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-ff166aa42b36abcd8ede41636fc3226247a7b4a44a796eb20a5a18ca47bc731c | A wheel makes 2.5 rotations about its axis. Find the angular displacement in radians of a point P on the wheel. | [
"\\(5\\pi\\)"
] | [
"A wheel makes 2.5 rotations about its axis. Find the angular displacement in radians of a point P on the wheel."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-cbf6f401c9ded4750bf061a7c163260162d496f17ae4827d41b0c8ec3d71d932 | A wheel of radius 1 is moving on a slope described by the equation \( y = \sin x \). Find a parametric equation for the curve on which the center of the wheel is moving. | [
"\\( x = t, y = \\sin t + 1 \\)"
] | [
"A wheel of radius 1 is moving on a slope described by the equation \\( y = \\sin x \\). Find a parametric equation for the curve on which the center of the wheel is moving."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-369c7c7a1f9b0a6ba3295155ca9c8a75ffa925439312030a2899d8a3e0b293e2 | A wheel of radius 5 and a wheel of radius 4 are connected at their centers by an axle of length 3. The larger wheel is soaked in ink and the two wheels are placed on a piece of paper and given a small push so that the ink traces a circle on the paper. How many revolutions will the wheels undergo before the larger wheel... | [
"\\(\\sqrt{10}\\)"
] | [
"A wheel of radius 5 and a wheel of radius 4 are connected at their centers by an axle of length 3. The larger wheel is soaked in ink and the two wheels are placed on a piece of paper and given a small push so that the ink traces a circle on the paper. How many revolutions will the wheels undergo before the larger ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2725f8df6a29fd038f50bdc1220cc9a405f100558ca7777e0a2299420597c73a | A wheel with a diameter of 60 feet makes one revolution in 3 minutes. Find the speed of a seat on the rim of the wheel. | [
"\\( 20 \\pi \\frac{\\text{ ft}}{\\text{ min}} \\)"
] | [
"A wheel with a diameter of 60 feet makes one revolution in 3 minutes. Find the speed of a seat on the rim of the wheel."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-1af5581249bd5d476e7504b5f5edcc6195b16fd4a42e96fc65271173b0204697 | A wheel with a diameter of 8.5 feet is rotating to pull cable cars at a speed of 9.55 mph. How fast is the wheel rotating? | [
"31.5 RPM"
] | [
"A wheel with a diameter of 8.5 feet is rotating to pull cable cars at a speed of 9.55 mph. How fast is the wheel rotating?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-01b1d17ce5cb50086f618053ab2f5c4b6a39eb08867292f99118b2bcc2df1a87 | A wheel with radius $1\text{ m}$ is rolled in a straight line through one complete revolution on a flat horizontal surface. How many meters did the center of the wheel travel horizontally from its starting location? | [
"2 \\pi"
] | [
"A wheel with radius $1\\text{ m}$ is rolled in a straight line through one complete revolution on a flat horizontal surface. How many meters did the center of the wheel travel horizontally from its starting location?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e96ab7174764de72daec3a01883fe33dad77b37a326ce4ce34ddef232e48a46 | A whispering gallery is constructed as part of the surface formed on rotation of the ellipse $\frac{x^2}{100} + \frac{y^2}{k}$ with $x$ and $y$ in yards. Each whisperer stands at a focus on the $x$-axis that is three feet from the nearest vertex. Find $k$. | [
"19"
] | [
"A whispering gallery is constructed as part of the surface formed on rotation of the ellipse $\\frac{x^2}{100} + \\frac{y^2}{k}$ with $x$ and $y$ in yards. Each whisperer stands at a focus on the $x$-axis that is three feet from the nearest vertex. Find $k$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-07f8cbf38b5b24ea130587ca7444dee69a5c23704547a1d5f67496e4a7f83368 | A white ball has finitely many disk-shaped black regions (closed) painted on its surface. Each black region has an area less than half of the surface area of the sphere, and no two black regions touch or overlap. Let $N$ be the number of diameters of the ball with two white endpoints. Calculate the minimum possible val... | [
"1"
] | [
"A white ball has finitely many disk-shaped black regions (closed) painted on its surface. Each black region has an area less than half of the surface area of the sphere, and no two black regions touch or overlap. Let $N$ be the number of diameters of the ball with two white endpoints. Calculate the minimum possibl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-20beb00fdd317690a5e6285900c7393d340feec273559deae44203848d0d86c0 | A white ball is on point \( p_1 \) and a red ball is on point \( p_2 \) on a billiard table. The white ball is shot towards the red ball, hitting 3 sides of the table first. Find the minimal distance the white ball must travel to reach the red ball. | [
"\\(28\\sqrt{2}\\)"
] | [
"A white ball is on point \\( p_1 \\) and a red ball is on point \\( p_2 \\) on a billiard table. The white ball is shot towards the red ball, hitting 3 sides of the table first. Find the minimal distance the white ball must travel to reach the red ball."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05535fc0d86608aa2341b0f5313deec37d26fee256736091b59f46d16c2281fa | A white bug sits in one corner square of a $1000 \times n$ chessboard, where $n$ is an odd positive integer and $n > 2020$. In the two nearest corner squares, there are two black chess bishops. On each move, the bug either steps into a square adjacent by side or moves as a chess knight. The bug wishes to reach the oppo... | [
"0"
] | [
"A white bug sits in one corner square of a $1000 \\times n$ chessboard, where $n$ is an odd positive integer and $n > 2020$. In the two nearest corner squares, there are two black chess bishops. On each move, the bug either steps into a square adjacent by side or moves as a chess knight. The bug wishes to reach th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d49d6f3210930670da7f2310cf6a0158d79c6ad35c10e0faa49dd053a3d0ee5 | A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, making two complete revolutions around it. What is the area of the stripe in square feet? | [
"240"
] | [
"A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, making two complete revolutions around it. What is the area of the stripe in square feet?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 4
} | 68 | 0 | 0 |
omr-ws-1b2034fdd4addd3890e1ec991dba7fe5ebb6e6e6968dacb5f6afdf0b50426428 | A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, making two complete revolutions around. What is the area of the stripe in square feet? | [
"240"
] | [
"A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, making two complete revolutions around. What is the area of the stripe in square feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-7b5603c7f2ac6fe81b8a39924c14d81b031054eb142d47cb5fd1e3943125c96f | A white piece called the "beetle" is placed in a corner cell of a $1000 \times n$ board, where $n$ is an odd natural number greater than $2020$. In the two nearest corners of the board, there are two black chess bishops. With each move, the beetle either moves to an adjacent cell by side or moves like a chess knight. T... | [
"0"
] | [
