank is partially filled with $9$ m$^3$ of water, and the plane of the free surface of the water cuts all lateral edges. One of the lateral edges has $2$ m under water. Determine the length of the part of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e18492acbbaeb20b58a333cac68f5ca8dc2ca9ec40477fa2a9c7f8dc1ecb4014 | A tank holds 500 gallons of brine. Brine containing 2 pounds per gallon of salt flows into the tank at the rate of 5 gallons per minute, and the mixture, kept uniform, flows out at the rate of 10 gallons per minute. If the maximum amount of salt is found in the tank at the end of 20 minutes, what was the initial salt c... | [
"375"
] | [
"A tank holds 500 gallons of brine. Brine containing 2 pounds per gallon of salt flows into the tank at the rate of 5 gallons per minute, and the mixture, kept uniform, flows out at the rate of 10 gallons per minute. If the maximum amount of salt is found in the tank at the end of 20 minutes, what was the initial s... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-65c032e33e7f3a6686d27f3b747dc5c4653b2a4cffe93225ce3052ed979ca62f | A tank in the form of a rectangular solid with a base area of 12 square feet has 8 inches of water in it. If a 1-foot gold cube is placed in the tank, by how many inches will the water rise? | [
"\\(\\frac{8}{11}\\)"
] | [
"A tank in the form of a rectangular solid with a base area of 12 square feet has 8 inches of water in it. If a 1-foot gold cube is placed in the tank, by how many inches will the water rise?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 19
} | 22 | 0 | 0 |
omr-ws-2675f9b8d06d448793a5544f2a2e82c081f8fba6669f15ac4eac194c1cec6d31 | A tank initially contains 1000 liters of brine with 0.15 kg of salt per liter. Fresh brine containing 0.25 kg of salt per liter flows into the tank at a rate of 4 liters per second, and the mixture flows out at the same rate. Let \( Q \) (in kg) be the amount of salt in the tank at time \( t \) (in seconds). Find the e... | [
"\\dfrac{dQ}{dt} = 1 - \\dfrac{Q}{250}"
] | [
"A tank initially contains 1000 liters of brine with 0.15 kg of salt per liter. Fresh brine containing 0.25 kg of salt per liter flows into the tank at a rate of 4 liters per second, and the mixture flows out at the same rate. Let \\( Q \\) (in kg) be the amount of salt in the tank at time \\( t \\) (in seconds). F... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-123a35fd3723575650e0464aaa3118db730ab2b72888a74e248400bd741aecb7 | A tank initially contains 1000 litres of brine with 15 kg of dissolved salt. Pure water is pumped into the tank at a rate of 10 litres per second, and liquid from the tank is pumped out at the same rate. The solution is kept thoroughly mixed. What is the concentration of salt at time \( t \)? | [
"\\( 0.015e^{-0.01t} \\)"
] | [
"A tank initially contains 1000 litres of brine with 15 kg of dissolved salt. Pure water is pumped into the tank at a rate of 10 litres per second, and liquid from the tank is pumped out at the same rate. The solution is kept thoroughly mixed. What is the concentration of salt at time \\( t \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-4c7bf0ee0a0fd4e76dc1c2fd53ab8d9110defb80915e7aac7afebe59d4c46399 | A tank initially contains 10kg of salt dissolved in 5000L of water. Brine containing 0.02kg of salt per liter enters the tank at a rate of 25L/min. The solution is thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after an hour? | [
"33.33"
] | [
"A tank initially contains 10kg of salt dissolved in 5000L of water. Brine containing 0.02kg of salt per liter enters the tank at a rate of 25L/min. The solution is thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after an hour?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-58e132051ee6edf1cd7cecf68771bb60d57fd9403e0484764450402b7dfeed49 | A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time as the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe. Find the time taken by the first pipe alone to fill the tank. | [
"15"
] | [
"A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time as the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe. Find the time taken by the first pipe alone to fill the t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3afe803801bab53a3e070d1a0c194394b99e87a14b7fdc51907e14638ff58d18 | A tank is filled with pure alcohol. Some of the alcohol is poured out and replaced by an equal amount of water; the same amount of the alcohol-water mixture thus obtained is then poured out, leaving 49 liters of pure alcohol in the tank. The tank has a capacity of 64 liters. How much alcohol was poured out for the firs... | [
"For the first time 8 liters of alcohol was poured out and for the second, 7 liters"
] | [
"A tank is filled with pure alcohol. Some of the alcohol is poured out and replaced by an equal amount of water; the same amount of the alcohol-water mixture thus obtained is then poured out, leaving 49 liters of pure alcohol in the tank. The tank has a capacity of 64 liters. How much alcohol was poured out for the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88929f772ccc9c2186e071bfbaa09b04c8de241a9a66d932bdf062971e62ae26 | A tank is filled with water from two taps. At the beginning the first tap was open for one third of the time which would haw been needed to fill the tank, if the second tap alone had been open. Then the second lap was open for one third of the time required to fill the tank, if the first tap alone were open. This done,... | [
"One of the taps fills the tank in 6 hours, the other, in 9 hours"
] | [
