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-16dca982e8333cd31909e57f2a21495e6a5b2e45747213408ed5e509d38183e1 | Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car? | [
"Aaron"
] | [
"Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 2
} | 65 | 0 | 0 |
omr-ws-b675b506bf6cc0020c81fd66445d82ce95252d645965d346bd42b23ab8ccd5a9 | Aasma devises an algorithm to find a husband among 1000 candidates by rejecting the first $k$ men and then selecting the next highest-ranked candidate. (a) What is the probability that the highest ranking prospect among all 1000 prospects is the $(m + 1)^{th}$ prospect? (b) Assume the highest ranking prospect is the $(... | [
"368"
] | [
"Aasma devises an algorithm to find a husband among 1000 candidates by rejecting the first $k$ men and then selecting the next highest-ranked candidate.\n\n(a) What is the probability that the highest ranking prospect among all 1000 prospects is the $(m + 1)^{th}$ prospect?\n\n(b) Assume the highest ranking prospec... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-538ff414f8a081f0831aaa10eff9154e21613f5a7943ad16927c8bd26d34b3ba | Aasma devises an algorithm to find a husband among 1000 candidates. She rejects the first $k$ men and then selects the next candidate with a rank higher than the highest rank of the first $k$ men. (a) What is the probability that the highest ranking prospect among all 1000 prospects is the $(m + 1)^{th}$ prospect? (b) ... | [
"368"
] | [
"Aasma devises an algorithm to find a husband among 1000 candidates. She rejects the first $k$ men and then selects the next candidate with a rank higher than the highest rank of the first $k$ men. \n\n(a) What is the probability that the highest ranking prospect among all 1000 prospects is the $(m + 1)^{th}$ prosp... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-56f5e9ceb0e92f6ac9461a5b93b2be32028a3bc4dcbaff75526edfc6b398e9ef | Abby and Brian play a game where they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins. Determine ... | [
"Abby has the winning strategy."
] | [
"Abby and Brian play a game where they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins. Deter... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-e0a8d639f3b3acaaf8a53e30c72992c6bb3c6ec223cacc4e45b7c4b5f8906780 | Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n+1$ or $2n$, provided that the number is not greater than $N$. The play... | [
"5"
] | [
"Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n+1$ or $2n$, provided that the number is not greater than $N$. The... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b832b3120d2b4fbc18a1a673f11bcca2279819e1c3727a51bbf48c50c7a48ce3 | Abby has 600 coins: 200 nickels (5 cents each), 200 dimes (10 cents each), and 200 quarters (25 cents each). She wants to buy as many bags of chips as possible with the coins she has, where each bag costs 45 cents and the vending machine only accepts exact change. How many bags of chips can she buy? | [
"150"
] | [
"Abby has 600 coins: 200 nickels (5 cents each), 200 dimes (10 cents each), and 200 quarters (25 cents each). She wants to buy as many bags of chips as possible with the coins she has, where each bag costs 45 cents and the vending machine only accepts exact change. How many bags of chips can she buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5b1e3f692f0b727307e546ecb73f168a37e59c9bba431256cb73bc3c3a44ea9e | Abby has a paper semicircle of radius 20. She folds 6 units of its diameter perfectly onto itself across a point, through which she draws a line. She marks the intersections of the line with the circumference, then unfolds and cuts out the triangle formed by the three points. What is the maximum possible area she can c... | [
"102"
] | [
