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-16a573753c4cdcc5a727cc2f91433e27f9e1e13d308d0301e4fd0b5e95240845 | 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 100 (such as 2012). The 200th 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 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th 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_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 17
} | 40 | 0 | 0 |
omr-ws-ea3eb633091680c9611ad2c12393690193e662367f13526a7290d3eb4438d957 | A year is considered lucky if there exists at least one date in the format day/month/year where the product of the month and the day equals the last two digits of the year. Determine how many years from 1901 to 2017 are not lucky. | [
"21"
] | [
"A year is considered lucky if there exists at least one date in the format day/month/year where the product of the month and the day equals the last two digits of the year. Determine how many years from 1901 to 2017 are not lucky."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-42dd2e5ae50082e5a27d8f4da89938ae2d1117f3b6d063e12d9f13fa1a7ed453 | A year is selected at random from the years 2000 to 2017. If the selected year has 53 Sundays, what is the probability that the year is a leap year? | [
"\\(\\frac{1}{2}\\)"
] | [
"A year is selected at random from the years 2000 to 2017. If the selected year has 53 Sundays, what is the probability that the year is a leap year?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-26a906f666e1a94a1b6e76ca2f49eca2f930721f13bcf654910f646378f1d430 | A yellow pig walks on a number line starting at $17$. Each step the pig has a probability of $\frac{8}{17}$ of moving $1$ unit in the positive direction and a probability of $\frac{9}{17}$ of moving $1$ unit in the negative direction. Find the expected number of steps until the yellow pig visits $0$. | [
"289"
] | [
"A yellow pig walks on a number line starting at $17$. Each step the pig has a probability of $\\frac{8}{17}$ of moving $1$ unit in the positive direction and a probability of $\\frac{9}{17}$ of moving $1$ unit in the negative direction. Find the expected number of steps until the yellow pig visits $0$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0a1a58d631a5afaaca4019f14885ed4b1a8ab1e6e940b6503c13907ded07e23f | A yellow pig walks on a number line starting at $17$. Each step the pig has probability $\frac{16}{17}$ of moving $1$ unit in the positive direction and probability $\frac{1}{17}$ of moving $1$ unit in the negative direction. The probability that the pig visits $0$ is $\frac{1}{2^n}$ for some positive integer $n$. Find... | [
"68"
] | [
"A yellow pig walks on a number line starting at $17$. Each step the pig has probability $\\frac{16}{17}$ of moving $1$ unit in the positive direction and probability $\\frac{1}{17}$ of moving $1$ unit in the negative direction. The probability that the pig visits $0$ is $\\frac{1}{2^n}$ for some positive integer $... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-4388d41f779df09ed5c023519a44f6c73ae3c3632ee99ba47b9df0f6e9560da9 | A yo-yo has an outer wheel with a diameter of 7 cm and an inner wheel with a diameter of 1 cm. The inner wheel is wound with 30 cm of string, and there is 40 cm of string left out. If the yo-yo is placed on a table and the string is pulled, how far and in what direction will the yo-yo move before the string is either s... | [
"210\\text{ cm in the direction of the pull}"
] | [
"A yo-yo has an outer wheel with a diameter of 7 cm and an inner wheel with a diameter of 1 cm. The inner wheel is wound with 30 cm of string, and there is 40 cm of string left out. If the yo-yo is placed on a table and the string is pulled, how far and in what direction will the yo-yo move before the string is eit... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-e914841447df95d0e5a33d1759a66e4c45b78f9736327568d2e4a3aee5bc2663 | A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available? | [
"60"
] | [
"A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f31b6ebbca0cc8bb3a9aa8d76709ae462d4136bcc48abf6ae33927125d97def | A young boy collects a million pebbles and repeatedly divides a pile into two smaller piles, noting the product of the sizes of the two new piles. He continues this process until he has a million piles of one pebble each. If he repeats this process many times, what is the mean value of the sum of the noted products? Wh... | [
"499,999,500,000 (mean value), 0 (variance)"
] | [
"A young boy collects a million pebbles and repeatedly divides a pile into two smaller piles, noting the product of the sizes of the two new piles. He continues this process until he has a million piles of one pebble each. If he repeats this process many times, what is the mean value of the sum of the noted product... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-6cff75da0b99cf3d151506a068d7372e95d556de6074e779ecb09d38e12e0260 | A young man wants to cross a 15-mile road in the desert. He can carry at most 15 bottles of water and needs to drink one bottle every mile. How many bottles of water are required for him to reach the end point? | [
"15"
] | [
"A young man wants to cross a 15-mile road in the desert. He can carry at most 15 bottles of water and needs to drink one bottle every mile. How many bottles of water are required for him to reach the end point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-6bef00fb2e00376bb314c17b9a7708569f0103b2d531f10aa40777bfe9e58bd0 | A young person with no initial capital invests \( k \) dollars per year in a retirement account at an annual rate of return of 0.06. Investments are made at the beginning of each year and the return is compounded continuously. Determine a formula for the sum \( S(t) \). | [
"S(t) = \\frac{k e^{0.06} (e^{0.06 t} - 1)}{e^{0.06} - 1}"
] | [
"A young person with no initial capital invests \\( k \\) dollars per year in a retirement account at an annual rate of return of 0.06. Investments are made at the beginning of each year and the return is compounded continuously. Determine a formula for the sum \\( S(t) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4eb0e6d77b15709220d64e0037be87c7567867807659b4ea333560eb5903a69 | A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from the six digits 2, 2, 2, 3, 7, and 9, and the rock's age begins with an odd digit. How many possibilities are there for the rock's age? | [
"60"
] | [
"A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from the six digits 2, 2, 2, 3, 7, and 9, and the rock's age begins with an odd digit.\n\nHow many possibilities are there for the rock's age?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1bd057545c207bea5b01c119da14e279d533d436c1825d366a650bf85382abc | A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find \( z_n \), the maximum number of regions into which \( n \) zig-zags can divide the plane. For example, \( z_1 = 2 \), \( z_2 = 12 \). Of these \( z_n \) regions, how many are bounded? Express your answers as polynomials in \( ... | [
"3n^2 - 3n"
] | [
"A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find \\( z_n \\), the maximum number of regions into which \\( n \\) zig-zags can divide the plane. For example, \\( z_1 = 2 \\), \\( z_2 = 12 \\). Of these \\( z_n \\) regions, how many are bounded? Express your answers as pol... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-4677f1cbf26e83b88c3334d76b4da07ebbef15ccc48427b2b62f181957dc7ff5 | A zip wire from a tree to the ground has an angle of depression of 18°. If the zip wire ends 250 feet from the base of the tree, how far up the tree does the zip wire start? | [
"\\( \\frac{250}{\\tan 72^\\circ} \\) or approximately 81.23 ft."