"A white piece called the \"beetle\" is placed in a corner cell of a $1000 \\times n$ board, where $n$ is an odd natural number greater than $2020$. In the two nearest corners of the board, there are two black chess bishops. With each move, the beetle either moves to an adjacent cell by side or moves like a chess k... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-d36e38dcddf12cdb4d3d7e2f4f7238eab6248953b6efa16e8464a49fb74e7be6 | A white plane is partitioned into cells. A finite number of cells are colored black, and each black cell has an even number (0, 2, or 4) of adjacent white cells. Suppose there are exactly 10 black cells. Let $N$ be the number of ways to color each white cell either green or red such that every black cell has an equal n... | [
"24"
] | [
"A white plane is partitioned into cells. A finite number of cells are colored black, and each black cell has an even number (0, 2, or 4) of adjacent white cells. Suppose there are exactly 10 black cells. Let $N$ be the number of ways to color each white cell either green or red such that every black cell has an eq... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5c145fcb9f52a6dac0e63ca0afa0574bbed7779446ebd58adf4ce8d07ef6612 | A white plane is partitioned into cells. A finite number of cells are colored black. Each black cell has an even (0, 2, or 4) adjacent (by the side) white cells. Determine the number of ways to color each white cell either green or red such that every black cell will have an equal number of red and green adjacent cells... | [
"256"
] | [
"A white plane is partitioned into cells. A finite number of cells are colored black. Each black cell has an even (0, 2, or 4) adjacent (by the side) white cells. Determine the number of ways to color each white cell either green or red such that every black cell will have an equal number of red and green adjacent ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-5cb4cde4fbac017feec30412612c9eea9922d6fc58a1aa4a92ead91e1007dedc | A white plane is partitioned into cells. A finite number of cells are colored black. Each black cell has an even number (0, 2, or 4) of adjacent white cells. If each white cell is colored either green or red, let $S$ be the sum of the absolute differences between the number of red and green adjacent cells for each blac... | [
"0"
] | [
"A white plane is partitioned into cells. A finite number of cells are colored black. Each black cell has an even number (0, 2, or 4) of adjacent white cells. If each white cell is colored either green or red, let $S$ be the sum of the absolute differences between the number of red and green adjacent cells for each... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-1bf092d06579a11651ab05cce33241c4c1a2b340a2c7b944aedf73d98bad61ea | A white rat is placed in a maze at time 0. At each time step \( n = 1, 2, \ldots \), the rat either moves to a new compartment or stays in the same compartment with probabilities \( \frac{1}{3} \) and \( \frac{2}{3} \) respectively. If it moves, it chooses one of the \( k \) possible compartments at random. If the rat ... | [
"2"
] | [
"A white rat is placed in a maze at time 0. At each time step \\( n = 1, 2, \\ldots \\), the rat either moves to a new compartment or stays in the same compartment with probabilities \\( \\frac{1}{3} \\) and \\( \\frac{2}{3} \\) respectively. If it moves, it chooses one of the \\( k \\) possible compartments at ran... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92808247c2dd347a7c9f4ccfab83ed3b5a6b407cef4b742a76e0175746f7e4d1 | A whiteboard has positive real numbers $1$ and $m$ written on it. Every second, if the numbers $x$ and $y$ are on the whiteboard, a ghost will replace those numbers with $|x^2-y^2|$ and $2xy$. The ghost stops as soon as one number on the whiteboard is once again $m$ times the other. For how many positive real numbers $... | [
"32768"
] | [
"A whiteboard has positive real numbers $1$ and $m$ written on it. Every second, if the numbers $x$ and $y$ are on the whiteboard, a ghost will replace those numbers with $|x^2-y^2|$ and $2xy$. The ghost stops as soon as one number on the whiteboard is once again $m$ times the other. For how many positive real numb... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-10110b5734d75a805a5aa01bee5dec9b383206f32c66d66adac142210da09d65 | A whiteboard has positive real numbers $1$ and $m$ written on it. Every second, if the numbers $x$ and $y$ are on the whiteboard, a ghost will replace those numbers with $|x^2-y^2|$ and $2xy$. The ghost stops once one number on the whiteboard is $m$ times the other. For how many positive real numbers $m$ does the ghost... | [
"65536"
] | [
"A whiteboard has positive real numbers $1$ and $m$ written on it. Every second, if the numbers $x$ and $y$ are on the whiteboard, a ghost will replace those numbers with $|x^2-y^2|$ and $2xy$. The ghost stops once one number on the whiteboard is $m$ times the other. For how many positive real numbers $m$ does the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-00ccd207370a54d9ae811358387284a5da6edc946ce455e0910fb10d330a0477 | A whole block of modeling clay is a right rectangular prism six inches by two inches by one inch. How many whole blocks need to be unwrapped to mold a cylindrical sculpture seven inches high and four inches in diameter? | [
"8"
] | [
"A whole block of modeling clay is a right rectangular prism six inches by two inches by one inch. How many whole blocks need to be unwrapped to mold a cylindrical sculpture seven inches high and four inches in diameter?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8409cfa42583ace80b5068e62b888aa436091b1621c91ed94d26838367095720 | A whole number decreases 9 times when one of its digits is deleted, and the resultant number is divisible by 9. Find all the whole numbers satisfying the condition of the problem. | [
"Start with proving the auxiliary proposition: the number in question decreases 9 times when the second (counting from right to left) digit 0 is deleted"
] | [
"A whole number decreases 9 times when one of its digits is deleted, and the resultant number is divisible by 9.\nFind all the whole numbers satisfying the condition of the problem."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b23f2917832ef3b849880445a6c45468add3f2a2dd56bbc75e949105aa4850d3 | A whole number decreases an integral number of times when its last digit is deleted. Find all such numbers. | [
"All the numbers ending with 0 and the two-digit numbers 11, 23, 33, 44, 55, 66, 77, 88, 99; 12, 24, 36, 48; 13, 26, 39; 14, 28; 15; 16; 17; 18; 19"
] | [
"A whole number decreases an integral number of times when its last digit is deleted. Find all such numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff9f96faa1d1298c839d5325b3ec11c9fba3b6febe0d14f889b20ef9435e0a90 | A whole number is equal to the arithmetic mean of all the numbers obtained from the given number with the aid of all the possible permutations of its digits (including, of course, the "identity permutation" under which all the digits retain their places). Find all whole numbers possessing this property. | [
"111, 222, 333,\\ldots, 999, 407, 518, 629, 370, 481, 592. {\\bf 44-45.} Consider the process of the addition of the given numbers written in a column"
] | [