"A tank is filled with water from two taps. At the beginning the first tap was open for one third of the time which would haw been needed to fill the tank, if the second tap alone had been open. Then the second lap was open for one third of the time required to fill the tank, if the first tap alone were open. This ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca5c11b08d749dd38dc9820265a88daeb459bfe606071df70e949bfb51c8f448 | A tank is filled with water through five pipes. Water flowing through the first pipe fills the tank in 40 minutes, through the second, third, and fourth pipes together in 10 minutes, through the second, third, and fifth pipes together in 20 minutes, and through the fourth, and fifth pipes jointly in 30 minutes. How muc... | [
"$\\frac{60}{7}$ minutes"
] | [
"A tank is filled with water through five pipes. Water flowing through the first pipe fills the tank in 40 minutes, through the second, third, and fourth pipes together in 10 minutes, through the second, third, and fifth pipes together in 20 minutes, and through the fourth, and fifth pipes jointly in 30 minutes. Ho... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ebe95b449003d84f6377e53adb1d1f328a300bb13e5a3014505b5ba15861bcc0 | A tank is to be filled with water. When the tank is one-sixth full, 130 gallons of water are added, making the tank three-fifths full. How many gallons does the tank contain when it is completely full? | [
"300"
] | [
"A tank is to be filled with water. When the tank is one-sixth full, 130 gallons of water are added, making the tank three-fifths full. How many gallons does the tank contain when it is completely full?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1815951964d085354cbb5f0dbcbbd67eea64e090b22689d45c8ef4e07ea83d13 | A tank of a volume of 425 m$^3$ was filled with water from two taps. One of the taps was open 5 hours longer than the other. If the first tap had been kept open as long as the second and the second tap as long as the first, then the first tap would have released one half the amount of water flowed out from the second. ... | [
"15 hours"
] | [
"A tank of a volume of 425 m$^3$ was filled with water from two taps. One of the taps was open 5 hours longer than the other. If the first tap had been kept open as long as the second and the second tap as long as the first, then the first tap would have released one half the amount of water flowed out from the sec... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40c543432f4408bcbafc9deb612fb01fcbcf8dc44d4e9c6cbcecdd89773135f5 | A tank travels 7 km on a bearing of 050° from a base \( O \) and then 6 km on a bearing of 140° to a bunker \( B \). Calculate the magnitude and direction of the displacement vector \( OB \). | [
"Magnitude \\( \\approx 9.22 \\) km, Direction \\( \\approx 90.6^\\circ \\)"
] | [
"A tank travels 7 km on a bearing of 050° from a base \\( O \\) and then 6 km on a bearing of 140° to a bunker \\( B \\). Calculate the magnitude and direction of the displacement vector \\( OB \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 29
} | 39 | 0 | 0 |
omr-ws-bc37213a62f88aba712c374c7b771716d238ed061a5e07df0c0c390897e08754 | A tank was being filled with water from two taps. One of the taps was first open during one third of the time required for filling the tank by the other tap alone. Then, conversely, the second tap was kept open for one third of the time required to fill the tank by using the first tap alone, after which the tank was $\... | [
"9 hours, 6 hours\n\nor\n\n360 min, 540 min\n\nSince the answer is given in two equivalent forms (hours and minutes), we can present the final answer as:\n\nt_1=9 \\text{ hours}, t_2=6 \\text{ hours}\n\nor\n\nt_1=360 \\text{ min}, t_2=540 \\text{ min} \n\nFor simplicity, we can choose one form:\n\nt_1=9 \\text{ hou... | [
"A tank was being filled with water from two taps. One of the taps was first open during one third of the time required for filling the tank by the other tap alone. Then, conversely, the second tap was kept open for one third of the time required to fill the tank by using the first tap alone, after which the tank w... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42f738b3248478a3b5839bb44877152ed3ecde3f98c93b3d0f6500f136d4392c | A tank was filled with petrol during the course of an integral number of hours, and the ratio of the amount added each succeeding hour to the amount added in the preceding hour was a constant. An hour before completion of the job there were 372 liters in the tank. If from a full tank we draw off the amount of petrol th... | [
"192 liters"
] | [
"A tank was filled with petrol during the course of an integral number of hours, and the ratio of the amount added each succeeding hour to the amount added in the preceding hour was a constant. An hour before completion of the job there were 372 liters in the tank. If from a full tank we draw off the amount of petr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c707883ae35332b6533c8ca3095a9db5aef5993c0b9a35108184726f284afe0 | A tanker truck was full of water, but it began to leak. Suppose the truck's fuel consumption is directly proportional to the weight it carries and the water flow and truck speed are constant. After traveling 200 km, the truck had half its water capacity and spent half a tank of fuel. If it were empty, the truck would w... | [
"\\(\\frac{11}{8}\\)"
] | [
"A tanker truck was full of water, but it began to leak. Suppose the truck's fuel consumption is directly proportional to the weight it carries and the water flow and truck speed are constant. After traveling 200 km, the truck had half its water capacity and spent half a tank of fuel. If it were empty, the truck wo... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-01b1a2778ec27d44274d96dd37d3d5b172888e70e5489f28acd9e182e1091323 | A tanker was filled with oil through two pipes, each of which having filled more than $\frac{1}{4}$ of its volume. If the amount of oil supplied per hour through the first pipe were increased by 1.5 times, and the amount of oil supplied per hour through the second pipe $\frac{1}{4}$ of its actual capacity, then the tim... | [