"Abby has a paper semicircle of radius 20. She folds 6 units of its diameter perfectly onto itself across a point, through which she draws a line. She marks the intersections of the line with the circumference, then unfolds and cuts out the triangle formed by the three points. What is the maximum possible area she ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ecdbe30636eec8ae6d92485f34b1b5f973513f64376223e64b5422f94432a0ba | Abby, Bart, Cindy and Damon weigh themselves in pairs. Together Abby and Bart weigh 260 pounds, Bart and Cindy weigh 245 pounds, and Cindy and Damon weigh 270 pounds. How many pounds do Abby and Damon weigh together? | [
"285"
] | [
"Abby, Bart, Cindy and Damon weigh themselves in pairs. Together Abby and Bart weigh 260 pounds, Bart and Cindy weigh 245 pounds, and Cindy and Damon weigh 270 pounds. How many pounds do Abby and Damon weigh together?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc451f72dbf1625fe938ee04298a025c66e56043632b606bd20b05ef60807d95 | Abby, Bernardo, Carl, and Debra each start with four coins. In each of four rounds, four balls (one green, one red, and two white) are placed in an urn. Each player draws a ball at random without replacement. The player who draws the green ball gives one coin to the player who draws the red ball. What is the probabilit... | [
"\\(\\frac{5}{192}\\)"
] | [
"Abby, Bernardo, Carl, and Debra each start with four coins. In each of four rounds, four balls (one green, one red, and two white) are placed in an urn. Each player draws a ball at random without replacement. The player who draws the green ball gives one coin to the player who draws the red ball. What is the proba... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 7,
"tir": 2
} | 27 | 0 | 0 |
omr-ws-fc5841686748c7cefcf887164658102efb705c7c8fa24506d7a0657a947e4957 | Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoeve... | [
"\\dfrac{1}{96}"
] | [
"Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to w... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-120228a1ba3597cb6f5d7673e260108def51dfbc61592cb8ec9eea52c9d7c909 | Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture. If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column? | [
"\\(\\frac{7}{15}\\)"
] | [
"Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture. If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 96,
"tir": 60
} | 156 | 0 | 0 |
omr-ws-d89841ba9d3b48a83a14c8ac4cc89aea2f5c35d6471b34f5303f92eb59500b46 | Abdul and Chiang are standing 48 feet apart in a field. Bharat is standing as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures \(60^\circ\). What is the square of the distance (in feet) between Abdul and Bharat? | [
"3072"
] | [
"Abdul and Chiang are standing 48 feet apart in a field. Bharat is standing as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures \\(60^\\circ\\). What is the square of the distance (in feet) between Abdul and Bharat?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-288027cb1ddef45784c6bf8753e57c3e238c2e22825689af7a739cf7b609fbca | Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together ... | [
"334"
] | [
"Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work toge... | [
"MATH_training_set",
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 45
} | 93 | 0 | 0 |
omr-ws-29447ae5264b6bd364791112fc9ff899dc5988843c678bb03c3e7fe5421ab472 | Abe flips a coin 3 times. Brenda rolls a regular die one time. Carl rolls a special 4-sided weighted die one time, where the probability for each number is proportional to the value of the number. Dane flips a weighted coin 3 times, with a 1/3 chance of landing heads and a 2/3 chance of landing tails. The value of Abe'... | [
"1"
] | [