"A whole number is equal to the arithmetic mean of all the numbers obtained from the given number with the aid of all the possible permutations of its digits (including, of course, the \"identity permutation\" under which all the digits retain their places). Find all whole numbers possessing this property."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7da1398ea54b4472eca4789ce7132b72d6fb4ed0d5d614f8b38187af915f957 | A whole number is said to be ''9-heavy'' if the remainder when the number is divided by 9 is greater than 5. What is the least three-digit 9-heavy whole number? | [
"105"
] | [
"A whole number is said to be ''9-heavy'' if the remainder when the number is divided by 9 is greater than 5. What is the least three-digit 9-heavy whole number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ac84d2687b495f7da5b7b7011145630cd21eca871673af2a1c97cf47767292a | A whole number is written on each square of a $2019 \times 2021$ board. If the number in each square is equal to the arithmetic mean of the numbers in two of its neighboring squares, how many different numbers can be written on the board at most? Two squares are neighbors if they share a common side. | [
"1"
] | [
"A whole number is written on each square of a $2019 \\times 2021$ board. If the number in each square is equal to the arithmetic mean of the numbers in two of its neighboring squares, how many different numbers can be written on the board at most? Two squares are neighbors if they share a common side."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-9b1ed928b6f6c736a0126fb53a6dadc31249f0d3090ad6682ef5aa29b27bf1dd | A whole number is written on each square of a $3 \times 2021$ board. If the number written in each square is greater than or equal to at least two of the numbers written in the neighboring squares, how many different numbers can be written on the board at most? Note: Two squares are neighbors when they have a common si... | [
"3"
] | [
"A whole number is written on each square of a $3 \\times 2021$ board. If the number written in each square is greater than or equal to at least two of the numbers written in the neighboring squares, how many different numbers can be written on the board at most? Note: Two squares are neighbors when they have a com... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-ceee1cbee83dd9fc23692f5cbbffb5d7b6ffda5121469ffa401ee6cb82131c62 | A whole number, $N$, is chosen so that $\frac{N}{3}$ is strictly between 7.5 and 8. What is the value of $N$ ? | [
"23"
] | [
"A whole number, $N$, is chosen so that $\\frac{N}{3}$ is strictly between 7.5 and 8. What is the value of $N$ ?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-84559df25e50d0d6e44573f8341b1911595ce73b020004eb9945bcfd0b4f8024 | A wholesaler sells sneakers at 20.00 reais per pair for orders of less than 50 pairs. For orders of 50 pairs or more (up to 600 pairs), the price per pair decreases by 0.02 reais per pair. What should be the ordered quantity to provide the highest possible revenue? | [
"500"
] | [
"A wholesaler sells sneakers at 20.00 reais per pair for orders of less than 50 pairs. For orders of 50 pairs or more (up to 600 pairs), the price per pair decreases by 0.02 reais per pair. What should be the ordered quantity to provide the highest possible revenue?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-1addf0274b4f09948aedf4936a7c6a54fe392bd49d9bbeba033561bd6c283b87 | A widget factory has four loading docks for storing crates of ready-to-ship widgets. Any time a loading dock contains at least 5 crates, a truck picks up 5 crates from that dock and ships them away. Suppose the factory produces 8 indistinguishable crates of widgets and sends each crate to one of the four loading docks.... | [
"85"
] | [
"A widget factory has four loading docks for storing crates of ready-to-ship widgets. Any time a loading dock contains at least 5 crates, a truck picks up 5 crates from that dock and ships them away. Suppose the factory produces 8 indistinguishable crates of widgets and sends each crate to one of the four loading d... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c315c72eaac400d453f2e80f4b9a9fffd3c08ae57cd5df254ed8ac4fb2f40f6 | A wind-up toy jumps two metres in a random direction, waits one second, and then jumps a fixed distance \( x \) in a random direction. The probability that it ends up within one metre of its original starting point is \( \frac{1}{6} \). What is \( x \)? | [
"\\( x = \\sqrt{3} \\)"
] | [
"A wind-up toy jumps two metres in a random direction, waits one second, and then jumps a fixed distance \\( x \\) in a random direction. The probability that it ends up within one metre of its original starting point is \\( \\frac{1}{6} \\). What is \\( x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 30
} | 44 | 0 | 0 |
omr-ws-5ea72b19355e885be340310fe42988b05e7b70a1b4a3bd5a57a30b8f2765a4e2 | A window consists of a rectangle and a regular triangle constructed on one of the sides of the rectangle. What should be the ratio of the sides of the rectangle so that the area of the window is the largest at the given perimeter? | [
"\\dfrac{3 + \\sqrt{3}}{3}"
] | [
"A window consists of a rectangle and a regular triangle constructed on one of the sides of the rectangle. What should be the ratio of the sides of the rectangle so that the area of the window is the largest at the given perimeter?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-3c760d2ef0d2094c7c7314c7e0a69ecb45bd3b5eca3d99b9de638cf1f9d39c21 | A window has a perimeter of 10 meters and is shaped like a rectangle with the top edge replaced by a semicircle. Find the dimensions of the rectangle that maximize the amount of light admitted through the window. | [
"\\(\\text{Height} = \\frac{10}{\\pi + 4}, \\text{Width} = \\frac{20}{\\pi + 4}\\)"
] | [
"A window has a perimeter of 10 meters and is shaped like a rectangle with the top edge replaced by a semicircle. Find the dimensions of the rectangle that maximize the amount of light admitted through the window."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 26
} | 39 | 0 | 0 |
omr-ws-b67eb05765c6d93f423906fc80b52759d18e03867b4e3803ed1e6dc278f0c719 | A window is formed by a rectangle topped by an equilateral triangle. The perimeter of the window is \(6 - \sqrt{3}\) and the area of the window is \(\frac{6 - \sqrt{3}}{4}\). If \(x\) is the length of the rectangle and \(y\) is the width, and the side length of the equilateral triangle is equal to \(x\), find \(x + y\)... | [
"\\(\\frac{5 - \\sqrt{3}}{2}\\)"
] | [
"A window is formed by a rectangle topped by an equilateral triangle. The perimeter of the window is \\(6 - \\sqrt{3}\\) and the area of the window is \\(\\frac{6 - \\sqrt{3}}{4}\\). If \\(x\\) is the length of the rectangle and \\(y\\) is the width, and the side length of the equilateral triangle is equal to \\(x\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-0f2e0a2e7ce0c109e75c90507460f38dbe65ece662f81f4fc76bd10c74847c0b | A window is formed by a rectangle topped by an equilateral triangle. The perimeter of the window is \(6 - \sqrt{3}\) and the area of the window is \(\frac{6 - \sqrt{3}}{4}\). If \(x\) is the length of the rectangle and the side length of the triangle, and \(y\) is the width of the rectangle, find \(x + y\). | [
"2"
] | [