"Twice as much delivers the second pipe"
] | [
"A tanker was filled with oil through two pipes, each of which having filled more than $\\frac{1}{4}$ of its volume. If the amount of oil supplied per hour through the first pipe were increased by 1.5 times, and the amount of oil supplied per hour through the second pipe $\\frac{1}{4}$ of its actual capacity, then ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b760c52bd70b2adb51de84936219511facca7e648620e6c2f3936a3274a3771e | A tap filled 40% of a tank in 2 hours. At that time, another tap with a flow rate 40% higher than the first was opened. How long did it take to completely fill the tank? | [
"1.25 hours"
] | [
"A tap filled 40% of a tank in 2 hours. At that time, another tap with a flow rate 40% higher than the first was opened. How long did it take to completely fill the tank?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-e3cc0966cc7903f1903a1944797c75b6f780481fefe9e136c14e29d2dccd99a8 | A tape recording is made of an 8-minute conversation that began sometime after the 21st minute on a 60-minute tape. If a 15-minute segment of the tape is erased randomly, find the probability that the entire 8-minute conversation was erased. | [
"\\dfrac{77}{558}"
] | [
"A tape recording is made of an 8-minute conversation that began sometime after the 21st minute on a 60-minute tape. If a 15-minute segment of the tape is erased randomly, find the probability that the entire 8-minute conversation was erased."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-19d970de46f02464daa9280207364b728e291f198ad651d583ceef4759fb6064 | A target consists of an equilateral triangle broken into 100 equilateral triangles of unit side length by three sets of parallel lines. A sniper shoots at the target repeatedly: he aims at one of the small triangles and then hits either that triangle or one of the small triangles which shares a side with it. He may cho... | [
"25"
] | [
"A target consists of an equilateral triangle broken into 100 equilateral triangles of unit side length by three sets of parallel lines. A sniper shoots at the target repeatedly: he aims at one of the small triangles and then hits either that triangle or one of the small triangles which shares a side with it. He ma... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-29c4d43829b50a49957584b595c8a25b2df7ee5ea27e120db584016c86248eae | A target has a diameter of 12 centimeters. You randomly fire 19 shots on the target, ensuring that none of the shots miss the target. Calculate the minimum possible distance between the two closest holes among these 19 shots. Express your answer in centimeters. | [
"3"
] | [
"A target has a diameter of 12 centimeters. You randomly fire 19 shots on the target, ensuring that none of the shots miss the target. Calculate the minimum possible distance between the two closest holes among these 19 shots. Express your answer in centimeters."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-a55e2d3560cda892dd14bccb8f6e72074ce4021667152d1c998f98edbb616937 | A target has the form of an equilateral triangle with a side length of 2. If the target is hit 17 times, what is the maximum minimal distance between any two holes? | [
"$1/2$",
"\\dfrac{2}{5}"
] | [
"A target has the form of an equilateral triangle with a side length of 2. If the target is hit 17 times, what is the maximum minimal distance between any two holes?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 48,
"genselect": 5
} | 53 | 0 | 0 |
omr-ws-4609211c228aa9f2831ee7d5bf42738a1b9bb1772e336885a109132ca840e163 | A target is formed by 30 concentric circles C1, C2, C3, ..., C30, with radii 1, 2, 3, ..., 30, respectively. Each impact on the target assigns points equal to the number of circles perforated by the impact. For example, an impact within C1 assigns 30 points, and an impact within C27 but outside C26 assigns 4 points. Th... | [
"465"
] | [
"A target is formed by 30 concentric circles C1, C2, C3, ..., C30, with radii 1, 2, 3, ..., 30, respectively. Each impact on the target assigns points equal to the number of circles perforated by the impact. For example, an impact within C1 assigns 30 points, and an impact within C27 but outside C26 assigns 4 point... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-34d2bca071d2e51d44a8c0d9c6d8fedd671fa66c4544f15d522149aa17379a5c | A task was set to deliver a building material from a railway station to a construction site in 8 hours. The material had to be delivered with 300 three-ton trucks. These trucks worked for two hours and then 9 five-ton trucks were added to help out. The task was completed in time. If the five-ton trucks had begun the op... | [
"$x = 300$; $y = 270$; 30 five-ton trucks will deliver all the material in $270 : 30 = 9$ hours"
] | [
"A task was set to deliver a building material from a railway station to a construction site in 8 hours. The material had to be delivered with 300 three-ton trucks. These trucks worked for two hours and then 9 five-ton trucks were added to help out. The task was completed in time. If the five-ton trucks had begun t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0498022d411fe66e70ae885e5dc614fa0e88b626f8c6eb946899bddab5983bbf | A taut belt connects two pulleys with radii 1 inch and 5 inches. The centers of the pulleys are 8 inches apart. What is the length of the belt? | [
"\\( 6\\pi + 8\\sqrt{5} \\)"
] | [
"A taut belt connects two pulleys with radii 1 inch and 5 inches. The centers of the pulleys are 8 inches apart. What is the length of the belt?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f1a7d3632a29b8daee5d4f36ed5ccd92ed8ebe7f8fbbb91295d234496a2b8b5 | A taxi ride costs $\$1.50$ plus $\$0.25$ per mile traveled. How much, in dollars, does a 5-mile taxi ride cost? | [
"\\$2.75"
] | [