] | [
"A zip wire from a tree to the ground has an angle of depression of 18°. If the zip wire ends 250 feet from the base of the tree, how far up the tree does the zip wire start?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7dc40d88e11a1db9c577ae2de0280b7432ba0743e46233c0c9415a4224b1cc35 | A zoo has a menagerie containing four pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the male giraffe, how many ways can he feed al... | [
"144"
] | [
"A zoo has a menagerie containing four pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the male giraffe, how many ways can he fe... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f893c6bceb826d5822ca5de2a0d0893e3b3586cf39e31e4a46070c9f59ac23ab | A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch of 10 cm by 5 cm. If 85 cm of new fence is available, what dimensions of the pen will maximize its area? No fence would be used along the walls of the animal house. | [
"25 cm by 25 cm (with adjusted dimensions of 20 cm and 15 cm due to the notch)"
] | [
"A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch of 10 cm by 5 cm. If 85 cm of new fence is available, what dimensions of the pen will maximize its area? No fence would be used along the walls of the animal house."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13075bc6e89a53dc800e79f7b737a03792b727b4f416fa0e464509ceda2483ee | A “No Passing Zone” sign has the shape of an isosceles triangle. The width of the sign is 7 inches greater than its height. The top and bottom edges of the sign are 44.38 inches. What is the height of the sign? | [
"34"
] | [
"A “No Passing Zone” sign has the shape of an isosceles triangle. The width of the sign is 7 inches greater than its height. The top and bottom edges of the sign are 44.38 inches. What is the height of the sign?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7502c51945b8296abb8e28a07faa1e1b3093c39313feaff0a3c4adba3b2f25b7 | A “continuous” roulette wheel has all real numbers from 0 to 1. Repeatedly spin the wheel until the numbers that arise add up to at least 1. What is the expected number of spins? | [
"\\( e \\)"
] | [
"A “continuous” roulette wheel has all real numbers from 0 to 1. Repeatedly spin the wheel until the numbers that arise add up to at least 1. What is the expected number of spins?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-1755f2f64074f73e1680c21157aad76dc86fb33017be7a9a1236d7fef44bac5b | A “word” is a sequence of letters such as $YALE$ and $AELY$. How many distinct $3$-letter words can be made from the letters in $BOOLABOOLA$ where each letter is used no more times than the number of times it appears in $BOOLABOOLA$? | [
"61"
] | [
"A “word” is a sequence of letters such as $YALE$ and $AELY$. How many distinct $3$-letter words can be made from the letters in $BOOLABOOLA$ where each letter is used no more times than the number of times it appears in $BOOLABOOLA$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3fcd9e4cf007af53d38ec10e9a6418c7d847c284fa553d791bbad8d867908e0a | A) Draw the line where \( x = 1 + 2h \) and \( y = -1 + 3h \). B) Find a parameter representation of the line through the points A (0,5) and B (2,3). Use parameter \( p \). C) Calculate the intersection point between the lines. | [
"(3, 2)"
] | [
"A) Draw the line where \\( x = 1 + 2h \\) and \\( y = -1 + 3h \\).\nB) Find a parameter representation of the line through the points A (0,5) and B (2,3). Use parameter \\( p \\).\nC) Calculate the intersection point between the lines."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a0d5228f4f27c9ff673fdb34d516c6a12c95630700fe6aff5e02f3ef5ff5d2c | A, B, C, D are consecutive vertices of a rectangle whose area is 2006. An ellipse with area $2006\pi$ passes through A and C and has foci at B and D. Determine the eccentricity of the ellipse. | [
"\\(\\frac{\\sqrt{3}}{2}\\)"
] | [
"A, B, C, D are consecutive vertices of a rectangle whose area is 2006. An ellipse with area $2006\\pi$ passes through A and C and has foci at B and D. Determine the eccentricity of the ellipse."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-e1c90bbbbe7a485b26db4dc1c036461f62d51809187af0b8f01bafb1e541fede | A, B, C, D are consecutive vertices of a rectangle whose area is 2006. An ellipse with area $2006\pi$ passes through A and C and has foci at B and D. Determine the perimeter of the rectangle. | [
"\\( 8\\sqrt{1003} \\)"
] | [
"A, B, C, D are consecutive vertices of a rectangle whose area is 2006. An ellipse with area $2006\\pi$ passes through A and C and has foci at B and D. Determine the perimeter of the rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-e817cd2e2a6101f5fe7e5d201dfeeb97e2e78380fa3d1bb298733b0a77f7a324 | A, B, C, D, and E are passing balls. Starting from A, the 5th pass returns to B, and there is at least 1 pass to A within these 5 passes. How many different passing methods are there? | [
"124"
] | [
"A, B, C, D, and E are passing balls. Starting from A, the 5th pass returns to B, and there is at least 1 pass to A within these 5 passes. How many different passing methods are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 20
} | 43 | 0 | 0 |
omr-ws-8f7f584e396f6eacc40a6cb6dc0d93ad52e4994e76390d7d9cca97da0e2f6bc4 | A, B, C, and D are four consecutive points on a circle such that $AB = 1$, $BC = 2$, $CD = 3$, and $\angle CDA = 60^{\circ}$. Determine all possible lengths of $DA$. | [
"\\( 1 \\) or \\( 2 \\)"
] | [
"A, B, C, and D are four consecutive points on a circle such that $AB = 1$, $BC = 2$, $CD = 3$, and $\\angle CDA = 60^{\\circ}$. Determine all possible lengths of $DA$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 7
} | 11 | 0 | 0 |
omr-ws-f7c37d6cb3599d291bb6e3724bf1509379c1d80f463256b94a63bf39ad383986 | A, B, C, and D are four distinct positive integers. The sum of any three of them is divisible by the fourth. How many possible values of A, B, C, and D exist? | [
"6"
] | [
"A, B, C, and D are four distinct positive integers. The sum of any three of them is divisible by the fourth. How many possible values of A, B, C, and D exist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-ce9092a52f129e709ec92d386aea6e12c4adc2f23c6774179a43b335de1d8a2d | A, B, and C are (distinct) vertices of an equilateral triangle, and D, E, and F are (distinct) midpoints of the sides AB, BC, and CA, in some order. Each vertex is colored red or blue. The 3 vertices of any equilateral triangle (there are 5 of them) are not allowed to be the same color. If rotations in the plane of ABC... | [
"12"
] | [
"A, B, and C are (distinct) vertices of an equilateral triangle, and D, E, and F are (distinct) midpoints of the sides AB, BC, and CA, in some order. Each vertex is colored red or blue. The 3 vertices of any equilateral triangle (there are 5 of them) are not allowed to be the same color. If rotations in the plane o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63fad2ecf440988e27d2d08ac30f5eb6ccefa274692df913a50d4cb34e36f0ed | A, B, and C are (distinct) vertices of an equilateral triangle, and D, E, and F are (distinct) midpoints of the sides AB, BC, and CA, in some order. Each vertex is colored red or blue. The 3 vertices of any equilateral triangle (there are 5 of them) are not allowed to be the same color. If rotations in the plane of ABC... | [
"12"
] | [
"A, B, and C are (distinct) vertices of an equilateral triangle, and D, E, and F are (distinct) midpoints of the sides AB, BC, and CA, in some order. Each vertex is colored red or blue. The 3 vertices of any equilateral triangle (there are 5 of them) are not allowed to be the same color. If rotations in the plane o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9892555094c58c3beef194744411bed192d815f22b0b07f8b6b328b09fb21de5 | A, B, and C are digits. The 3-digit number ACB is divisible by 3, BAC is divisible by 4, BCA is divisible by 5, and CBA has an odd number of factors. Find the 3-digit number ABC. | [
"522"
] | [
"A, B, and C are digits. The 3-digit number ACB is divisible by 3, BAC is divisible by 4, BCA is divisible by 5, and CBA has an odd number of factors. Find the 3-digit number ABC."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-225a60266dc9661dbaa8af21860218dff53e67b96b308261a2f470c0e2922a0d | A, B, and C are playing a game on an integer axis. The marker starts at the origin. A moves the marker to any integer, then B moves it to any integer that has an absolute difference of at most $k$ with the current integer, and C moves it to any integer that has an absolute difference of at most $k$ with the current int... | [
"\\( k = 1 \\)"
] | [
"A, B, and C are playing a game on an integer axis. The marker starts at the origin. A moves the marker to any integer, then B moves it to any integer that has an absolute difference of at most $k$ with the current integer, and C moves it to any integer that has an absolute difference of at most $k$ with the curren... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-206127fe40078b33312e2db214346d1726d207f63784f1a1acb638c4209a64f0 | A, B, and C are playing a game on an integer axis. The marker starts at the origin. A moves the marker to any integer, then B moves it to any other integer, and C moves it to any integer that has an absolute difference of at most $k$ with the current integer. The marker may not visit an integer that has been previously... | [
"[3, \\infty)"
] | [