"A window is formed by a rectangle topped by an equilateral triangle. The perimeter of the window is \\(6 - \\sqrt{3}\\) and the area of the window is \\(\\frac{6 - \\sqrt{3}}{4}\\). If \\(x\\) is the length of the rectangle and the side length of the triangle, and \\(y\\) is the width of the rectangle, find \\(x +... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-210f726e322c018176a4d250ead136ed6f399cc1311a26945a64d5b7189de704 | A wine glass has the shape defined by the equation $z = x^2 + y^2$ for $z \le 1$. The glass is filled to a height $h$. When the glass is tilted by $45$ degrees, the wine just starts spilling out. Determine the value of $h$. | [
"\\( \\frac{1}{4} \\)"
] | [
"A wine glass has the shape defined by the equation $z = x^2 + y^2$ for $z \\le 1$. The glass is filled to a height $h$. When the glass is tilted by $45$ degrees, the wine just starts spilling out. Determine the value of $h$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 5
} | 23 | 0 | 0 |
omr-ws-21cc2828471a6c7c8b277f9210e06fd7931e744938dc06abadf46dd04c9ea47b | A wineglass is full of water, and an iron ball is dipped into it such that water spills over. The iron ball rests inside with its top level with the rim of the wineglass. What is the maximum angle \( A \) (the angle subtended by the wineglass' side with its vertical axis) that maximizes the spill over of water? | [
"60^\\circ"
] | [
"A wineglass is full of water, and an iron ball is dipped into it such that water spills over. The iron ball rests inside with its top level with the rim of the wineglass. What is the maximum angle \\( A \\) (the angle subtended by the wineglass' side with its vertical axis) that maximizes the spill over of water?"... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-a604f9ff82f9abfbcf883e2dc2e7df99806d3c65a27273079591484e3f65a150 | A wire $0.25$ cm thick is cut into $100$ equal parts, with which $100$ hoops of $2$ cm in diameter are made. How long is the wire used to form the chain? | [
"175.5 cm"
] | [
"A wire $0.25$ cm thick is cut into $100$ equal parts, with which $100$ hoops of $2$ cm in diameter are made. How long is the wire used to form the chain?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a36dd2f5ace2c386f5a374621ac6a26bf52911e6c604a50cb583d0c82700b44e | A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped like a circle. Determine the domain of the function \( A \) that represents the total area enclosed by the pieces of wire. | [
"\\( (0, \\frac{10}{3}) \\)"
] | [
"A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped like a circle. Determine the domain of the function \\( A \\) that represents the total area enclosed by the pieces of wire."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-24b172114480065a823ba466dae7cb22afdd4a62161b374be74be50837152eae | A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$? | [
"\\frac{\\sqrt6}2"
] | [
"A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\\frac{a}{b}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65be8337ff9bec49010d9de029f29c92e8024b57741f92170a4810071394fd05 | A wire of constant mass per unit length is bent to follow the path given by \( x = t - \sin t \) and \( y = 1 - \cos t \) for \( 0 \le t \le 2\pi \). Compute the center of mass of this wire. | [
"\\((\\pi, 1)\\)"
] | [
"A wire of constant mass per unit length is bent to follow the path given by \\( x = t - \\sin t \\) and \\( y = 1 - \\cos t \\) for \\( 0 \\le t \\le 2\\pi \\). Compute the center of mass of this wire."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a48cae4325949e65680a89906c2a3d4e9dd7499b94139a8fb9cf238960a25487 | A wire triangle has angles \(x\), \(y\), and \(z\). Each side of the triangle is bent at some point by \(1\) degree, resulting in a non-convex hexagon with angles \(x - 1\), \(181^\circ\), \(y - 1\), \(181^\circ\), \(z - 1\), and \(181^\circ\). If the original triangle has sides of length \(a\), \(b\), and \(c\) opposi... | [
"5"
] | [
"A wire triangle has angles \\(x\\), \\(y\\), and \\(z\\). Each side of the triangle is bent at some point by \\(1\\) degree, resulting in a non-convex hexagon with angles \\(x - 1\\), \\(181^\\circ\\), \\(y - 1\\), \\(181^\\circ\\), \\(z - 1\\), and \\(181^\\circ\\). If the original triangle has sides of length \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-654ca707feeb7aa35188f27ac416d9875c96149a2445818d23bee40e679af290 | A wire with length $L$ cm will be cut into three pieces. One piece will be curved into a circle, another into a square, and the third into an equilateral triangle. How should the wire be cut to minimize the combined area of the three figures? | [
"\\( \\left( \\frac{L \\pi}{\\pi + 4 + 3\\sqrt{3}}, \\frac{4L}{\\pi + 4 + 3\\sqrt{3}}, \\frac{3\\sqrt{3}L}{\\pi + 4 + 3\\sqrt{3}} \\right) \\)"
] | [
"A wire with length $L$ cm will be cut into three pieces. One piece will be curved into a circle, another into a square, and the third into an equilateral triangle. How should the wire be cut to minimize the combined area of the three figures?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 29
} | 37 | 0 | 0 |
omr-ws-d4ad02a37faf55557b06541b72ceba214d843b0fd213a374eb87381e9c1d0ca3 | A wire-cable has a cross-section of diameter $1\frac{3}{4}$ inches. Find the volume in cubic feet of each yard of cable. | [
"\\(\\frac{147\\pi}{64}\\) cubic feet."
] | [
"A wire-cable has a cross-section of diameter $1\\frac{3}{4}$ inches. Find the volume in cubic feet of each yard of cable."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c519547e7a23a1de36b23021a07b5362722163922a6d861d638e4ef6510b1f1 | A wire-cable has a cross-section of diameter $1\frac{3}{4}$ inches. Find the weight in tons of $3\frac{1}{2}$ miles of the cable, whose material weighs $175$ lb. per cubic foot. | [
"27.01"
] | [
"A wire-cable has a cross-section of diameter $1\\frac{3}{4}$ inches. Find the weight in tons of $3\\frac{1}{2}$ miles of the cable, whose material weighs $175$ lb. per cubic foot."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-34baa03d48d7f2b5f1b560c08efe974a0d797f1b994d2185a8f2db77d72b75f4 | A wizard lands on a random point inside a square-shaped arena. The sides of the arena are laced with poisonous chemicals, and in order to survive, the wizard must land closer to the center of the square, which houses a fountain of regeneration, than the sides of the arena. The probability that the wizard survives can b... | [
"11"
] | [
"A wizard lands on a random point inside a square-shaped arena. The sides of the arena are laced with poisonous chemicals, and in order to survive, the wizard must land closer to the center of the square, which houses a fountain of regeneration, than the sides of the arena. The probability that the wizard survives ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-128954435b26845833110fdc25d2c49470d411682c34c40824b44fd7ed4e19d3 | A wizard thinks of a number from $1$ to $n$. You can ask the wizard any number of yes/no questions about the number, but the wizard may answer them in any order. What is the least number of questions that must be asked to determine the number for sure? (Express your answer in terms of $n$.) | [
"\\( n-1 \\)"
] | [
"A wizard thinks of a number from $1$ to $n$. You can ask the wizard any number of yes/no questions about the number, but the wizard may answer them in any order. What is the least number of questions that must be asked to determine the number for sure? (Express your answer in terms of $n$.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 5,
"tir": 1
} | 44 | 0 | 0 |