"A taxi ride costs $\\$1.50$ plus $\\$0.25$ per mile traveled. How much, in dollars, does a 5-mile taxi ride cost?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b4281352a8e19acb5b9ac49f6a8d6dbbd97a71545b3b965c61f722b0fbf48335 | A taxi ride costs \$1.50 plus \$0.25 per mile traveled. How much does a 5-mile taxi ride cost? | [
"\\$2.75"
] | [
"A taxi ride costs \\$1.50 plus \\$0.25 per mile traveled. How much does a 5-mile taxi ride cost?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-9c7db357eb8896aca2057c2d427cdbe3d237390a4b55772a3baa1d111e8b05a3 | A taxi travels between four villages, $W$, $X$, $Y$, and $Z$, situated at the corners of a rectangle. The distances between the villages are as follows: $W$ to $X$ and $Y$ to $Z$ are $5$ miles, and $W$ to $Z$ and $Y$ to $X$ are $10$ miles. The taxi receives calls from any of the villages with probability $1/4$ and each... | [
"} \\mathbb{E}(D) = 10, \\textup{Var}(D) = \\frac{50}{3}\n\\]"
] | [
"A taxi travels between four villages, $W$, $X$, $Y$, and $Z$, situated at the corners of a rectangle. The distances between the villages are as follows: $W$ to $X$ and $Y$ to $Z$ are $5$ miles, and $W$ to $Z$ and $Y$ to $X$ are $10$ miles. The taxi receives calls from any of the villages with probability $1/4$ and... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-9e95911134b3fd0e481fce6bf0fd42f3d555602dd82a317fc45c5ffa505bc080 | A teacher asked each student of a class to write down the number of classmates with the same surname as his, as well as the number of classmates whose birthday is the same as his. Among the answers received, each of the numbers 0, 1, 2, …, 10 has appeared. Find the minimum number of students in the class. | [
"11"
] | [
"A teacher asked each student of a class to write down the number of classmates with the same surname as his, as well as the number of classmates whose birthday is the same as his. Among the answers received, each of the numbers 0, 1, 2, …, 10 has appeared. Find the minimum number of students in the class."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3412a5788e5ab3aa98e6a884a8fdc58d05ceb8bbd2a2755d4793f867900af199 | A teacher asks Annie, Basil, Cara, Dave, and Elli, in that order, to each multiply a pair of (possibly equal) single digits. Except for Annie, each of them gets a product that is $50\%$ more than the previous student. What product did Dave get? | [
"54"
] | [
"A teacher asks Annie, Basil, Cara, Dave, and Elli, in that order, to each multiply a pair of (possibly equal) single digits. Except for Annie, each of them gets a product that is $50\\%$ more than the previous student. What product did Dave get?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-c370e4ef79d08400455e203a7921644eb4f46abdd23850f1c78763f7053bc330 | A teacher asks her class what the value of $\left(5^2-4^2\right)^3$ is. What is the value of the expression? | [
"729"
] | [
"A teacher asks her class what the value of $\\left(5^2-4^2\\right)^3$ is. What is the value of the expression?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2f28cea3d1d9ba4f2fc46ff52665fca9779a0715f5c895ca892d287860d55ec | A teacher assigns 6 different tasks to 4 students, where each task is to be assigned to one student only and all students are assigned tasks. How many ways of assignment are there? | [
"1560"
] | [
"A teacher assigns 6 different tasks to 4 students, where each task is to be assigned to one student only and all students are assigned tasks. How many ways of assignment are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-40fb1458e8bc7653f387a203a2bc51ebb0cc9ad41792d80d0048ddfc8182ae48 | A teacher chose a set $S$ of four integers, each given to a different student. The numbers in $S$ are four consecutive two-digit positive integers, with one number divisible by $6$ and a different number divisible by $7$. Initially, none of the students could deduce the set $S$. However, upon hearing that all four stud... | [
"258"
] | [
"A teacher chose a set $S$ of four integers, each given to a different student. The numbers in $S$ are four consecutive two-digit positive integers, with one number divisible by $6$ and a different number divisible by $7$. Initially, none of the students could deduce the set $S$. However, upon hearing that all four... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 11
} | 21 | 0 | 0 |
omr-ws-335165cf8603c8e3ac74cd4ce7d810a570dfc81746691293583627d1a8d92ffa | A teacher computed all the remainders of the division of 365 by 1, 2, 3, ..., 365, summed them up, and wrote the result on a blackboard. A student computed all the remainders of the division of 366 by 1, 2, 3, ..., 366, summed them up, and wrote the result on a scratch. Whose number is bigger? | [
"Teacher"
] | [
"A teacher computed all the remainders of the division of 365 by 1, 2, 3, ..., 365, summed them up, and wrote the result on a blackboard. A student computed all the remainders of the division of 366 by 1, 2, 3, ..., 366, summed them up, and wrote the result on a scratch. Whose number is bigger?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f2898331a9bb765e70f24f7203d9200d75ae65f54f12282589e2131db17700ad | A teacher counts 40 heads and 100 legs at a park. How many kids and dogs are there? Find one more solution besides 30 kids and 10 dogs. | [
"29 \\text{ kids and } 10 \\text{ dogs}"
] | [
"A teacher counts 40 heads and 100 legs at a park. How many kids and dogs are there? Find one more solution besides 30 kids and 10 dogs."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-4920812fbb549d6b2f4671d656b2bc6bf0c2d162290bef41489ede61e3cfd478 | A teacher created groups of three with two girls and one boy in each group, but 14 boys were left without a group. When the teacher instead created groups with two boys and one girl in each group, 1 girl and 1 boy were left without a group. How many students does the teacher have? | [
"29"
] | [