"Abe flips a coin 3 times. Brenda rolls a regular die one time. Carl rolls a special 4-sided weighted die one time, where the probability for each number is proportional to the value of the number. Dane flips a weighted coin 3 times, with a 1/3 chance of landing heads and a 2/3 chance of landing tails. The value of... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5961cfb0d157956c1c59c71dd55fb4daed67aa087cdb37e27a31a739c69e250e | Abe holds 1 green and 1 red jelly bean, while Bea holds 1 green, 1 yellow, and 2 red jelly beans. Each randomly picks a jelly bean to show the other. What is the probability that the colors match? | [
"\\(\\frac{3}{8}\\)"
] | [
"Abe holds 1 green and 1 red jelly bean, while Bea holds 1 green, 1 yellow, and 2 red jelly beans. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-41c14fafddeb3a65e91a2e22c02cb57b71108a957b8fd27c4c1034b0c33078c4 | Abe, Bobby, Charles, Devin and Edwin are the participants in a race. How many different 1st-2nd-3rd place outcomes are possible if there are no ties? Two different outcomes to include are Bobby-Devin-Edwin and Devin-Bobby-Edwin. | [
"60"
] | [
"Abe, Bobby, Charles, Devin and Edwin are the participants in a race. How many different 1st-2nd-3rd place outcomes are possible if there are no ties? Two different outcomes to include are Bobby-Devin-Edwin and Devin-Bobby-Edwin."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ee4c8f077577495c16d8c8a4f1586efa07ec4e8f49db30cc9ba69c6b0e86ec2 | Abhik has 3 accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. If the initial amounts in the three accounts are 100, 200, and 300 dollars respectively, what is the minimum number of transfers... | [
"2"
] | [
"Abhik has 3 accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. If the initial amounts in the three accounts are 100, 200, and 300 dollars respectively, what is the minimum number of tran... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-540b738190a6892b76c19fd0115cd26a81785fbf1e6e51cbf2894a81fc1560ac | Abhinav's marks in comprehension and grammar sum up to a certain number and their product is another number. Bharani knows the sum and Chandran knows the product. Initially, Bharani says that neither he nor Chandran knows the individual marks. However, after Chandran says he now knows the marks, Bharani also figures ou... | [
"1 \\text{ and } 4"
] | [
"Abhinav's marks in comprehension and grammar sum up to a certain number and their product is another number. Bharani knows the sum and Chandran knows the product. Initially, Bharani says that neither he nor Chandran knows the individual marks. However, after Chandran says he now knows the marks, Bharani also figur... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-c7d5812b9ec7a4a35073d4c28741d77c81046ed94da0858ed20e579cd4d92bec | Abhishek is choosing positive integer factors of \(2021 \times 2^{2021}\) with replacement. He writes down a factor each minute. He stops the first time the product of all numbers written down is a perfect square. Find the expected number of minutes it takes for him to stop. | [
"8"
] | [
"Abhishek is choosing positive integer factors of \\(2021 \\times 2^{2021}\\) with replacement. He writes down a factor each minute. He stops the first time the product of all numbers written down is a perfect square. Find the expected number of minutes it takes for him to stop."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-49d0c1aebdf239e1c7030653fd5ad83ec056bbadc339e067ff35310137e53ea7 | Abigail is thinking of an integer \( A \) between \( 1 \) and \( 99 \), inclusive. She tells Bruno the first digit after the decimal point of the number \( A/11 \) (where this digit is \( 0 \) if \( A/11 \) is an integer). Then she tells Chris the first digit after the decimal point of the number \( A/99 \). Bruno says... | [
"\\( N = 3 \\) and the possibilities for \\( A \\) are \\( 10, 14, 15 \\)."
] | [