omr-ws-9e268240a6addb3bc58b5c030798690727acc6c6053460ee154f9b8131f5c211 | A wolf and 50 sheep are placed on a plane. Player 1 moves the wolf up to one meter, then player 2 moves a sheep up to one meter, and they alternate turns. Let $d$ be the minimum initial distance between the wolf and the closest sheep such that the wolf can guarantee to catch a sheep regardless of the sheep's movements.... | [
"4"
] | [
"A wolf and 50 sheep are placed on a plane. Player 1 moves the wolf up to one meter, then player 2 moves a sheep up to one meter, and they alternate turns. Let $d$ be the minimum initial distance between the wolf and the closest sheep such that the wolf can guarantee to catch a sheep regardless of the sheep's movem... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-76b0f76656b74dcc8503f2f1445d692d1ed841040a44125eaa0b98ee9d062ddc | A wolf is chasing a rabbit. The rabbit is \( s2 \) away from its hole, and the wolf is \( s1 \) away from the line connecting the rabbit and the hole, perpendicular to it. The wolf runs at speed \( v1 \) and the rabbit runs at speed \( v2 \). The wolf always runs directly towards the rabbit. Determine the conditions on... | [
"v_1 \\geq v_2 \\sqrt{1 + \\left( \\frac{s_1}{s_2} \\right)^2 }"
] | [
"A wolf is chasing a rabbit. The rabbit is \\( s2 \\) away from its hole, and the wolf is \\( s1 \\) away from the line connecting the rabbit and the hole, perpendicular to it. The wolf runs at speed \\( v1 \\) and the rabbit runs at speed \\( v2 \\). The wolf always runs directly towards the rabbit. Determine the ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-5a8320ea4e4cf6863c04902e63de646276821e30a4e95b524e52e1d7ae85d092 | A wolf sits at one of the vertices of a cube and is not visible to the hunters. $N$ hunters can shoot at any $N$ vertices of the cube in one volley. If they do not hit the wolf, the wolf can move to one of the three adjacent vertices before the next volley. Determine the minimum number of volleys needed to ensure the w... | [
"2, 2, 2, 2, 3"
] | [
"A wolf sits at one of the vertices of a cube and is not visible to the hunters. $N$ hunters can shoot at any $N$ vertices of the cube in one volley. If they do not hit the wolf, the wolf can move to one of the three adjacent vertices before the next volley. Determine the minimum number of volleys needed to ensure ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b4e0a2809e1b4768bc296926a8616692193429dd1903247af770eb62524d712f | A woman and her husband are walking up an escalator. The woman counts 21 steps, while the husband, walking twice as fast, counts 28 steps. How many steps are there on the escalator at any given time? | [
"35"
] | [
"A woman and her husband are walking up an escalator. The woman counts 21 steps, while the husband, walking twice as fast, counts 28 steps. How many steps are there on the escalator at any given time?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0831b33a1de13f2f3ce87a1745b159b8d34de05ea5308451bd98e3a8d52dcbc4 | A woman arrives at an airport between 4:00 and 5:00 and stays for 10 minutes. Her husband arrives at the airport between 4:00 and 5:00 and stays for 20 minutes. What is the probability that they will meet each other? | [
"\\(\\frac{31}{72}\\)"
] | [
"A woman arrives at an airport between 4:00 and 5:00 and stays for 10 minutes. Her husband arrives at the airport between 4:00 and 5:00 and stays for 20 minutes. What is the probability that they will meet each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-38c9237de7e4a0d34820c91c00c120222dfe797ed4ce59e4a7f5a8d57440c599 | A woman borrows $3000 at a 12% interest rate compounded semi-annually for 1.5 years and $5000 at a 12% interest rate compounded monthly for 3 years. What single amount due in 3.5 years at a 16% interest rate compounded quarterly would be required to pay off both obligations at the same time? | [
"6161.29"
] | [
"A woman borrows $3000 at a 12% interest rate compounded semi-annually for 1.5 years and $5000 at a 12% interest rate compounded monthly for 3 years. What single amount due in 3.5 years at a 16% interest rate compounded quarterly would be required to pay off both obligations at the same time?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17cbfa44364d3b6a842c3406ef956f89a1187d69f179b1894499570deb43358f | A woman buys \( x \) flowers for \( y \) dollars. The clerk then offers her 18 more flowers for a total of 6 dollars. If she accepts, she will be paying 60 cents less per dozen. If \( x \) and \( y \) are integers, find the values of \( x \) and \( y \). | [
"\\( x = 30, y = 4 \\)"
] | [
"A woman buys \\( x \\) flowers for \\( y \\) dollars. The clerk then offers her 18 more flowers for a total of 6 dollars. If she accepts, she will be paying 60 cents less per dozen. If \\( x \\) and \\( y \\) are integers, find the values of \\( x \\) and \\( y \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2683916379d9563edb4d7176bb41fe03932fad9bfb4261d6bd3e8f2e2d84a855 | A woman drives a car starting with the odometer reading $abc$ miles where $a \geq 1$ and $a + b + c \leq 7$. She drives for a whole number of hours at 55 miles per hour. At the end, her odometer reads $cba$ miles. Find $a^2 + b^2 + c^2$. | [
"37"
] | [
"A woman drives a car starting with the odometer reading $abc$ miles where $a \\geq 1$ and $a + b + c \\leq 7$. She drives for a whole number of hours at 55 miles per hour. At the end, her odometer reads $cba$ miles. Find $a^2 + b^2 + c^2$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-e9c874020a507deb665a02525cd3b0bff285335159a3f5c512b4e6d62d42635f | A woman gave birth to a quintuplet. The probability of giving birth to a boy is 0.5. What is the probability that the woman will give birth to 2 boys and 3 girls given that one of her children is a boy? | [
"\\(\\frac{10}{31}\\)"
] | [
"A woman gave birth to a quintuplet. The probability of giving birth to a boy is 0.5. What is the probability that the woman will give birth to 2 boys and 3 girls given that one of her children is a boy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-cb37677b163ead34ceb5f8c22a75011eb36ac02c4be838b3ef6eb3e3cbac90df | A woman has $\$1.58$ in pennies, nickels, dimes, quarters, half-dollars, and silver dollars. If she has a different number of coins of each denomination, how many coins does she have? | [
"11"
] | [
"A woman has $\\$1.58$ in pennies, nickels, dimes, quarters, half-dollars, and silver dollars. If she has a different number of coins of each denomination, how many coins does she have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-ffdd1621f17f34df445291509c16f1d8a3c51e1b6fc12ce19045e3f6f6a234e3 | A woman has 10 keys and tries them one by one to open a lock, setting aside any key that does not work. What is the probability that the seventh key she tries opens the lock? | [
"\\(\\frac{1}{10}\\)"
] | [
"A woman has 10 keys and tries them one by one to open a lock, setting aside any key that does not work. What is the probability that the seventh key she tries opens the lock?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-de49653043e95b2182ebe289402a8de5b2569b2f7dabeda515966e1ed1623a7e | A woman has at least $2$ children and more than $\$300$ in the bank. If the product of her age, the number of dollars in her account, and the number of children she has (all integers) is $31310$, how many children does she have, what is her age, and how much money does she have in the bank? | [
"31 children, 2 years old, $505 in the bank."