"A teacher created groups of three with two girls and one boy in each group, but 14 boys were left without a group. When the teacher instead created groups with two boys and one girl in each group, 1 girl and 1 boy were left without a group. How many students does the teacher have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-679fadec4545386e96fadc297d2bc3e24ee86f20f0b1b26c568f4eeb66fd4730 | A teacher does not know the names of any of her 10 students. Each of these students has completed a homework assignment, and 3 of the students have the same name and answers, so their assignments are identical. If the teacher hands back each of the assignments to a random student, what is the expected number of student... | [
"9"
] | [
"A teacher does not know the names of any of her 10 students. Each of these students has completed a homework assignment, and 3 of the students have the same name and answers, so their assignments are identical. If the teacher hands back each of the assignments to a random student, what is the expected number of st... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-7d4400e476e1255b46eeeb1c79ebccc13844cf2a4cbea703cbd8057ac99e692b | A teacher finds that when she offers candy to her class of 30 students, the mean number of pieces taken by each student is 5. If every student takes some candy, what is the greatest number of pieces one student could have taken? | [
"121"
] | [
"A teacher finds that when she offers candy to her class of 30 students, the mean number of pieces taken by each student is 5. If every student takes some candy, what is the greatest number of pieces one student could have taken?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e34d62f83d95e240af4b57c5c560d41d44826cb35911a826da834dfea2804d6 | A teacher has 16 apples to distribute among 4 kids in a line. The teacher wants to ensure that no kid can correctly guess the difference in the number of apples between the two kids in front of them. Find the number of distinct ways the teacher can distribute the apples under these conditions, assuming each kid receive... | [
"144"
] | [
"A teacher has 16 apples to distribute among 4 kids in a line. The teacher wants to ensure that no kid can correctly guess the difference in the number of apples between the two kids in front of them. Find the number of distinct ways the teacher can distribute the apples under these conditions, assuming each kid re... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1df71523fc01f806f7279bd7c48618bec7caf807b8558e6a4db23619f7a6e6a6 | A teacher has 2010 questions and wants to divide them into three folders of 670 questions each. Each folder must be given to a student who solved all 670 questions in that folder. Determine the minimum number of students in the class that makes this possible, given that at most two students did not solve any given ques... | [
"3",
"6"
] | [
"A teacher has 2010 questions and wants to divide them into three folders of 670 questions each. Each folder must be given to a student who solved all 670 questions in that folder. Determine the minimum number of students in the class that makes this possible, given that at most two students did not solve any given... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 25,
"genselect": 3
} | 28 | 0 | 0 |
omr-ws-bf766abb4f478fc22bda5eec793756e0585e48c4faf2509a82312c088a73b4dc | A teacher has 2013 candles of 11 different types and distributes them to students such that no student receives two candles of the same type. Each pair of students writes down the number of candle types they both have. Let \( M \) be the sum of these numbers. Find the minimum value of \( M \). | [
"183183"
] | [
"A teacher has 2013 candles of 11 different types and distributes them to students such that no student receives two candles of the same type. Each pair of students writes down the number of candle types they both have. Let \\( M \\) be the sum of these numbers. Find the minimum value of \\( M \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36d8978699ef64b4b72392ba56fded4b7929c75e46d314d70733cefdaacbaf0f | A teacher has \( n \) students and writes each student's name and their project subject on separate pieces of paper. What strategy should the teacher use to maximize the probability that each student receives their own paper? | [
"1"
] | [
"A teacher has \\( n \\) students and writes each student's name and their project subject on separate pieces of paper. What strategy should the teacher use to maximize the probability that each student receives their own paper?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-de971c4d6935fb87efed2caae2b75fca3132458fec9f5170222276b990964556 | A teacher has a class with $24$ students in it. If she wants to split the students into equal groups of at most $10$ students each, what is the least number of groups that she needs? | [
"3"
] | [
"A teacher has a class with $24$ students in it. If she wants to split the students into equal groups of at most $10$ students each, what is the least number of groups that she needs?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0bc7fb547ca4649174e28b50c22efbee640562a4e043db69d5df15ebfabbdc5a | A teacher has established an order for testing 30 students but has not communicated it. Students can form groups of 3 and ask the teacher who is the first or the last in the testing order within that group. Determine the largest integer \( n \) such that students can be confident of determining the order of testing for... | [
"15"
] | [
"A teacher has established an order for testing 30 students but has not communicated it. Students can form groups of 3 and ask the teacher who is the first or the last in the testing order within that group. Determine the largest integer \\( n \\) such that students can be confident of determining the order of test... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-f9bf76cbbf7c525441693cca3dd675dfd1fa7f35b99ffb53d9562661626ee5f6 | A teacher plays at least one game of chess each day but no more than 10 games in a week. Suppose the teacher continues playing chess for 28 days. Calculate the total number of games played in a continuous period of days where the teacher plays exactly 21 games, given that such a period exists. If there are multiple suc... | [