"A, B, and C are playing a game on an integer axis. The marker starts at the origin. A moves the marker to any integer, then B moves it to any other integer, and C moves it to any integer that has an absolute difference of at most $k$ with the current integer. The marker may not visit an integer that has been previ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-886f9a273592ef300da0d670f1e1f61d7ad971106fb240c58f4033bb0a4b8795 | A, B, and C are playing a game with a coin by flipping it several times. If A flips heads, then the coin goes to B, if he flips tails, it goes to C. If B flips heads, then the coin goes to C, if he flips tails, it goes to A. If C flips heads, then the coin goes to A, if he flips tails, it goes to B. After each flip, th... | [
"\\dfrac{21}{64}"
] | [
"A, B, and C are playing a game with a coin by flipping it several times. If A flips heads, then the coin goes to B, if he flips tails, it goes to C. If B flips heads, then the coin goes to C, if he flips tails, it goes to A. If C flips heads, then the coin goes to A, if he flips tails, it goes to B. After each fli... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-df53d595e2a4ca1242282da16c7c01839bb21eef96d317658c9c93860c3afe4b | A, B, and C are points on a circle $\Omega$ with radius $1$. Three circles are drawn outside triangle $ABC$ and tangent to $\Omega$ internally. These three circles are also tangent to $BC$, $CA$, and $AB$ at their respective midpoints $D$, $E$, and $F$. If the radii of two of these three circles are $\frac{2}{3}$ and $... | [
"\\(\\frac{1}{33}\\)"
] | [
"A, B, and C are points on a circle $\\Omega$ with radius $1$. Three circles are drawn outside triangle $ABC$ and tangent to $\\Omega$ internally. These three circles are also tangent to $BC$, $CA$, and $AB$ at their respective midpoints $D$, $E$, and $F$. If the radii of two of these three circles are $\\frac{2}{3... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 7
} | 12 | 0 | 0 |
omr-ws-8b0b38164b40906a71149d10b8382c50a829d20d6108d84c768779832514f985 | A, B, and C are seated in a row such that A can see B and C, and B can see C. There are 2 blue caps and 3 red caps available. Caps are placed on their heads, and they do not know the color of their own cap. When asked about the color of their caps, A and B cannot answer, but C answers correctly. Determine the color of ... | [
"A: Blue, B: Red, C: Red"
] | [
"A, B, and C are seated in a row such that A can see B and C, and B can see C. There are 2 blue caps and 3 red caps available. Caps are placed on their heads, and they do not know the color of their own cap. When asked about the color of their caps, A and B cannot answer, but C answers correctly. Determine the colo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-bc4b1b6fe79cf94654127ff927cef76d0717d42bd082608a355509485816e58c | A, B, and C are three towns forming a triangle. A man has to walk from one to the next, ride thence to the next, and drive thence to his starting point. He can walk, ride, and drive a mile in a, b, and c minutes, respectively. If he starts from B, he takes \(a+c-b\) hours; if he starts from C, he takes \(b+a-c\) hours;... | [
"60"
] | [
"A, B, and C are three towns forming a triangle. A man has to walk from one to the next, ride thence to the next, and drive thence to his starting point. He can walk, ride, and drive a mile in a, b, and c minutes, respectively. If he starts from B, he takes \\(a+c-b\\) hours; if he starts from C, he takes \\(b+a-c\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-700031fdc37e3a767dd07b5537b583aaf9d3c93aec73a0053a1514defa6d9afd | A, B, and C are throwing a die. A wins if he throws a 1, 2, or 3; B wins if he throws a 4 or 5; C wins if he throws a 6. The die rotates from A to B to C to A, etc., until one player wins. Calculate the probability that C wins. | [
"\\(\\frac{1}{13}\\)"
] | [
"A, B, and C are throwing a die. A wins if he throws a 1, 2, or 3; B wins if he throws a 4 or 5; C wins if he throws a 6. The die rotates from A to B to C to A, etc., until one player wins. Calculate the probability that C wins."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-7876c77f32965c71fa0350fa1b19598a73e761fb91d96158b75673e398318da0 | A, B, and C can walk at 5 km/hr. They have a car that can accommodate any two of them and travels at 50 km/hr. Can they all reach a point 62 km away in less than 3 hours? | [
"Yes"
] | [
"A, B, and C can walk at 5 km/hr. They have a car that can accommodate any two of them and travels at 50 km/hr. Can they all reach a point 62 km away in less than 3 hours?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b84c958d956e33d1b0037e6b181b022501a015c4ed6e18d77025c0c4b41e2495 | A, B, and C each had a bag of candies. A gave B candies equal to half of what C had. Then B gave C candies equal to half of what A now had. B ended up with 17 candies. A now had a number of candies equal to what C originally had, while C now had a number of candies equal to what A originally had. If there were 57 candi... | [
"16"
] | [
"A, B, and C each had a bag of candies. A gave B candies equal to half of what C had. Then B gave C candies equal to half of what A now had. B ended up with 17 candies. A now had a number of candies equal to what C originally had, while C now had a number of candies equal to what A originally had. If there were 57 ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-17d5c4d56708241d55456950dfed04bca3b71715d5900182b98a35ac787f195f | A, B, and C listen to four different songs. No song is liked by all three. Furthermore, for each of the three pairs of the boys, there is at least one song liked by those two boys but disliked by the third. If the number of ways this can happen is \(2^k + 4\), find \(k\). | [
"7"
] | [
"A, B, and C listen to four different songs. No song is liked by all three. Furthermore, for each of the three pairs of the boys, there is at least one song liked by those two boys but disliked by the third. If the number of ways this can happen is \\(2^k + 4\\), find \\(k\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0e9df92c7e2e471d508619f17616fc0f637fdfc32afdf0083184cf9d2ea9f74 | A, B, and C paint a long line of fence-posts. A paints the first post, then every \(a\)-th post. B paints the second post, then every \(b\)-th post. C paints the third post, then every \(c\)-th post. Every post gets painted exactly once. Find all possible triples \((a, b, c)\). | [
"(3, 3, 3) and (4, 2, 4)"
] | [
"A, B, and C paint a long line of fence-posts. A paints the first post, then every \\(a\\)-th post. B paints the second post, then every \\(b\\)-th post. C paints the third post, then every \\(c\\)-th post. Every post gets painted exactly once. Find all possible triples \\((a, b, c)\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-526dc24a1846ddbb3ddeeb2eca991c8f2644049779f234349f385b964fa958d3 | A, B, and C run around a circular track of length 750 meters at speeds of 3 m/sec, 6 m/sec, and 18 m/sec, respectively. If all three start from the same point, simultaneously, and run in the same direction, when will they meet for the first time after they start the race? | [
"250 seconds"
] | [
"A, B, and C run around a circular track of length 750 meters at speeds of 3 m/sec, 6 m/sec, and 18 m/sec, respectively. If all three start from the same point, simultaneously, and run in the same direction, when will they meet for the first time after they start the race?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-a48101508cd908fef69ab5683f72afc52897275dac6f125238eb2f43781e08e8 | A, B, and C take turns drawing from a bag containing 4 white chips and 2 blue chips, in that order, without replacement. The winner is the first person to draw a blue chip. If no one wins, the chips are returned to the bag. What is the probability that C will win the game? | [
"\\(\\frac{1}{4}\\)"
] | [
"A, B, and C take turns drawing from a bag containing 4 white chips and 2 blue chips, in that order, without replacement. The winner is the first person to draw a blue chip. If no one wins, the chips are returned to the bag. What is the probability that C will win the game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-e67de409bba743ddacbc075b5464f53626dc9201c9357f666cf5600633f0ca64 | A, B, and C want to help each other build houses. A will spend half his time on his own house and a quarter of his time on each of the houses of B and C. B will spend one third of his time on each of the three houses under construction. C will spend one sixth of his time on A's house, one third on B's house, and one ha... | [
"7.5"
] | [
"A, B, and C want to help each other build houses. A will spend half his time on his own house and a quarter of his time on each of the houses of B and C. B will spend one third of his time on each of the three houses under construction. C will spend one sixth of his time on A's house, one third on B's house, and o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea7679a2e12a2bc93274b0baf40eeb4fbb279235b47c69c7275e7171bc750171 | A, B, and C watered 100 pots of flowers at the same time. A watered 78 pots, B watered 68 pots, and C watered 58 pots. What is the minimum number of pots that all three have watered? | [
"4"
] | [
"A, B, and C watered 100 pots of flowers at the same time. A watered 78 pots, B watered 68 pots, and C watered 58 pots. What is the minimum number of pots that all three have watered?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-6c756b216727c5e6b5e81ebf17a569a3d5b83a44489d0e409412463a6d29cce5 | A1A2...An is a regular polygon. An equilateral triangle A1BA2 is constructed outside the polygon. What is the largest n for which BA1An can be consecutive vertices of a regular polygon? | [
"42"
] | [
"A1A2...An is a regular polygon. An equilateral triangle A1BA2 is constructed outside the polygon. What is the largest n for which BA1An can be consecutive vertices of a regular polygon?"
] | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b51745a61cc497453d282e4ba39c4f1f77e9b996dd937ed16c0c92137a8cdaf3 | AB and CD are chords of a circle with radius 4 on opposite sides of the center. The length of arc CD is four times the length of arc AB, and the length of arc CD is \( \frac{8\pi}{5} \). Find the measure of angle <APB, where CB and DA produced meet at P outside the circle. | [
"\\(\\frac{3}{20}\\pi\\) or 27 degrees"
] | [
"AB and CD are chords of a circle with radius 4 on opposite sides of the center. The length of arc CD is four times the length of arc AB, and the length of arc CD is \\( \\frac{8\\pi}{5} \\). Find the measure of angle <APB, where CB and DA produced meet at P outside the circle."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 18
} | 35 | 0 | 0 |
omr-ws-a6870e324a6f3fa08fc608659e16064712c2534ab71ebb7878df464c13c94980 | AB and CD are perpendicular chords of a circle that intersect at E. Given AE=4, EB=5, CE=10, and ED=2, what is the diameter of the circle? | [
"\\(\\sqrt{145}\\)"
] | [
"AB and CD are perpendicular chords of a circle that intersect at E. Given AE=4, EB=5, CE=10, and ED=2, what is the diameter of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-d2bb8ff17136d9b619dee8cf0155c7623701947378cca38b3706591d2ecefcc5 | AB and CD are perpendicular to a diameter of circle O. Let CM be a chord that intersects AB at E, such that CE = 6 and EM = 5. Find the circumference of the circle. | [
"3\\sqrt{14}\\pi"
] | [
"AB and CD are perpendicular to a diameter of circle O. Let CM be a chord that intersects AB at E, such that CE = 6 and EM = 5. Find the circumference of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-1241816bd2d8e4688312cdddcfbc593c3b97628fbb9542448365ac58685c8a33 | AB and CD are two chords on a circle with center O. They intersect perpendicularly at point P. Given $AP = 6$, $PD = 3$, and $PN = \frac{1}{2}$, where OM and ON are perpendicular to AB and CD respectively, find the difference between the areas of triangles APC and BPD. | [
"9"
] | [
"AB and CD are two chords on a circle with center O. They intersect perpendicularly at point P. Given $AP = 6$, $PD = 3$, and $PN = \\frac{1}{2}$, where OM and ON are perpendicular to AB and CD respectively, find the difference between the areas of triangles APC and BPD."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-2b2eafb4db6ea1b88f56faf7aba4a169b9f3b63c90692232cf470cba813c8638 | AB is a chord measuring 6 cm in a circle with center O and radius 8 cm. As the chord AB is rotated around circle O, its trisection point describes a circular path. Find the ratio of the area of the smaller circle to the area of the larger circle. | [
"\\(\\frac{7}{8}\\)"
] | [
"AB is a chord measuring 6 cm in a circle with center O and radius 8 cm. As the chord AB is rotated around circle O, its trisection point describes a circular path. Find the ratio of the area of the smaller circle to the area of the larger circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-64dc2fdf91bddeee91332e5d6c261617aaf165f39b553aa80f99212f9211cf28 | AB is a chord of length 6 of a circle centered at O with radius 5. Let PQRS denote the square inscribed in the sector OAB such that P is on the radius OA, S is on the radius OB, and Q and R are points on the arc of the circle between A and B. Find the area of PQRS. | [
"1"
] | [
"AB is a chord of length 6 of a circle centered at O with radius 5. Let PQRS denote the square inscribed in the sector OAB such that P is on the radius OA, S is on the radius OB, and Q and R are points on the arc of the circle between A and B. Find the area of PQRS."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57ff41c8ca26c79a6bc3341050456b0f0c21bd4d3b466d7ec68043aa46900cba | AB is a diameter of a circle, and AD and BC are tangents to the circle at points A and B, respectively. AC and BD intersect at a point on the circle. If AD = a and BC = b, and a ≠ b, find the length of AB. | [
"\\(\\sqrt{ab}\\)"
] | [
"AB is a diameter of a circle, and AD and BC are tangents to the circle at points A and B, respectively. AC and BD intersect at a point on the circle. If AD = a and BC = b, and a ≠ b, find the length of AB."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-2e46e8d685d3e477b992f9eaee1fe5d7ead0b64ff134b863b4474e2678df9180 | AB is the diameter of a circle S with center O. We construct circle C(0) to be tangent to AB at O and also internally tangent to S. Circle C(1) is constructed to be tangent to AB, internally tangent to S, and externally tangent to C(0). If AB = 4, find the radius of C(1). | [
"\\( \\frac{1}{2} \\)"
] | [
"AB is the diameter of a circle S with center O. We construct circle C(0) to be tangent to AB at O and also internally tangent to S. Circle C(1) is constructed to be tangent to AB, internally tangent to S, and externally tangent to C(0). If AB = 4, find the radius of C(1)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-38102d0d114f4a13aebc08d786d8452af74d6e15dd49c15936c008c56357ac1b | AB is the diameter of a circle. A chord CD is perpendicular to diameter AB at H. The length of AB is a 2-digit number and the number obtained by interchanging the digits is the length of chord CD. Find the length of AB if OH is a rational number, where O is the center of the circle. | [
"65"
] | [
"AB is the diameter of a circle. A chord CD is perpendicular to diameter AB at H. The length of AB is a 2-digit number and the number obtained by interchanging the digits is the length of chord CD. Find the length of AB if OH is a rational number, where O is the center of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-617e7746ca7af3c4df3c36c80a6bdb9adf0917fc00d4bbb0e1acf93231933384 | AB is the diameter of a circle. A chord CD is perpendicular to diameter AB at H. The length of AB is a 2-digit number and the number obtained by interchanging the digits is the length of chord CD. Find the length of CD. | [
"65"
] | [
"AB is the diameter of a circle. A chord CD is perpendicular to diameter AB at H. The length of AB is a 2-digit number and the number obtained by interchanging the digits is the length of chord CD. Find the length of CD."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad749b2eb72988cf058b61251b757796fedd2035ce45d105f0f3fcfac9122847 | AB is the diameter of a circular path. Alex moves from point A and Betsy moves from point B, both at the same time and towards each other. They meet for the first time at point C and for the second time at point D. Given AC = 80m and BD = 60m, what is the circumference of the circle? | [
"280m"
] | [
"AB is the diameter of a circular path. Alex moves from point A and Betsy moves from point B, both at the same time and towards each other. They meet for the first time at point C and for the second time at point D. Given AC = 80m and BD = 60m, what is the circumference of the circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58fffd360df2543f1ac8c469a2fa81a719fa8fcea8374cf30d91332d4caa1b11 | ABCD and AEGF are squares. H is a point on CE such that AH is perpendicular to BF and AH = 4. Determine the length of BF. | [