] | [
"A woman has at least $2$ children and more than $\\$300$ in the bank. If the product of her age, the number of dollars in her account, and the number of children she has (all integers) is $31310$, how many children does she have, what is her age, and how much money does she have in the bank?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bfbac2f9703155fef1bfe4a2a8d6c9fd203c8e98ca72a003ea7c9849606c331c | A woman invested $10,000 in two accounts. One account pays 8% interest and the other pays 15%. How much is invested in each account if the total interest earned is $1,080? | [
"$6000 in the 8% account, $4000 in the 15% account"
] | [
"A woman invested $10,000 in two accounts. One account pays 8% interest and the other pays 15%. How much is invested in each account if the total interest earned is $1,080?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 31
} | 32 | 0 | 0 |
omr-ws-08d06e7c4df1e9a5b0b108dcb7e116adf91f30159a6a155eff48163a778aebf4 | A woman labels the squares of a very large chessboard $1$ through $64$. On each square $k$, the woman puts $2^k$ grains of rice. How many more grains of rice are placed on the $10^{th}$ square than on the first $8$ squares combined? | [
"514"
] | [
"A woman labels the squares of a very large chessboard $1$ through $64$. On each square $k$, the woman puts $2^k$ grains of rice. How many more grains of rice are placed on the $10^{th}$ square than on the first $8$ squares combined?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d35c0f8a947f979c7b2d51e8afda48b3ec8c201a452ab980a31d80940b016574 | A woman mistook the hour hand for the minute hand and the minute hand for the hour hand. Now, the time she reads is 45 minutes advanced than the actual time. What is the actual time? | [
"2:15"
] | [
"A woman mistook the hour hand for the minute hand and the minute hand for the hour hand. Now, the time she reads is 45 minutes advanced than the actual time. What is the actual time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-d7369526b4b6d405e98410afd8649952eecc53e276077d0e07d0fcb1b4741b33 | A woman owes $4500 due in three years and $3000 due in five years. She wants to settle her obligations with a single payment in four years. Find the size of the payment if money is worth a 6% discount rate compounded quarterly. | [
"7602.69"
] | [
"A woman owes $4500 due in three years and $3000 due in five years. She wants to settle her obligations with a single payment in four years. Find the size of the payment if money is worth a 6% discount rate compounded quarterly."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9a8879b22382ef3537492ab8ede99fe92967a73b3dd7d1d522b79f2e37fdcd9 | A woman states that the product of her sons' ages is 36 and the sum of their ages is her age 20 years ago. The census taker needs additional information and the woman mentions that her eldest son knows how to play the violin. Determine the ages of the children. | [
"2, 2, 9"
] | [
"A woman states that the product of her sons' ages is 36 and the sum of their ages is her age 20 years ago. The census taker needs additional information and the woman mentions that her eldest son knows how to play the violin. Determine the ages of the children."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 5
} | 16 | 0 | 0 |
omr-ws-ab58258133b49120fbe2095fce78d2b3aa53999260965d945eb59882afbff6c3 | A woman threw a rubber ball downstream in a river. The dog caught up to the ball in 1.5 minutes and took 2.5 minutes to swim back with the ball. How long would it have taken for the ball to float from the woman to the point where the dog caught up to it? | [
"7.5"
] | [
"A woman threw a rubber ball downstream in a river. The dog caught up to the ball in 1.5 minutes and took 2.5 minutes to swim back with the ball. How long would it have taken for the ball to float from the woman to the point where the dog caught up to it?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3940886cc9a9c3a7fb994ca6d797da3702935a5823b626493d1b1e6940af3929 | A woman, while writing the numbers from 400 to 599, mistakenly wrote 8 every time instead of 4. How many times did she write the digit 8? Choose the correct option: 1. 300 2. 180 3. 230 4. 360 | [
"180"
] | [
"A woman, while writing the numbers from 400 to 599, mistakenly wrote 8 every time instead of 4. How many times did she write the digit 8? \n\nChoose the correct option:\n1. 300\n2. 180\n3. 230\n4. 360"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-340e1b19e148fcb2e5bf96ba14ed215ad194cf270f71071706a0e88829aa141b | A wooden beam $EFGHABCD$ is cut into 8 smaller beams with three cuts, each parallel to one of the three pairs of opposite sides. Each pair of cuts is perpendicular to each other. The smaller beams at the corners $A, C, F,$ and $H$ have capacities of $9, 12, 8,$ and $24$ respectively. Calculate the volume of the entire ... | [
"105"
] | [
"A wooden beam $EFGHABCD$ is cut into 8 smaller beams with three cuts, each parallel to one of the three pairs of opposite sides. Each pair of cuts is perpendicular to each other. The smaller beams at the corners $A, C, F,$ and $H$ have capacities of $9, 12, 8,$ and $24$ respectively. Calculate the volume of the en... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-b25726f1f639f9a7c9c8556a9348a12fe07482a324166c6695a3073c490b5e9d | A wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into sixteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number? [asy] size(4cm,4cm); pair A,B,C,D,E,F,G,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r; A=(0.5,0.1);... | [
"8"
] | [
"A wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into sixteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?\n\n[asy]\n\nsize(4cm,4cm);\n\npair A,B,C,D,E,F,G,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b368dbd6f0240fc63ec39147a580a5ff38aab63d00e74a2829ac470e9e3c34a | A wooden box $80$ cm high and $210$ cm long was placed against a wall. About that box and next to the wall, another box was placed that is $160$ cm high and $90$ cm long. Then a ladder was placed that is supported by the floor, the wall, and the two boxes. Calculate the length of the ladder, in cm. | [
"450"
] | [
"A wooden box $80$ cm high and $210$ cm long was placed against a wall. About that box and next to the wall, another box was placed that is $160$ cm high and $90$ cm long. Then a ladder was placed that is supported by the floor, the wall, and the two boxes. Calculate the length of the ladder, in cm."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 11
} | 14 | 0 | 0 |
omr-ws-0072b6ee03450b9f29844c37aea069dc4c6b99e535dad25d42a1e8d016cb2ffd | A wooden box in the shape of a rectangular parallelepiped has dimensions \( 3 \times 4 \times 6 \). It is on the floor with one of its faces completely resting on the floor. A source of light emits parallel rays of light forming \( 45^\circ \) with the ground. Considering only this source of light, what is the area of ... | [
"24"
] | [
"A wooden box in the shape of a rectangular parallelepiped has dimensions \\( 3 \\times 4 \\times 6 \\). It is on the floor with one of its faces completely resting on the floor. A source of light emits parallel rays of light forming \\( 45^\\circ \\) with the ground. Considering only this source of light, what is ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-9501a22ed4414c2334cf648ef7a00a9690ea3f1fa6869d02e9002b7cc8ddfc3d | A wooden cube \( n \) units on a side is painted red on all six faces and then cut into \( n^3 \) unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is \( n \)? | [
"4"
] | [
"A wooden cube \\( n \\) units on a side is painted red on all six faces and then cut into \\( n^3 \\) unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is \\( n \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-210cf3624834a577d190be1f73e70eaefadb71fa281651b82691fe56ce8054f5 | A wooden cube has edges of length 3 meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. Calculate the entire surface area, including the inside, in square meters. | [
"72"
] | [
"A wooden cube has edges of length 3 meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. Calculate the entire surface area, including the inside, in square meters."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 10
} | 33 | 0 | 0 |
omr-ws-5a4db2ab2ad96849c215a2b6e69e0bef155d80aed5261052c6aa9905a4f1ff4e | A wooden cube with 9 cm edges is completely painted, then broken into 27 equal cubes with 3 cm edges. What is the total area of the unpainted faces of these small cubes? | [
"972"
] | [