"168"
] | [
"A teacher plays at least one game of chess each day but no more than 10 games in a week. Suppose the teacher continues playing chess for 28 days. Calculate the total number of games played in a continuous period of days where the teacher plays exactly 21 games, given that such a period exists. If there are multipl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-e10276f04ca128d8697e37d8b0e40af8df50a824e2e7d161605365705d3ae6ca | A teacher rounds test scores to the nearest integer. Adam received a score of 99 on his test by getting only one question wrong. However, the teacher then removed a problem that Adam got right, and his score changed to 98. How many questions were on the original test? | [
"67"
] | [
"A teacher rounds test scores to the nearest integer. Adam received a score of 99 on his test by getting only one question wrong. However, the teacher then removed a problem that Adam got right, and his score changed to 98. How many questions were on the original test?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-32f0de748c994743a131eac130cb0232b10a192e3a2461f5739045d2d3184542 | A teacher selects a set $S$ of four distinct integers and tells the class that some numbers in $S$ are divisible by 4, some by 6, and some by 7. Initially, no student can determine $S$, but when one student realizes $S$ after hearing that no one else can determine it, find the sum of the elements of $S$. | [
"29"
] | [
"A teacher selects a set $S$ of four distinct integers and tells the class that some numbers in $S$ are divisible by 4, some by 6, and some by 7. Initially, no student can determine $S$, but when one student realizes $S$ after hearing that no one else can determine it, find the sum of the elements of $S$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-faa34ef6b41180aa618649b26bd9bef6b33bd8f84e132f23d40007d181293ff6 | A teacher stands at $(0,10)$ and has some students, where there is exactly one student at each integer position in the following triangle: \[ \begin{array}{c} \text{(triangle extends until and includes the column (21,y))} \end{array} \] Here, the circle denotes the teacher at $(0,10)$ and the triangle extends until and... | [
"141"
] | [
"A teacher stands at $(0,10)$ and has some students, where there is exactly one student at each integer position in the following triangle:\n\\[\n\\begin{array}{c}\n\\text{(triangle extends until and includes the column (21,y))}\n\\end{array}\n\\]\nHere, the circle denotes the teacher at $(0,10)$ and the triangle e... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-ef19d7f67600a108fb52bcfe2e01f78c3884842dd65e532a542cc5fb9c438276 | A teacher thinks of a monic polynomial \( P(x) \) with integer coefficients and degree 2017. The teacher provides \( k \) integers \( n_1, n_2, \ldots, n_k \) and the value of \( P(n_1)P(n_2) \cdots P(n_k) \). Determine the minimum value of \( k \) such that there exists at least one case where the students can uniquel... | [
"2018"
] | [
"A teacher thinks of a monic polynomial \\( P(x) \\) with integer coefficients and degree 2017. The teacher provides \\( k \\) integers \\( n_1, n_2, \\ldots, n_k \\) and the value of \\( P(n_1)P(n_2) \\cdots P(n_k) \\). Determine the minimum value of \\( k \\) such that there exists at least one case where the stu... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-00782fe8c863f56f3de22cc4a933545bd7eb34b96ebd2192b664e8146a98212c | A teacher thinks of a positive integer \( n \). Students can name infinitely many positive integers \( k \). The teacher will respond as follows: - "Yes" if both \( n+k \) and \( nk \) are perfect squares. - "No" if both \( n+k \) and \( nk \) are not perfect squares. - Anything else if one of \( n+k \) or \( nk \) is ... | [
"No"
] | [
"A teacher thinks of a positive integer \\( n \\). Students can name infinitely many positive integers \\( k \\). The teacher will respond as follows:\n- \"Yes\" if both \\( n+k \\) and \\( nk \\) are perfect squares.\n- \"No\" if both \\( n+k \\) and \\( nk \\) are not perfect squares.\n- Anything else if one of \... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-31b93ed1157c895f709081c693cd6a5c7426c73d5e838e41b91c7373b07c6e64 | A teacher thought of two positive integers (which may or may not be different) and told Steven their sum and Peter their product. Steven said to Peter, "There is no way you can determine what number I was given." Peter then replied, "Now I know that you were given the number 136." Using sound reasoning, determine the t... | [
"\\( 17 \\) and \\( 8 \\)"
] | [
"A teacher thought of two positive integers (which may or may not be different) and told Steven their sum and Peter their product. Steven said to Peter, \"There is no way you can determine what number I was given.\" Peter then replied, \"Now I know that you were given the number 136.\" Using sound reasoning, determ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5617415b90d301b6161d775ab68aafb9b3d79b512a26bc8becc4522a25ea8b6a | A teacher walks home against a river current with speed $1.5v$. He accidentally drops a wooden club into the river and continues walking. After some distance, he realizes his mistake, turns around, and runs back at speed $2v$ to retrieve a hat that also fell into the river. After catching the hat, he returns to his ori... | [
"\\(3t_1 + \\frac{7}{90}\\)"
] | [