"Abigail is thinking of an integer \\( A \\) between \\( 1 \\) and \\( 99 \\), inclusive. She tells Bruno the first digit after the decimal point of the number \\( A/11 \\) (where this digit is \\( 0 \\) if \\( A/11 \\) is an integer). Then she tells Chris the first digit after the decimal point of the number \\( A... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-812a6d898d38accbc7ed709c3763ae9ecba77eb456ff2762ebe803691cee5b45 | Abigail, Beatrice, and Carson combine their eggs to sell them at the market. If Abigail has 37 eggs, Beatrice has 49 eggs, and Carson has 14 eggs, and if eggs can only be sold in cartons of 12, how many eggs will be left over if all cartons are sold? | [
"4"
] | [
"Abigail, Beatrice, and Carson combine their eggs to sell them at the market. If Abigail has 37 eggs, Beatrice has 49 eggs, and Carson has 14 eggs, and if eggs can only be sold in cartons of 12, how many eggs will be left over if all cartons are sold?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b87976a6f4dfa84e255074869352dfe7da1e4fe7d0ffd4d7e888010c98603d73 | About a circle a trapezoid is circumscribed, whose nonparallel sides form acute angles $\alpha $ and $\beta $ with the larger of the parallel sides. Determine the radius of the circle, if the area of the trapezoid is equal to $Q$. | [
"$R=\\frac{1}{2}\\sqrt {\\frac{Q\\sin \\alpha \\sin \\beta }{\\sin\\frac{\\alpha +\\beta }{2} \\cos\\frac{\\alpha -\\beta }{2}}}$"
] | [
"About a circle a trapezoid is circumscribed, whose nonparallel sides form acute angles $\\alpha $ and $\\beta $ with the larger of the parallel sides. Determine the radius of the circle, if the area of the trapezoid is equal to $Q$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b7c156ef2a73fb0c7ca53b3fb90a9b179fd9588c9bd0c305c5f2ce3ea8dff9c1 | About a circle of radius $r$ a right-angled trapezoid is circumscribed, whose least side is equal to $3r/2$. Find the area of the trapezoid. | [
"$S=\\frac{9r^2}{2}$"
] | [
"About a circle of radius $r$ a right-angled trapezoid is circumscribed, whose least side is equal to $3r/2$. Find the area of the trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88476e3d67e427fec1160c36702ef63169028692439392d4895113d89cbb2c94 | About a given circle circumscribe a triangle with given angle and given side opposite this angle. Find the solvability condition for the problem. | [
"\\(\\frac{2r}{a} \\ge \\frac{1-\\sin \\frac{\\alpha }{2}}{\\cos\\frac{\\alpha }{2}}\\)"
] | [
"About a given circle circumscribe a triangle with given angle and given side opposite this angle. Find the solvability condition for the problem."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e75a0fb683ade8fabbb9925315ac6c9f55411cf10683a045bddbcef6898fd04 | About a given rectangle circumscribe a new one with given area $m^2$. For what $m$ is the problem solvable? | [
"\\(\\sqrt{ab} \\le m \\le \\frac{a+b}{\\sqrt{2}}\\)"
] | [
"About a given rectangle circumscribe a new one with given area $m^2$. For what $m$ is the problem solvable?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e5fe18928e2c6d2f78dfad5e9ee2ee6c67b80a591141a547230ba68e9f91619 | About a sphere of radius $r$ a regular $n$-gonal pyramid with dihedral angle at the base $\alpha $ is circumscribed. Find the ratio of the volume of the sphere to that of the pyramid. | [
"\\(\\frac{4\\pi }{n}\\tan^3 \\frac{\\alpha }{2} \\cot\\alpha \\cot\\frac{\\pi }{n}\\)"
] | [
"About a sphere of radius $r$ a regular $n$-gonal pyramid with dihedral angle at the base $\\alpha $ is circumscribed. Find the ratio of the volume of the sphere to that of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8cda7ac94903a1cfabc8aea573148a7a8dcdb2882f0478ea3b2b26717d0a7a6 | Above the plane ceiling of a hal1 having the form of a square with side $a$ a roof is made which is constructed in the following way: each pair of adjacent vertices of the square forming the ceiling is joined by straight lines with the midpoint of the opposite side and on each of the four triangles thus obtained a pyra... | [
"\\frac{a^2h}{2}"