"A wooden cube with 9 cm edges is completely painted, then broken into 27 equal cubes with 3 cm edges. What is the total area of the unpainted faces of these small cubes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-fe62e2a0300ba7e2b31b6d94d85bc852f4807fde57ed73b84f709dccefe309dc | A wooden cube with a side length of 4 cm is painted blue on its entire outer surface. After making horizontal and vertical cuts, 64 smaller cubes of 1 cm on each side are obtained. How many of these smaller cubes have 3, 2, 1, and 0 blue faces, respectively? | [
"[hide]There are 8 cubes with 3 painted faces, 24 with 2 and 1, and 8 unpainted cubes[/hide]\n\nTherefore, the answer is:"
] | [
"A wooden cube with a side length of 4 cm is painted blue on its entire outer surface. After making horizontal and vertical cuts, 64 smaller cubes of 1 cm on each side are obtained. How many of these smaller cubes have 3, 2, 1, and 0 blue faces, respectively?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 24
} | 28 | 0 | 0 |
omr-ws-25c3ac751b751fc8a952080a27e7014f5fda5658e61e3604b52a9f1a76396e3e | A wooden cube with edges of 1 centimeter rests on a horizontal surface. A point source of light is $x$ centimeters directly above an upper vertex of the cube, causing the cube to cast a shadow on the horizontal surface. The area of the shadow, excluding the area beneath the cube, is 48 square centimeters. Find the grea... | [
"166"
] | [
"A wooden cube with edges of 1 centimeter rests on a horizontal surface. A point source of light is $x$ centimeters directly above an upper vertex of the cube, causing the cube to cast a shadow on the horizontal surface. The area of the shadow, excluding the area beneath the cube, is 48 square centimeters. Find the... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd04331ca4d4dd5915c2b55ecbdb6533729dcac4c95f891aa4e79df053ec3efe | A wooden cube with edges one centimeter long rests on a horizontal surface. A point source of light is x centimeters directly above an upper vertex of the cube, casting a shadow on the horizontal surface. The area of the shadow, excluding the area beneath the cube, is 48 square centimeters. Find the greatest integer th... | [
"166"
] | [
"A wooden cube with edges one centimeter long rests on a horizontal surface. A point source of light is x centimeters directly above an upper vertex of the cube, casting a shadow on the horizontal surface. The area of the shadow, excluding the area beneath the cube, is 48 square centimeters. Find the greatest integ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7345859996f1375acc40237b72578244484ee3e4b431a975a2ca85e7dd70f0ef | A wooden cube with edges one centimeter long rests on a horizontal surface. A point source of light is x centimeters directly above an upper vertex of the cube, causing it to cast a shadow on the horizontal surface. The area of the shadow, excluding the area beneath the cube, is 48 square centimeters. Find the greatest... | [
"166"
] | [
"A wooden cube with edges one centimeter long rests on a horizontal surface. A point source of light is x centimeters directly above an upper vertex of the cube, causing it to cast a shadow on the horizontal surface. The area of the shadow, excluding the area beneath the cube, is 48 square centimeters. Find the gre... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-7e0960f2e9ecbb0f8f0f8c5abeeb301c2193c8b76172318e1a73427d929724be | A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the area beneath the cube is 48 square centi... | [
"166"
] | [
"A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the area beneath the cube is 48 square ... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 7
} | 38 | 0 | 0 |
omr-ws-c656e2498e6c62942fb3bcb56cade4009578eb0609a595779b152c31fe167ab4 | A wooden cylinder with a radius of 10 inches sits on a table with its flat side on the bottom. A quarter with a diameter of 1 inch rotates around the cylinder. How many rotations does the quarter make to go all the way around the cylinder? | [
"20"
] | [
"A wooden cylinder with a radius of 10 inches sits on a table with its flat side on the bottom. A quarter with a diameter of 1 inch rotates around the cylinder. How many rotations does the quarter make to go all the way around the cylinder?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fb706d5c40fa477be75826f5f2bfc4da650f68e2f6062233a6f2bbffd019437 | A wooden model of a square pyramid has a base edge of 12 cm and an altitude of 8 cm. A cut is made parallel to the base of the pyramid that separates it into two pieces: a smaller pyramid and a frustum. Each base edge of the smaller pyramid is 6 cm and its altitude is 4 cm. How many cubic centimeters are in the volume ... | [
"336"
] | [
"A wooden model of a square pyramid has a base edge of 12 cm and an altitude of 8 cm. A cut is made parallel to the base of the pyramid that separates it into two pieces: a smaller pyramid and a frustum. Each base edge of the smaller pyramid is 6 cm and its altitude is 4 cm. How many cubic centimeters are in the vo... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7fb7f9adfc2d0c603699ce218a132ef54b2b9af0bc1ba2c1ff6261213b9e5c59 | A wooden unit cube is cut along every plane that is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces are obtained? | [
"96"
] | [
"A wooden unit cube is cut along every plane that is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces are obtained?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 1
} | 1 | 0 | 0 |
omr-ws-413fa6fafdc2e0cdc11bfdaafe061d11ad4b6baa300b980e8eda382a42c0cadd | A wooden unit cube rests on a horizontal surface. A point light source a distance \(x\) above an upper vertex casts a shadow of the cube on the surface. The area of the shadow (excluding the part under the cube) is 48. Find \(x\). | [
"\\( \\frac{1}{6} \\)"
] | [
"A wooden unit cube rests on a horizontal surface. A point light source a distance \\(x\\) above an upper vertex casts a shadow of the cube on the surface. The area of the shadow (excluding the part under the cube) is 48. Find \\(x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-4e867f518c1c666a33e3039871b44dd4a1f1ad99e5da97964845846f083661d2 | A woodworker received 20 feet of wood on Sunday. Starting from Monday, the woodworker makes one cut 2 feet into the wood every day, except on weekends (Saturday and Sunday). On what day of the week does the woodworker make the last cut? | [
"Thursday"
] | [
"A woodworker received 20 feet of wood on Sunday. Starting from Monday, the woodworker makes one cut 2 feet into the wood every day, except on weekends (Saturday and Sunday). On what day of the week does the woodworker make the last cut?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a01f3401f685e19800b99cb00f8dc4f5d74dd84c66e36e15fcdf6348406143f2 | A word $W$ consists of $n$ $A$s and $n$ $B$s. Find the largest possible number of palindromic substrings of $W$. | [
"n(n + 1)"
] | [
"A word $W$ consists of $n$ $A$s and $n$ $B$s. Find the largest possible number of palindromic substrings of $W$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a846b4a2dbfdbc77dc2a7fa9c3ca852e4e7313b07f4b641c6392315dfe713eaf | A word is a combination of 8 letters, each either A or B. Let x and y be two words that differ in exactly 3 places. How many words differ from each of x and y in at least 5 places? | [
"38"
] | [
"A word is a combination of 8 letters, each either A or B. Let x and y be two words that differ in exactly 3 places. How many words differ from each of x and y in at least 5 places?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 7
} | 28 | 0 | 0 |
omr-ws-9465cb70f6d1d7bd548efaafdeb8d24e8a38076baa4040597fdec908571e1bb4 | A word is initially written with $n$ different letters. At each step, a new word of $n$ letters is written by reversing the longest initial subword that does not produce a word already written in the previous steps. Let $f(n)$ be the total number of distinct words generated when starting with $n$ different letters. Cal... | [
"120"
] | [
"A word is initially written with $n$ different letters. At each step, a new word of $n$ letters is written by reversing the longest initial subword that does not produce a word already written in the previous steps. Let $f(n)$ be the total number of distinct words generated when starting with $n$ different letters... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7bb80ddac0b4a6ae4a7d2a7175b195eea8842098ded080861e31538468daff44 | A word of 5 letters is formed at random using the letters of the word "COLLEGE". What is the probability that all the repeated letters (i.e., both 'L's) are used in the word? | [
"\\(\\frac{10}{21}\\)"
] | [
"A word of 5 letters is formed at random using the letters of the word \"COLLEGE\". What is the probability that all the repeated letters (i.e., both 'L's) are used in the word?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d4a268588950746be42a9966d3387e303a1aa52ee499e8483696d47b75910e9 | A word of 6 letters is formed from a set of 16 different letters of the English alphabet (with replacement). Find the probability that exactly 2 letters are repeated. | [