"A teacher walks home against a river current with speed $1.5v$. He accidentally drops a wooden club into the river and continues walking. After some distance, he realizes his mistake, turns around, and runs back at speed $2v$ to retrieve a hat that also fell into the river. After catching the hat, he returns to hi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-819aab6ca29105dc0e7698c8b9a5bb861e12244af3da452733fbf9c2633fe49d | A teacher wants to arrange 3 copies of Introduction to Geometry and 4 copies of Introduction to Number Theory on a bookshelf. In how many ways can he do that? | [
"35"
] | [
"A teacher wants to arrange 3 copies of Introduction to Geometry and 4 copies of Introduction to Number Theory on a bookshelf. In how many ways can he do that?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b23d83c25e3f5084c75ce7832a0d097597ba7ac8924a0f017133467cb9007e1 | A teacher wants to divide un-graded papers into a collection of several stacks, each consisting of the same number of papers. He knows that there is somewhere between 1000 and 2000 papers. He tries dividing the stack into groups of 2, 3, 4, 5, 6, 7, and 8, but repeatedly gets 1 paper left over. How many papers are ther... | [
"1681"
] | [
"A teacher wants to divide un-graded papers into a collection of several stacks, each consisting of the same number of papers. He knows that there is somewhere between 1000 and 2000 papers. He tries dividing the stack into groups of 2, 3, 4, 5, 6, 7, and 8, but repeatedly gets 1 paper left over. How many papers are... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2590b72952a75f808e0fb60f6873d05d749b1aaadc1335677cfcc5a879386d33 | A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $... | [
"258"
] | [
"A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 7,
"tir": 14
} | 23 | 0 | 0 |
omr-ws-244048174c1c4272a2cf7509b6ce0f42c171876a4eb00e9cd60db0923c7ffd7d | A teacher was leading a class of four perfectly logical students. The teacher chose a set \(S\) of four integers and gave a different number in \(S\) to each student. Then the teacher announced to the class that the numbers in \(S\) were four consecutive two-digit positive integers, that some number in \(S\) was divisi... | [
"258"
] | [
"A teacher was leading a class of four perfectly logical students. The teacher chose a set \\(S\\) of four integers and gave a different number in \\(S\\) to each student. Then the teacher announced to the class that the numbers in \\(S\\) were four consecutive two-digit positive integers, that some number in \\(S\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-ab8ed74c00af26c22d2b617ed465ec6179b5bd6bfcd5ff33d4e058fdf192d371 | A teacher whispers a positive integer \( p \) to student \( A \), \( q \) to student \( B \), and \( r \) to student \( C \). The students don't know one another's numbers, but they know the sum of the numbers is 14. The following is the sequence of their statements: - A says "I know that B and C have different numbers... | [
"42"
] | [
"A teacher whispers a positive integer \\( p \\) to student \\( A \\), \\( q \\) to student \\( B \\), and \\( r \\) to student \\( C \\). The students don't know one another's numbers, but they know the sum of the numbers is 14. The following is the sequence of their statements:\n\n- A says \"I know that B and C h... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 7,
"tir": 21
} | 73 | 0 | 0 |
omr-ws-7bbc91513b9e5b8184636826764f69a7bd90619db0a4236ab1f8e49ec3747f12 | A teacher wishes to distribute 20 identical pieces of candy among 7 students, so that the number of pieces of candy for any two students differs by at most 2. In how many different ways can the teacher distribute the candy? | [
"357"
] | [
"A teacher wishes to distribute 20 identical pieces of candy among 7 students, so that the number of pieces of candy for any two students differs by at most 2. In how many different ways can the teacher distribute the candy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-d9af363fd1b9cf22ed6eda1da621680b9d95681f623311da686473a507557fd6 | A teacher with a class of 7 students has 10 pieces of candy. If each piece of candy is given to a student at random, what is the probability that every student gets at least one piece of candy? | [
"\\(\\frac{3}{286}\\)"
] | [
"A teacher with a class of 7 students has 10 pieces of candy. If each piece of candy is given to a student at random, what is the probability that every student gets at least one piece of candy?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f9a1f8ee1aa55dcb52648ca54cfae43c12394efdff6199697dfd398726682af | A teacher writes a positive integer less than fifty thousand on the board. One student claims it is an exact multiple of 2; a second student says it is an exact multiple of 3; and so on, until the twelfth student says that it is an exact multiple of 13. The teacher observes that all but two of the students were right, ... | [
"25740"
] | [
"A teacher writes a positive integer less than fifty thousand on the board. One student claims it is an exact multiple of 2; a second student says it is an exact multiple of 3; and so on, until the twelfth student says that it is an exact multiple of 13. The teacher observes that all but two of the students were ri... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 58
} | 90 | 0 | 0 |
omr-ws-d54dadc57a65aeccb1791c1df834ab16f2a62bc3adf9e3e9e545a954a19e8269 | A teacher writes the positive integers from $1$ to $12$ on a blackboard. Every minute, they choose a number $k$ uniformly at random from the written numbers, subtract $k$ from each number $n \geq k$ on the blackboard (without touching the numbers $n < k$), and erase every $0$ on the board. Estimate the expected number ... | [
"24"
] | [