"8"
] | [
"ABCD and AEGF are squares. H is a point on CE such that AH is perpendicular to BF and AH = 4. Determine the length of BF."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-c350138c738f9970dacd6c55b6d82ea1c08d340d62ce4a3a4f7587f5817bbb12 | ABCD is a convex quadrilateral with diagonals AC and BD intersecting at M. If the areas of triangles AMB and CMD are 4 and 9 respectively, what is the minimum area of quadrilateral ABCD? | [
"25"
] | [
"ABCD is a convex quadrilateral with diagonals AC and BD intersecting at M. If the areas of triangles AMB and CMD are 4 and 9 respectively, what is the minimum area of quadrilateral ABCD?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-59de04a1284eb3200042ddb364136f665f92d4761031f979d560553f4fb80a84 | ABCD is a cyclic quadrilateral with side AD = d, where d is the diameter of the circle. AB = a, BC = a, and CD = b. If a, b, and d are integers where a is not equal to b, find the minimum value of d such that d is not a prime number. | [
"8"
] | [
"ABCD is a cyclic quadrilateral with side AD = d, where d is the diameter of the circle. AB = a, BC = a, and CD = b. If a, b, and d are integers where a is not equal to b, find the minimum value of d such that d is not a prime number."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-7e45f5424e115bbeb86ae5d84ed1e437fbfc9a57e5135a53e7db0e4d323fd37f | ABCD is a cyclic quadrilateral. The tangent at A meets the ray CB at K, and the tangent at B meets the ray DA at M, such that BK = BC and AM = AD. If the lengths of the sides AB and CD are 10 and 14 respectively, find the length of side AD. | [
"2\\sqrt{15}"
] | [
"ABCD is a cyclic quadrilateral. The tangent at A meets the ray CB at K, and the tangent at B meets the ray DA at M, such that BK = BC and AM = AD. If the lengths of the sides AB and CD are 10 and 14 respectively, find the length of side AD."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-c608644a578741785a2a5cc151ff60fcfb12e2617de021f53dbe5ee269c7d2cc | ABCD is a cyclic quadrilateral. The tangent at A meets the ray CB at K, and the tangent at B meets the ray DA at M, such that BK=BC and AM=AD. If the length of side AB is 10 units, find the length of side CD. | [
"10"
] | [
"ABCD is a cyclic quadrilateral. The tangent at A meets the ray CB at K, and the tangent at B meets the ray DA at M, such that BK=BC and AM=AD. If the length of side AB is 10 units, find the length of side CD."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-c21f644cf3d4030f9cd9fa247c4050aae193304c63e54e10495aaa01cbe41bba | ABCD is a cyclic quadrilateral. The tangent at A meets the ray CB at K, and the tangent at B meets the ray DA at M, such that BK=BC and AM=AD. If the lengths of the sides AB and CD are 10 and 14 respectively, find the length of side AD. | [
"2\\sqrt{15}"
] | [
"ABCD is a cyclic quadrilateral. The tangent at A meets the ray CB at K, and the tangent at B meets the ray DA at M, such that BK=BC and AM=AD. If the lengths of the sides AB and CD are 10 and 14 respectively, find the length of side AD."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-960e28fbe634a62acfed7d1a0300f6bf72287fb5a63f84d39fda3b1b00a826e9 | ABCD is a kite with an area of 143. Each vertex is reflected across the opposite vertex to create a new kite. Compute the area of this new kite. | [
"1287"
] | [
"ABCD is a kite with an area of 143. Each vertex is reflected across the opposite vertex to create a new kite. Compute the area of this new kite."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-617d4040ec33c372081b31699f7c262b546948bd736a661e769becb0e6207cdd | ABCD is a rectangle. X is a point on AD, Y is a point on AB, and N is a point outside ABCD such that XYNC is also a rectangle and YN intersects BC at its midpoint M. ∠BYM = 45°. If MN = 5, what is the sum of the areas of ABCD and XYNC? | [
"225"
] | [
"ABCD is a rectangle. X is a point on AD, Y is a point on AB, and N is a point outside ABCD such that XYNC is also a rectangle and YN intersects BC at its midpoint M. ∠BYM = 45°. If MN = 5, what is the sum of the areas of ABCD and XYNC?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-49cc2e8d6fd0309b80387e7864ce8cd7285b2e527125f6e2ad87ace2e9635bba | ABCD is a rectangular sheet of paper that has been folded so that corner B is matched with point B' on edge AD. The crease is EF, where E is on AB and F is on CD. The dimensions AE = 8, BE = 17, and CF = 3 are given. The perimeter of rectangle ABCD is \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive... | [
"293"
] | [
"ABCD is a rectangular sheet of paper that has been folded so that corner B is matched with point B' on edge AD. The crease is EF, where E is on AB and F is on CD. The dimensions AE = 8, BE = 17, and CF = 3 are given. The perimeter of rectangle ABCD is \\(\\frac{m}{n}\\), where \\(m\\) and \\(n\\) are relatively pr... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 16
} | 64 | 0 | 0 |
omr-ws-3eb5f96088aedca9e8ebc4a4406122faa13184d950bdb8261d653b656e8d3ceb | ABCD is a rhombus with side length equal to 28. Circle Γ1 has center A and passes through C. Circle Γ2 has center B and passes through D. Given that Γ1 and Γ2 are tangential at T, what is the area of the rhombus? | [
"588"
] | [
"ABCD is a rhombus with side length equal to 28. Circle Γ1 has center A and passes through C. Circle Γ2 has center B and passes through D. Given that Γ1 and Γ2 are tangential at T, what is the area of the rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-d1c1f02306a75ee2df97c383ae2eeb32b926c019fed4aed8a8059b27a34993a0 | ABCD is a rhombus. The circumradii of triangles ABD and ACD are 12.5 and 25, respectively. Find the area of the rhombus. | [
"400"
] | [
"ABCD is a rhombus. The circumradii of triangles ABD and ACD are 12.5 and 25, respectively. Find the area of the rhombus."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
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"cot",
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"cot": 31,
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} | 38 | 0 | 0 |
omr-ws-8a4ceaaa3668bbfb2cc5374c72f9dcbd8537d6d060bd87333313cb34a4421283 | ABCD is a square drawn inside a circle with center 'O'. The center of the square coincides with 'O' and the diagonal AC is horizontal. If AP and DQ are vertical and AP = 45 cm, PQ = 25 cm, find the radius of the circle, the side of the square, and the area of the shaded region PQAD. | [
"Radius of the circle = 70 cm, Side of the square = \\(70\\sqrt{2} \\, \\text{cm}\\), Area of the shaded region PQAD = \\(1750\\sqrt{2} \\, \\text{cm}^2\\)."