] | [
"Above the plane ceiling of a hal1 having the form of a square with side $a$ a roof is made which is constructed in the following way: each pair of adjacent vertices of the square forming the ceiling is joined by straight lines with the midpoint of the opposite side and on each of the four triangles thus obtained a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3d86b71c222100fae318adc4b6625c09e764f57972c079401345b86a17ef393 | Above the planet, which has the shape of a ball, $37$ satellites fly. At any given moment, let $N$ be the maximum number of satellites visible from any point on the planet's surface. Find the minimum possible value of $N$. | [
"19"
] | [
"Above the planet, which has the shape of a ball, $37$ satellites fly. At any given moment, let $N$ be the maximum number of satellites visible from any point on the planet's surface. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cee4a85e514e164ef8597eed6542d31eef6e30a168392545405b08bf116e5aad | Abra the magician is randomly creating letters $A, B, C, D, R$ with each letter having equal probability. Each second he generates a new letter. The expected number of seconds that passes before he first sees the string $ABRACADABRA$ can be written in the form $p^a + p^b + p^c$, where $p$ is a prime and $a, b, c$ are d... | [
"21"
] | [
"Abra the magician is randomly creating letters $A, B, C, D, R$ with each letter having equal probability. Each second he generates a new letter. The expected number of seconds that passes before he first sees the string $ABRACADABRA$ can be written in the form $p^a + p^b + p^c$, where $p$ is a prime and $a, b, c$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-7c9a807200aeaab8ac5347b3f0d981e3668c12796ee611a09fe3949caa87d13c | Abraham Van Helsing and Jonathan Harker play a game against Count Dracula with the following rules: - Count Dracula places 1000 lockets into 1000 coffins, each engraved with a unique number from 1 to 1000. - Van Helsing can switch the lockets in any two coffins. - Van Helsing leaves, and Jonathan Harker enters. - Dracu... | [
"1"
] | [
"Abraham Van Helsing and Jonathan Harker play a game against Count Dracula with the following rules:\n- Count Dracula places 1000 lockets into 1000 coffins, each engraved with a unique number from 1 to 1000.\n- Van Helsing can switch the lockets in any two coffins.\n- Van Helsing leaves, and Jonathan Harker enters.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-a21356f182ea28eac3c985cc8a0152e3fc155946868d202148dc858a585d0c1b | Abran, Alisa, Ava, and Alejandro are walking home in the middle of the night with one lantern. The bridge they need to cross can support at most two people at a time. Alisa can cross in 1 minute, Abran in 2 minutes, Ava in 5 minutes, and Alejandro in 10 minutes. The bridge will collapse after 18 minutes. Can all four f... | [
"17"
] | [
"Abran, Alisa, Ava, and Alejandro are walking home in the middle of the night with one lantern. The bridge they need to cross can support at most two people at a time. Alisa can cross in 1 minute, Abran in 2 minutes, Ava in 5 minutes, and Alejandro in 10 minutes. The bridge will collapse after 18 minutes. Can all f... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-ff85d5daaf3e79e2f917a467f882038fe877c4d1fd8f12beb4809d8977de2b90 | According to Engel's book, Problem Solving Strategies, $(a,p)=(2,341)$ is a counterexample to the converse of Fermat's Little Theorem. However, Lehmer's theorem states otherwise. Clarify the understanding of this discrepancy. | [
"\\((2, 341)\\) is a counterexample to the converse of Fermat's Little Theorem."
] | [
"According to Engel's book, Problem Solving Strategies, $(a,p)=(2,341)$ is a counterexample to the converse of Fermat's Little Theorem. However, Lehmer's theorem states otherwise. Clarify the understanding of this discrepancy."