"A teacher writes the positive integers from $1$ to $12$ on a blackboard. Every minute, they choose a number $k$ uniformly at random from the written numbers, subtract $k$ from each number $n \\geq k$ on the blackboard (without touching the numbers $n < k$), and erase every $0$ on the board. Estimate the expected n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-443006bd0fca8d322f3d13cfee45217e452503a1d064ec0201c611b8c7e79f92 | A teacher writes the positive integers from 1 to 10 on the board. Then, he takes two random numbers, erases them, and replaces them with their average. The teacher repeats this process until only one number (not necessarily an integer) is left. What is the expected value of this last number? | [
"5.5"
] | [
"A teacher writes the positive integers from 1 to 10 on the board. Then, he takes two random numbers, erases them, and replaces them with their average. The teacher repeats this process until only one number (not necessarily an integer) is left. What is the expected value of this last number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9d86993037c5c22307757a4ace253973e5305bd7b641a22f44a97162dfb57af0 | A teacher wrote down three positive real numbers on the blackboard and told Dima to decrease one of them by $3\%$, decrease another by $4\%$, and increase the last by $5\%$. Dima wrote down the results in his notebook. It turned out that he wrote down the same three numbers that are on the blackboard, just in a differe... | [
"3"
] | [
"A teacher wrote down three positive real numbers on the blackboard and told Dima to decrease one of them by $3\\%$, decrease another by $4\\%$, and increase the last by $5\\%$. Dima wrote down the results in his notebook. It turned out that he wrote down the same three numbers that are on the blackboard, just in a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e35d5ad54e22bd828d77a84bb3f50b8a1a084390b48f31cf14536b9b4270d9ff | A team contains five boys and five girls. For the team banquet, they select seats at random around a circular table that seats ten people. What is the probability that some two girls will seat next to one another? Express your answer as a rational number in lowest terms. | [
"\\(\\frac{125}{126}\\)"
] | [
"A team contains five boys and five girls. For the team banquet, they select seats at random around a circular table that seats ten people. What is the probability that some two girls will seat next to one another? Express your answer as a rational number in lowest terms."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-96f929ed44d5ddc84d1f1eb76720f524d20f482a4ba5a6d8cafe3c5c06a07261 | A team contest between \( n \) participants is to be held. Each participant has exactly \( k \) enemies among the other participants. (If \( A \) is an enemy of \( B \), then \( B \) is an enemy of \( A \). Nobody is his own enemy.) Assume that there are no three participants such that every two of them are enemies of ... | [
"\\dfrac{nk}{2}"
] | [
"A team contest between \\( n \\) participants is to be held. Each participant has exactly \\( k \\) enemies among the other participants. (If \\( A \\) is an enemy of \\( B \\), then \\( B \\) is an enemy of \\( A \\). Nobody is his own enemy.) Assume that there are no three participants such that every two of the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-7a6a613c596593b999c0b03ec8037f04d5fb354603b0a5715194720c021f706e | A team is going to play 8 games this season. The chance of them winning each game is $\frac{2}{3}$. Find the probability that they will win four games in a row at some point. | [
"\\(\\frac{112}{243}\\)"
] | [
"A team is going to play 8 games this season. The chance of them winning each game is $\\frac{2}{3}$. Find the probability that they will win four games in a row at some point."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 22
} | 35 | 0 | 0 |
omr-ws-c306675dc5c087f25b383417882930ada58550ac93fd76ea7edeee8ea9f0f72e | A team of 10 people are planning to take a math competition consisting of two tests each for algebra, geometry, combinatorics, number theory, and calculus. Each teammate is supposed to do one test, and each person is good at two subjects with no two people being good at the same two subjects. Find the number of ways th... | [
"24"
] | [
"A team of 10 people are planning to take a math competition consisting of two tests each for algebra, geometry, combinatorics, number theory, and calculus. Each teammate is supposed to do one test, and each person is good at two subjects with no two people being good at the same two subjects. Find the number of wa... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-3be3b9a6295097fd2b5fb7609bc4fe7d3282d1a59cf247d62a881f2903181944 | A team of 6 distinguishable students competed at a math competition. They each scored an integer amount of points, and the sum of their scores was 170. Their highest score was a 29, and their lowest score was a 27. How many possible ordered 6-tuples of scores could they have scored? | [
"90"
] | [
"A team of 6 distinguishable students competed at a math competition. They each scored an integer amount of points, and the sum of their scores was 170. Their highest score was a 29, and their lowest score was a 27. How many possible ordered 6-tuples of scores could they have scored?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f7f3a6f98b3eb49085e12b12ada9d1bfb6975afb78dbc0eb41b5b4cd14a02da | A team of \( n \) players is allowed a strategy session before being placed in a room with an infinite tower of hats on each of their heads. Each hat is either black or white, chosen by independent flips of a fair coin. After seeing all hats except their own, each player must simultaneously name a positive integer. The... | [
"\\(\\frac{1}{n+1}\\)"
] | [
"A team of \\( n \\) players is allowed a strategy session before being placed in a room with an infinite tower of hats on each of their heads. Each hat is either black or white, chosen by independent flips of a fair coin. After seeing all hats except their own, each player must simultaneously name a positive integ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
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