] | [
"ABCD is a square drawn inside a circle with center 'O'. The center of the square coincides with 'O' and the diagonal AC is horizontal. If AP and DQ are vertical and AP = 45 cm, PQ = 25 cm, find the radius of the circle, the side of the square, and the area of the shaded region PQAD."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-747555eb19d20b8556ce3cd3449b35f54b305702016e12e7aaced77bb0ce7b79 | ABCD is a square inscribed in a circle. Point E is on the circle. $\overline{AE}$ intersects $\overline{BC}$ at G, and $\overline{CE}$ intersects $\overline{AB}$ at F. Given that $\overline{CE} = 3$ and $\overline{EF} = 2$, find the length of $\overline{GE}$. | [
"1"
] | [
"ABCD is a square inscribed in a circle. Point E is on the circle. $\\overline{AE}$ intersects $\\overline{BC}$ at G, and $\\overline{CE}$ intersects $\\overline{AB}$ at F. Given that $\\overline{CE} = 3$ and $\\overline{EF} = 2$, find the length of $\\overline{GE}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-91d8e99c0067c8f2e3f91f91f313ed4779fb9e07172b5be83b6233f046e372a8 | ABCD is a square of side length 1, M is the midpoint of CD, suppose AC intersects BM at P, find the radius of the incircle of ABP. | [
"\\(\\frac{1}{\\sqrt{2}+\\frac{\\sqrt{5}}{2}+\\frac{3}{2}}\\)"
] | [
"ABCD is a square of side length 1, M is the midpoint of CD, suppose AC intersects BM at P, find the radius of the incircle of ABP."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 12
} | 31 | 0 | 0 |
omr-ws-2e3b01be82e4be80a10e764cadcf2aab5a44cfa99986e3089ba8d143329afe59 | ABCD is a square with area 1. An equilateral triangle is inscribed in the square such that one vertex is at A, another vertex lies on side BC, and the third vertex lies on side CD. What is the perimeter of this equilateral triangle? | [
"\\( 3\\sqrt{6} - 3\\sqrt{2} \\)"
] | [
"ABCD is a square with area 1. An equilateral triangle is inscribed in the square such that one vertex is at A, another vertex lies on side BC, and the third vertex lies on side CD. What is the perimeter of this equilateral triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-f6cbc57df1e4b6596a5464423aa17e2b5363d1c2fbc8ca81f84c7b222edde64b | ABCD is a square with side length 1 unit. Two circles \( C_1 \) and \( C_2 \) of equal radii are drawn inside the square such that they are touching each other, \( C_1 \) is touching sides AB and BC, and \( C_2 \) is touching sides CD and DA. Find the radius of the circles. | [
"\\(\\frac{2-\\sqrt{2}}{2}\\)"
] | [
"ABCD is a square with side length 1 unit. Two circles \\( C_1 \\) and \\( C_2 \\) of equal radii are drawn inside the square such that they are touching each other, \\( C_1 \\) is touching sides AB and BC, and \\( C_2 \\) is touching sides CD and DA. Find the radius of the circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-eb4d70a7d6134053bb6c28bf6f0c9184ddb1f315fa12d5ddef31855085309a98 | ABCD is a square with side length 1. M is the midpoint of AB, and N is the midpoint of BC. The lines CM and DN intersect at I. Find the area of triangle CIN. | [
"\\(\\frac{1}{12}\\)"
] | [
"ABCD is a square with side length 1. M is the midpoint of AB, and N is the midpoint of BC. The lines CM and DN intersect at I. Find the area of triangle CIN."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7343501523033d7e01bd5b05dd49c982bb2a4d756572f3449baac4840ea95ed1 | ABCD is a square with side length 10 cm. AD and DC are the diameters of two semicircles, and AB is the radius of a quadrant. Find the area of the shaded portion. | [
"\\(\\frac{25}{4}\\bigg(3\\pi-4-6\\sin^{-1}\\bigg(\\frac{3}{5}\\bigg)\\bigg)\\)"
] | [
"ABCD is a square with side length 10 cm. AD and DC are the diameters of two semicircles, and AB is the radius of a quadrant. Find the area of the shaded portion."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fac7a9fbbe792d2f6efed29527d4f5a10d40cd82bb85aaf9fdc2928f40a29b4f | ABCD is a square with side length 10 units. Arcs of circles with centers at A, B, C, and D are drawn inside the square. Find the area of the central region bounded by the four arcs. | [
"\\( 100\\left(1 - \\sqrt{3} + \\frac{\\pi}{3}\\right) \\)"
] | [
"ABCD is a square with side length 10 units. Arcs of circles with centers at A, B, C, and D are drawn inside the square. Find the area of the central region bounded by the four arcs."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 6
} | 16 | 0 | 0 |
omr-ws-592bfb7bff7961e1c55c2b4fcde60e293755c0b7817208816d5ef909aaa00a00 | ABCD is a square with side length 4. A semicircle O is inscribed in ABCD such that its diameter XY is a maximum. Find the length of the diameter XY. | [
"\\( 16 - \\sqrt{128} \\)"
] | [
"ABCD is a square with side length 4. A semicircle O is inscribed in ABCD such that its diameter XY is a maximum. Find the length of the diameter XY."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 10
} | 22 | 0 | 0 |
omr-ws-f171c928ed340c261818dc2da7b5840ce4ee796eeca3622581a50b0f79f48b58 | ABCD is a square with side length \(6\sqrt{2}\). EF is parallel to the square and has length \(12\sqrt{2}\). The faces BCF and ADE are equilateral triangles. What is the volume of the solid ABCDEF? | [
"288"
] | [
"ABCD is a square with side length \\(6\\sqrt{2}\\). EF is parallel to the square and has length \\(12\\sqrt{2}\\). The faces BCF and ADE are equilateral triangles. What is the volume of the solid ABCDEF?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 13
} | 33 | 0 | 0 |
omr-ws-62edecad0817fa7f8a9560723ccbe080113f09c74106af011fe5519eafd4ae5e | ABCD is a square, with points E and F on sides BC and CD respectively such that AE intersects the diagonal BD at G and FG is perpendicular to AE. Point K is on FG such that AK = EF. Find the measure of angle EKF. | [
"\\(135^\\circ\\)"
] | [
"ABCD is a square, with points E and F on sides BC and CD respectively such that AE intersects the diagonal BD at G and FG is perpendicular to AE. Point K is on FG such that AK = EF. Find the measure of angle EKF."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 7
} | 25 | 0 | 0 |
omr-ws-5e87c7ca963e46b95d53fbc5e5810a448d171f6284a3546969efbbea965c6849 | ABCD is a square. A circle with radius AB and center A is drawn. This circle intersects the perpendicular bisector of BC at two points, of which X is the closest to C. Find the measure of angle AXC. | [
"135 degrees"
] | [
"ABCD is a square. A circle with radius AB and center A is drawn. This circle intersects the perpendicular bisector of BC at two points, of which X is the closest to C. Find the measure of angle AXC."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-9d84b57b7e21bf3b153c73b91049dc73495228f0c4cee3b073515796773c0b25 | ABCD is inscribed in a circle with diameter 3. AC is a diameter. AC intersects BD at P, AB = BD, and PC = 3/5. Find the perimeter of ABCD. | [
"6"
] | [
"ABCD is inscribed in a circle with diameter 3. AC is a diameter. AC intersects BD at P, AB = BD, and PC = 3/5. Find the perimeter of ABCD."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-53577413773f579973e581a2813628782c702a7d8665e0579bc7cef2c5e5bee4 | ABCD is the rectangular base of a pyramid whose vertex is P. If PA = 670, PB = 938, and PC = 737, find PD. | [