"\\dfrac{135825}{1048576}"
] | [
"A word of 6 letters is formed from a set of 16 different letters of the English alphabet (with replacement). Find the probability that exactly 2 letters are repeated."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-d63069074adfeb1e8df72c656351416588fbfd740e897982cbb8c9c53e5d101b | A worker carries two bags, each initially containing $N$ nails. Whenever the worker needs a nail, he takes it from a bag picked at random. At some point, the worker picks an empty bag. Find the probability that the other bag contains exactly $m$ nails. | [
"\\(\\frac{\\binom{2N - m}{N}}{2^{2N - m}}\\)"
] | [
"A worker carries two bags, each initially containing $N$ nails. Whenever the worker needs a nail, he takes it from a bag picked at random. At some point, the worker picks an empty bag. Find the probability that the other bag contains exactly $m$ nails."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-f28f60db1cfea08b50e39a03ce74b25a47c68e018e9a3b7764f5e8666740cac6 | A worker carries two bags, each initially containing N nails. Whenever the worker needs a nail, he takes it from a bag picked at random. At some point, the worker picks an empty bag. Find the probability that the other bag contains exactly m nails. | [
"\\dbinom{2N - m}{N} \\left(\\frac{1}{2}\\right)^{2N - m}"
] | [
"A worker carries two bags, each initially containing N nails. Whenever the worker needs a nail, he takes it from a bag picked at random. At some point, the worker picks an empty bag. Find the probability that the other bag contains exactly m nails."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-0c71ce1ef22e53472a5d6770258167bb30f3a7f1311726393514cbcb6312f72c | A worker made a certain number of identical parts in a specified time. If he had produced 10 parts more every day, he would have completed the job $4\frac{1}{2}$ days ahead of schedule, and if he had produced 5 parts less every day, he would have been 3 days behind time. How many parts did he make and in what time? | [
"The worker made 1350 parts in 27 days"
] | [
"A worker made a certain number of identical parts in a specified time. If he had produced 10 parts more every day, he would have completed the job $4\\frac{1}{2}$ days ahead of schedule, and if he had produced 5 parts less every day, he would have been 3 days behind time. How many parts did he make and in what tim... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-28832212f52c0fe151afd435ba77dc0be47a865e6b8b552a28ef7bbbe2ab8973 | A worker starts at point X on a straight line XY that is 100 meters long. There are 15 heavy stones placed 1 meter apart starting from X. The worker needs to transport all the stones to point Y, carrying only one stone at a time. What is the minimum distance the worker has to travel to accomplish this task? | [
"2790"
] | [
"A worker starts at point X on a straight line XY that is 100 meters long. There are 15 heavy stones placed 1 meter apart starting from X. The worker needs to transport all the stones to point Y, carrying only one stone at a time. What is the minimum distance the worker has to travel to accomplish this task?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fcf454f3dd33a884e42c708a43f7255dcf07f7a6023f79795b5d7766f30d461 | A worm makes a tunnel of length 10 in an apple (sphere) of radius 6. Determine the maximum possible volume of the rotten part of the apple if it is cut into two equal halves such that only one half is considered to be rotten. Express your answer in terms of $\pi$. | [
"144\\pi"
] | [
"A worm makes a tunnel of length 10 in an apple (sphere) of radius 6. Determine the maximum possible volume of the rotten part of the apple if it is cut into two equal halves such that only one half is considered to be rotten. Express your answer in terms of $\\pi$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-eb7ba013df45a3b0fdc5e958a2632d0799f3f6517dc171fb5ac93b3f248881a5 | A worm starts at vertex $A$ of a rectangular box $ABCD.A'B'C'D'$. It moves along the edges of the box to an adjacent vertex at each step. Calculate the probability that after nine moves, the worm is at vertex $C'$. | [
"\\dfrac{1640}{6561}"
] | [
"A worm starts at vertex $A$ of a rectangular box $ABCD.A'B'C'D'$. It moves along the edges of the box to an adjacent vertex at each step. Calculate the probability that after nine moves, the worm is at vertex $C'$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-13b7a24d8a3a716acc3183a9819ae1efe2c2c2ed8f7e07094ea581e84e0a2a22 | A wounded king strolls the battlefield, encountering an \( 8 \times 8 \) chessboard that he must cross. The chessboard is then set on fire, with each square having a 50% chance of burning up and becoming impassable. The king, being wounded, can only move in the four cardinal directions, as well as up-right or bottom-le... | [
"\\(\\frac{1}{2}\\)"
] | [
"A wounded king strolls the battlefield, encountering an \\( 8 \\times 8 \\) chessboard that he must cross. The chessboard is then set on fire, with each square having a 50% chance of burning up and becoming impassable. The king, being wounded, can only move in the four cardinal directions, as well as up-right or b... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-0dfcfaff7dd9ce86e689fd7f20e37f40ca1600389959c65578fc0c8b8d5114ca | A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction $<0,3,-4>$ at the end of the wrench. Find the magnitude of the force needed to supply 100 J of torque to the bolt. | [
"416.67 N"
] | [
"A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction $<0,3,-4>$ at the end of the wrench. Find the magnitude of the force needed to supply 100 J of torque to the bolt."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-55683abb04b6f8ef189748a891880a6bc7ada534b5512b5f6a1bcfe772f4c3ec | A writer receives 35% of the money earned from the sale of his book. The book sells for $7. After 3000 books are sold, how much does the writer receive? | [
"7350"
] | [
"A writer receives 35% of the money earned from the sale of his book. The book sells for $7. After 3000 books are sold, how much does the writer receive?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a23836a12f3a3cbfb78e489d8f44938998b22052911fd32ac4ecf98914761f4b | A writes two real numbers between 1 and 1000 and seals each of them in separate envelopes. B selects one envelope and sees the number inside. B must then predict whether the number he has seen is the smaller or the larger of the two numbers. Devise a strategy for B to win with a probability greater than 0.5. | [
">0.5\n\nThe strategy described in Post 3 ensures that B wins with a probability strictly greater than 0.5. This is achieved by choosing a random number $r$ from a continuous probability distribution supported on the whole interval [1, 1000]. If $r$ falls between the two numbers chosen by A, B will always win. If $... | [
"A writes two real numbers between 1 and 1000 and seals each of them in separate envelopes. B selects one envelope and sees the number inside. B must then predict whether the number he has seen is the smaller or the larger of the two numbers. Devise a strategy for B to win with a probability greater than 0.5."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-bcef6243cb945dc26a5af39ec2b61cfec88cef4d628ea6a67acfcf46121a5df3 | A writes two real numbers between 1 and 1000 and seals each of them in separate envelopes. B selects one envelope, sees the number inside, and must predict whether it is the smaller or larger of the two numbers. Devise a strategy for B to win with a probability greater than 0.5. | [
"B can win with a probability greater than 0.5 using the described strategy."
] | [
"A writes two real numbers between 1 and 1000 and seals each of them in separate envelopes. B selects one envelope, sees the number inside, and must predict whether it is the smaller or larger of the two numbers. Devise a strategy for B to win with a probability greater than 0.5."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-c399aab003ac0c57e5e3d309e36c80aa7d62b1f62428f4ad52648efbe9ef4083 | A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born? | [
"Wednesday"
] | [
"A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8b4ec9c0f40d8001ef8b4f4a895a00063a510aa14e33369c95209087ea8397df | A year is a leap year if and only if the year number is divisible by $400$ or is divisible by $4$ but not by $100$. The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born? | [
"Friday"
] | [
"A year is a leap year if and only if the year number is divisible by $400$ or is divisible by $4$ but not by $100$. The $200\\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 26
} | 46 | 0 | 0 |
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