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a911855b5e510ac1f660e9c6e1f2a06728cc2a456708311dccf5118673fe5852 | According to a law, for every 10 sheep a shepherd must plow and sow one acre of land, and for every 4 sheep, one acre of pasture is allocated. A wealthy shepherd has 7000 acres of land. How many sheep can he keep according to this statute? | [
"20000"
] | [
"According to a law, for every 10 sheep a shepherd must plow and sow one acre of land, and for every 4 sheep, one acre of pasture is allocated. A wealthy shepherd has 7000 acres of land. How many sheep can he keep according to this statute?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-1329ad61322eae721541471744fc1e591f2af6512777768d1783007d3a9f303a | According to a survey, at least 70% of people like apples, at least 75% like bananas, and at least 80% like cherries. What can you say about the percentage of people who like all three fruits? | [
"25%"
] | [
"According to a survey, at least 70% of people like apples, at least 75% like bananas, and at least 80% like cherries. What can you say about the percentage of people who like all three fruits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-ec18203bec81da7b9ef389a8999351dd9aa1d53e401b0b2b547445969ab74c4e | According to an ancient belief, when a friend visits a sick person, \(\frac{1}{70}\) of the person's sickness is taken away. What is the least number of friends that need to visit to take away at least 86% of a person's illness? | [
"137"
] | [
"According to an ancient belief, when a friend visits a sick person, \\(\\frac{1}{70}\\) of the person's sickness is taken away. What is the least number of friends that need to visit to take away at least 86% of a person's illness?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b83db4b6d504bc9fa0b32c72375fb16cec5e86d6df8198136315f6db4336a84c | According to an industry report, the mean monthly cell phone bill is $55.90 with a standard deviation of $9.60. A random sample of 400 households was selected, and the mean bill for the 88 households with cell phones was $59.00. What is the probability of getting a mean bill of $57.00 or more if the industry report's n... | [
"0.1401"
] | [
"According to an industry report, the mean monthly cell phone bill is $55.90 with a standard deviation of $9.60. A random sample of 400 households was selected, and the mean bill for the 88 households with cell phones was $59.00. What is the probability of getting a mean bill of $57.00 or more if the industry repor... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-54513155a740a4ca8b3d74f04a5e003489f7177eeb587877a837e0f30f043105 | According to company records, in 1980 the company earned 3.2 million in earnings and in 1985 earned 4.7 million. Assume a linear model for earnings in millions of dollars, \( E \), versus time in years, \( t \). Let 1980 be year \( t = 0 \). a. Write an equation for \( E \) as a function of \( t \). b. Estimate the com... | [
"E = 0.3t + 3.2 (for part a), 9.2 million (for part b), 0.3 (for part c), and the slope conveys that the earnings are increasing over time (for part d)."
] | [
"According to company records, in 1980 the company earned 3.2 million in earnings and in 1985 earned 4.7 million. Assume a linear model for earnings in millions of dollars, \\( E \\), versus time in years, \\( t \\). Let 1980 be year \\( t = 0 \\).\n\na. Write an equation for \\( E \\) as a function of \\( t \\).\n... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-d89e34451bff1299dae1bec55f9a98a8acb057d51358d4fbb5c52726dcd097de | According to schedule, a team of lumbermen had to stock 216 m$^3$ firewood in several days. For the first three days the team worked according to the schedule, and then each day they chopped 8 m$^3$ of firewood more than was planned. Therefore, a day before the assigned date, they had a stock of 232 m$^3$ of wood. How ... | [
"24 m$^3$ a day"
] | [
"According to schedule, a team of lumbermen had to stock 216 m$^3$ firewood in several days. For the first three days the team worked according to the schedule, and then each day they chopped 8 m$^3$ of firewood more than was planned. Therefore, a day before the assigned date, they had a stock of 232 m$^3$ of wood.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c8645511ed57ff280cde7a2dc588cf17b79eb03b24e07e8aca00731b74d17d2 | According to the Constitution of the Kingdom of Nepal, the shape of the flag is constructed as follows: Draw a line $AB$ of the required length from left to right. From $A$ draw a line $AC$ perpendicular to $AB$ making $AC$ equal to $AB$ plus one third $AB$. From $AC$ mark off $D$ making line $AD$ equal to line $AB$. J... | [
"\\dfrac{3\\sqrt{2} + 5}{12}"
] | [
"According to the Constitution of the Kingdom of Nepal, the shape of the flag is constructed as follows: Draw a line $AB$ of the required length from left to right. From $A$ draw a line $AC$ perpendicular to $AB$ making $AC$ equal to $AB$ plus one third $AB$. From $AC$ mark off $D$ making line $AD$ equal to line $A... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-33263218be416637c433e6c9a5a9d492645693a25e3d19a89a80bf8930f6ceed | According to the Remainder and Factor Theorems, what can you conclude about \( x = 2 \) if \( P(x) \) is divided by \( (x - 2) \) and the remainder is 10? | [
"\\( P(2) = 10 \\) and \\( x - 2 \\) is not a factor of \\( P(x) \\)."