"335"
] | [
"ABCD is the rectangular base of a pyramid whose vertex is P. If PA = 670, PB = 938, and PC = 737, find PD."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 15
} | 47 | 0 | 0 |
omr-ws-f45d8f0b8be7c9816ec9598f5f89e21be52c76dda32d2ee95178bd244e886087 | ABCDE is a cyclic pentagon symmetric about the diameter through A. The chord CD is twice as far from A as the chord BE. Calculate the value of \( \frac{BC + BD}{BE} \). | [
"\\sqrt{2}"
] | [
"ABCDE is a cyclic pentagon symmetric about the diameter through A. The chord CD is twice as far from A as the chord BE. Calculate the value of \\( \\frac{BC + BD}{BE} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27e6f564d259a38dd61370acbd5da81f80e9c32877e56721e39349855c56cc26 | ABCDE is a cyclic pentagon with BC = CD = DE. The diagonals AC and BE intersect at M. N is the foot of the altitude from M to side AB. We have MA = 25, MD = 113, and MN = 15. The area of triangle ABE can be expressed as \( \frac{m}{n} \) where \( m \) and \( n \) are relatively prime positive integers. Determine the re... | [
"727"
] | [
"ABCDE is a cyclic pentagon with BC = CD = DE. The diagonals AC and BE intersect at M. N is the foot of the altitude from M to side AB. We have MA = 25, MD = 113, and MN = 15. The area of triangle ABE can be expressed as \\( \\frac{m}{n} \\) where \\( m \\) and \\( n \\) are relatively prime positive integers. Dete... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3a43e6e779894b5f21a5193652cf6c9500e83493d685f23c2ae5dbb249dd9b4 | ABCDE is a pentagon with \( AB = BC = CD = \sqrt{2} \), \( \angle ABC = \angle BCD = 120 \) degrees, and \( \angle BAE = \angle CDE = 105 \) degrees. Find the area of triangle \( \triangle BDE \). Your answer in its simplest form can be written as \( \frac{a + \sqrt{b}}{c} \), where \( a, b, c \) are integers and \( b ... | [
"18"
] | [
"ABCDE is a pentagon with \\( AB = BC = CD = \\sqrt{2} \\), \\( \\angle ABC = \\angle BCD = 120 \\) degrees, and \\( \\angle BAE = \\angle CDE = 105 \\) degrees. Find the area of triangle \\( \\triangle BDE \\). Your answer in its simplest form can be written as \\( \\frac{a + \\sqrt{b}}{c} \\), where \\( a, b, c \... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-3e23a9714086e836d19e350909043b2b858df4fe6090b6bb209b5034db86c26f | ABCDEF is a convex hexagon with all its sides equal. Also, \( A + C + E = B + D + F \). If \( A = x^\circ \), \( B = (x + 10)^\circ \), \( C = (x + 20)^\circ \), \( D = (x + 30)^\circ \), \( E = (x + 40)^\circ \), and \( F = (x + 50)^\circ \), find the value of \( x \). | [
"95"
] | [
"ABCDEF is a convex hexagon with all its sides equal. Also, \\( A + C + E = B + D + F \\). If \\( A = x^\\circ \\), \\( B = (x + 10)^\\circ \\), \\( C = (x + 20)^\\circ \\), \\( D = (x + 30)^\\circ \\), \\( E = (x + 40)^\\circ \\), and \\( F = (x + 50)^\\circ \\), find the value of \\( x \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b2720bdeb902f3786d898af670a7b8a9fd4876248271f5276afe21bbb342ee6d | ABCDEF is a convex hexagon with all sides equal and opposite sides parallel. Angle FAB = 120 degrees. A = (0,0) and B = (b,2), and the y-coordinates of the other vertices are 4, 6, 8, 10 in some order. Find the area of the hexagon. | [
"\\( 56 \\sqrt{3} \\)"
] | [
"ABCDEF is a convex hexagon with all sides equal and opposite sides parallel. Angle FAB = 120 degrees. A = (0,0) and B = (b,2), and the y-coordinates of the other vertices are 4, 6, 8, 10 in some order. Find the area of the hexagon."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-abcf9edfc881dc7281aaf7466db6aa7350954c95253f3e49311a36fd36c9b29a | ABCDEF is a convex hexagon with all sides equal and opposite sides parallel. Angle FAB = 120 degrees. The y-coordinates of A and B are 0 and 2, respectively, and the y-coordinates of the other vertices are 4, 6, 8, and 10 in some order. Find the area of the hexagon. | [
"48\\sqrt{3}"
] | [
"ABCDEF is a convex hexagon with all sides equal and opposite sides parallel. Angle FAB = 120 degrees. The y-coordinates of A and B are 0 and 2, respectively, and the y-coordinates of the other vertices are 4, 6, 8, and 10 in some order. Find the area of the hexagon."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-989a50cbfd149ab888e8b6e3d399321d1ee0b44189da4716004893083f323865 | ABCDEF is a regular hexagon with area 1. The intersection of triangle ACE and triangle BDF forms a smaller hexagon. What is the area of the smaller hexagon? | [
"\\(\\frac{1}{3}\\)"
] | [
"ABCDEF is a regular hexagon with area 1. The intersection of triangle ACE and triangle BDF forms a smaller hexagon. What is the area of the smaller hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 23
} | 42 | 0 | 0 |
omr-ws-bd96612d6aa929e43a796581080e7487a04d146e60662c05e9bac2a4f4fb0213 | ABCDEF is a regular hexagon with side length 6. Let \( m \) be the midpoint of \( EF \). What is the area of pentagon \( BCDEm \)? | [
"\\( 36\\sqrt{3} \\)"
] | [
"ABCDEF is a regular hexagon with side length 6. Let \\( m \\) be the midpoint of \\( EF \\). What is the area of pentagon \\( BCDEm \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-84e3fa6ab4d0a99d3f01ae996794c303cf5ae7a927e73dbbc4748bd61595fbae | ABCDEFGH is a regular octagon with side length 8. Diagonals AE and CG meet at X. Point M is the midpoint of AB. A) Find the area of ABCDEFGH. B) Find XM, the apothem of the octagon. C) Find XC. | [
"A) \\( 32[2+\\sqrt{2}] \\), B) \\( 7.2 \\), C) \\( 10.45 \\)"
] | [
"ABCDEFGH is a regular octagon with side length 8. Diagonals AE and CG meet at X. Point M is the midpoint of AB.\nA) Find the area of ABCDEFGH.\nB) Find XM, the apothem of the octagon.\nC) Find XC."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea1d70a0eecd36f8b879083189838edd35159ca1684c063cc6b4f372566ead92 | ABE is a right triangle with right angle E. C is a point on AE. D is a point on AB. CD forms a perpendicular line to AE. Given \( AC = 8 \), \( AD = 10 \), \( CD = 6 \), and \( \angle BCD = 30^\circ \). If \( BD = k(4\sqrt{3} + 3) \), find \( k \). | [
"\\dfrac{10}{13}"
] | [
"ABE is a right triangle with right angle E. C is a point on AE. D is a point on AB. CD forms a perpendicular line to AE. Given \\( AC = 8 \\), \\( AD = 10 \\), \\( CD = 6 \\), and \\( \\angle BCD = 30^\\circ \\). If \\( BD = k(4\\sqrt{3} + 3) \\), find \\( k \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-63221a309762ab797ea1c5cd70db5160f9f3dea40fafe620558772f4f7583bcf | AD is the angle bisector of triangle ABC. Line \( l \) is tangent to the circumcircles of triangles ADB and ADC at points M and N, respectively. Let P, Q, and R be the midpoints of BD, DC, and MN, respectively. If the radius of the circle passing through P, Q, and R is \( r \), find the value of \( r \) given that AB =... | [
"\\dfrac{7}{2}"
] | [