] | [
"According to the Remainder and Factor Theorems, what can you conclude about \\( x = 2 \\) if \\( P(x) \\) is divided by \\( (x - 2) \\) and the remainder is 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 7,
"tir": 28
} | 39 | 0 | 0 |
omr-ws-2217bc0b0d98c737871ac4b245d0cf9656b91460a5dc2f4fe4c6fe685218d10d | According to the bijection constructed between $\mathbb{N} \times \mathbb{N} \times \mathbb{N}$ and $\mathbb{N}$, which triple will be in location $45$? | [
"(1, 3, 1)"
] | [
"According to the bijection constructed between $\\mathbb{N} \\times \\mathbb{N} \\times \\mathbb{N}$ and $\\mathbb{N}$, which triple will be in location $45$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-4b9e01bc2e2169d96fe046b9da29475d7ac5e41b53dd4e2c11bf43854e18503b | According to the data shown, how much cheaper, in cents, is the cheaper store's price for Camera $X$ compared to the more expensive? \begin{tabular}{|l|l|} \hline \textbf{Store}& \textbf{Sale Price for Camera $X$} \\ \hline Super Savers & $\$9$~off the list price~$\$39.96$ \\ \hline Penny Wise & $25\%$~off the list pri... | [
"99"
] | [
"According to the data shown, how much cheaper, in cents, is the cheaper store's price for Camera $X$ compared to the more expensive? \\begin{tabular}{|l|l|}\n\\hline\n\\textbf{Store}& \\textbf{Sale Price for Camera $X$} \\\\ \\hline\nSuper Savers & $\\$9$~off the list price~$\\$39.96$ \\\\ \\hline\nPenny Wise & $2... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-df403d7fc5f149c63fc03f42c64d99ae8fa8529e00355fbbdaa99094ead6db9e | According to the fundamental theorem of algebra, a polynomial of degree \( n \) has \( n \) roots. For the polynomial \( f(x) = x^n \), what are the roots and how do they align with the theorem? | [
"The roots of \\( f(x) = x^n \\) are \\( 0, 0, 0, \\ldots, 0 \\) (n times)."
] | [
"According to the fundamental theorem of algebra, a polynomial of degree \\( n \\) has \\( n \\) roots. For the polynomial \\( f(x) = x^n \\), what are the roots and how do they align with the theorem?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 29
} | 47 | 0 | 0 |
omr-ws-30fd8f36dfce65784c68d0a25edd48e9fca0cd7721ee74af1834338b752fbccc | According to the nutritional information on a package of General Mill's Honey Nut Cheerios® brand cereal, each 1-ounce serving of Cheerios contains 3 grams of protein and 24 grams of carbohydrates, and each half-cup serving of enriched skim milk contains 4 grams of protein and 6 grams of carbohydrates. (a) Plan a meal ... | [
"(a) 2 servings of Cheerios, 5 servings of milk\n(b) \\( x = \\frac{\\frac{2B}{3} - A}{13} \\), \\( y = \\frac{8A - B}{26} \\)"
] | [
"According to the nutritional information on a package of General Mill's Honey Nut Cheerios® brand cereal, each 1-ounce serving of Cheerios contains 3 grams of protein and 24 grams of carbohydrates, and each half-cup serving of enriched skim milk contains 4 grams of protein and 6 grams of carbohydrates.\n(a) Plan a... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-700504da5ea6498c6693f9d60dd75936ef4e031a8f2937214c5d6899430f496b | According to the nutritional information on a package of General Mill's Honey Nut Cheerios® brand cereal, each 1-ounce serving of Cheerios contains 3 grams of protein and 24 grams of carbohydrates, and each half-cup serving of enriched skim milk contains 4 grams of protein and 6 grams of carbohydrates. (a) Plan a meal ... | [
"(a) 2 servings of Cheerios and 5 servings of milk.\n(b) \\( x = \\frac{2B - 3A}{39} \\) servings of Cheerios and \\( y = \\frac{5B + 12A}{39} \\) servings of milk."