"AD is the angle bisector of triangle ABC. Line \\( l \\) is tangent to the circumcircles of triangles ADB and ADC at points M and N, respectively. Let P, Q, and R be the midpoints of BD, DC, and MN, respectively. If the radius of the circle passing through P, Q, and R is \\( r \\), find the value of \\( r \\) give... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f24744d9f07395072bf7a58b8584c1989404adf1fb926e2166fae49c00faa01b | AJ is standing in the center of an equilateral triangle with vertices labelled $A$, $B$, and $C$. They begin by moving to one of the vertices and recording its label; afterwards, each minute, they move to a different vertex and record its label. Suppose that they record $21$ labels in total, including the initial one. ... | [
"88"
] | [
"AJ is standing in the center of an equilateral triangle with vertices labelled $A$, $B$, and $C$. They begin by moving to one of the vertices and recording its label; afterwards, each minute, they move to a different vertex and record its label. Suppose that they record $21$ labels in total, including the initial ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-7fa6effb50822a8fcddeb58dd3c96552ff695457fca8a8d5aecb9b0797c7e8c5 | Aarav has infinitely many blocks that are $1$, $2$, $3$, and $4$ units high. Each block has a width and length of $1$. Aarav randomly stacks these blocks into a $1\times1$ tower. How many whole, individual blocks are there on average in the bottom $4\times1\times1$ section? | [
"\\(\\frac{369}{256}\\)"
] | [
"Aarav has infinitely many blocks that are $1$, $2$, $3$, and $4$ units high. Each block has a width and length of $1$. Aarav randomly stacks these blocks into a $1\\times1$ tower. How many whole, individual blocks are there on average in the bottom $4\\times1\\times1$ section?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-4d29c1a574c2808eb91b5914581ebde752a8a16f5d4d02385ce5603704caa89c | Aaron and Bob are playing a game of ping pong where they play up to $m$ points, and the game ends when one player reaches $m$ points. After each point, each player's score is divided by the greatest common factor of their two scores. Let $f(m,n)$ denote the probability that after a total of $n$ points (or less if the g... | [
"1 - 2^{1 - n} \\sum_{k=0}^{m-1} \\binom{n-1}{k}"
] | [
"Aaron and Bob are playing a game of ping pong where they play up to $m$ points, and the game ends when one player reaches $m$ points. After each point, each player's score is divided by the greatest common factor of their two scores. Let $f(m,n)$ denote the probability that after a total of $n$ points (or less if ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2537fba4175751b98ebe6d87942384ff3a628478f5100926054af072dbb2461d | Aaron has two bags of marbles, each containing 2 blue marbles, 2 red marbles, and 3 green marbles. He opens one bag and puts all the marbles into a single pile. He then removes marbles without replacement until only two marbles are left. He then opens the second bag and adds these marbles to the pile. He continues remo... | [
"\\dfrac{4}{63}"
] | [
"Aaron has two bags of marbles, each containing 2 blue marbles, 2 red marbles, and 3 green marbles. He opens one bag and puts all the marbles into a single pile. He then removes marbles without replacement until only two marbles are left. He then opens the second bag and adds these marbles to the pile. He continues... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-adfa6c401421e0033f247b9258f70d3db994e1d5f35b89e77cc5010621993248 | Aaron has two counters, one starting at 15 and the other at 10. Every minute, he can press one of them randomly, reducing its number by 1. What is the expected value of the number of minutes until one of the counters reaches 0? | [
"12"
] | [
"Aaron has two counters, one starting at 15 and the other at 10. Every minute, he can press one of them randomly, reducing its number by 1. What is the expected value of the number of minutes until one of the counters reaches 0?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d4bd9fc2a74102a8d0697ea91c8d5d68cc81a1e7b186a532644715943332c43 | Aaron is driving to work in a $5 \times 5 \times 5$ cubic car with vertices at $(5,5,5)$, $(5,0,5)$, $(5,5,0)$, $(5,0,0)$, $(0,5,0)$, $(0,0,5)$, $(0,5,5)$, and $(0,0,0)$. Aaron is initially at $(4,3,4)$ and the spider is at $(0,0,0)$. Every minute, the spider moves $1$ unit in any of the six possible directions (up, do... | [
"0"
] | [
"Aaron is driving to work in a $5 \\times 5 \\times 5$ cubic car with vertices at $(5,5,5)$, $(5,0,5)$, $(5,5,0)$, $(5,0,0)$, $(0,5,0)$, $(0,0,5)$, $(0,5,5)$, and $(0,0,0)$. Aaron is initially at $(4,3,4)$ and the spider is at $(0,0,0)$. Every minute, the spider moves $1$ unit in any of the six possible directions ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-4c89a19594629915024c5572b777eb21ea90c6423ebfb70cfc8828a2b4081f85 | Aaron is driving to work in a $5 \times 5 \times 5$ cubic car with vertices at $(5,5,5)$, $(5,0,5)$, $(5,5,0)$, $(5,0,0)$, $(0,5,0)$, $(0,0,5)$, $(0,5,5)$, and $(0,0,0)$. Aaron is initially at $(4,3,4)$, and a spider is at $(0,0,0)$. Every minute, the spider moves $1$ unit in any of the six possible directions (up, dow... | [
"0"
] | [
"Aaron is driving to work in a $5 \\times 5 \\times 5$ cubic car with vertices at $(5,5,5)$, $(5,0,5)$, $(5,5,0)$, $(5,0,0)$, $(0,5,0)$, $(0,0,5)$, $(0,5,5)$, and $(0,0,0)$. Aaron is initially at $(4,3,4)$, and a spider is at $(0,0,0)$. Every minute, the spider moves $1$ unit in any of the six possible directions (... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-62b458d2b70a5bb9cbc0958fdd2e5e6403252d4455bdeb2f42ab34bcf913ab77 | Aaron randomly selects a pair of real numbers $(x,y)$ satisfying $\sin\sin\sin x=\sin \sin \sin y$ with $-30\pi\le x,y\le 30\pi$. What is the probability that $x=y$? | [
"\\(\\frac{1}{60}\\)"
] | [
"Aaron randomly selects a pair of real numbers $(x,y)$ satisfying $\\sin\\sin\\sin x=\\sin \\sin \\sin y$ with $-30\\pi\\le x,y\\le 30\\pi$. What is the probability that $x=y$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 5
} | 23 | 0 | 0 |
omr-ws-5c0ec22e84ce4cf8f9882dceebd990c3fd761eb29ac23178b1cbb5bc19b52d11 | Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+... | [
"(13, -22)"
] | [
"Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\\circ$ left and walk one unit to an unvisited point... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 61,
"genselect": 16,
"tir": 2
} | 79 | 0 | 0 |
omr-ws-c04fe74d56c43aa3e2fcde9d19a28e279721ce9f0404602dc550c6cf23f98ab3 | Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin \( p_0 = (0,0) \) facing east and walks one unit, arriving at \( p_1 = (1,0) \). For \( n = 1, 2, 3, \ldots \), right after arriving at the point \( p_n \), if Aaron can turn \( 90^\circ \) left and walk one unit to an... | [
"(13, -22)"
] | [
"Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin \\( p_0 = (0,0) \\) facing east and walks one unit, arriving at \\( p_1 = (1,0) \\). For \\( n = 1, 2, 3, \\ldots \\), right after arriving at the point \\( p_n \\), if Aaron can turn \\( 90^\\circ \\) left and wa... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
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