] | [
"According to the nutritional information on a package of General Mill's Honey Nut Cheerios® brand cereal, each 1-ounce serving of Cheerios contains 3 grams of protein and 24 grams of carbohydrates, and each half-cup serving of enriched skim milk contains 4 grams of protein and 6 grams of carbohydrates.\n(a) Plan a... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-86ceb76962051e271a6258bcd01da1e491dcd1c430f4d38712426fb676839dac | According to the pattern below, what is the next number in the sequence? \[2, 6, 15, 28, 55, 78, \ldots\] | [
"119"
] | [
"According to the pattern below, what is the next number in the sequence?\n\\[2, 6, 15, 28, 55, 78, \\ldots\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-b496476c952763dfc754b4b9a4c22e0a85f70b427acbd324b032ff19dccce4de | According to the schedule, the train always covers a certain 120 km span at the same speed. Yesterday the train covered half the span with that speed and had to stop for 5 min. To reach the point of destination on time, the engine-driver had to increase the speed by 10 km/h on the second half of the span. Today, the tr... | [
"100 km/h"
] | [
"According to the schedule, the train always covers a certain 120 km span at the same speed. Yesterday the train covered half the span with that speed and had to stop for 5 min. To reach the point of destination on time, the engine-driver had to increase the speed by 10 km/h on the second half of the span. Today, t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c0263100292e14866b9d581b5d2e1c14a3094a59e6ccb67788e963af02c24d9 | According to the standard convention for exponentiation, $2^{2^{2^{2}}} = 2^{(2^{(2^{2})})} = 2^{16} = 65,536$. If the order in which the exponentiations are performed is changed, how many other values are possible? | [
"1"
] | [
"According to the standard convention for exponentiation, $2^{2^{2^{2}}} = 2^{(2^{(2^{2})})} = 2^{16} = 65,536$. If the order in which the exponentiations are performed is changed, how many other values are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c00bc8b5b8d45540d518dcd6ad7205eac66fd5e56853a2500efe91ec3b5150a | According to the standard convention for exponentiation, \( 3^{3^{3^3}} = 3^{\left(3^{\left(3^3\right)}\right)} \). If the order in which the exponentiations are performed is changed, how many other values are possible? | [
"5"
] | [
"According to the standard convention for exponentiation, \\( 3^{3^{3^3}} = 3^{\\left(3^{\\left(3^3\\right)}\\right)} \\). If the order in which the exponentiations are performed is changed, how many other values are possible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c0f69a9e459f51019f2c4b196edcfdc7d732201f82a20766fca5191f996ec9c6 | According to the standard convention for exponentiation, \(2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65,536\). If the order in which the exponentiations are performed is changed, how many other values are possible? | [
"1"
] | [
"According to the standard convention for exponentiation, \\(2^{2^{2^2}} = 2^{\\left(2^{\\left(2^2\\right)}\\right)} = 2^{16} = 65,536\\). If the order in which the exponentiations are performed is changed, how many other values are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b096aa87e690de67f06f737e10dc04f072b37155be964e1b6c92c6d4fae50c15 | According to the timetable, a train is to cover the distance of 20 km between $A$ and $B$ at a constant speed. The train covered half the distance at that speed and then stopped for 3 minutes; in order to arrive at $B$ on schedule it had to increase the speed by 10 km/h on the remaining half of the trip. Another time t... | [
"60 km/h"
] | [
"According to the timetable, a train is to cover the distance of 20 km between $A$ and $B$ at a constant speed. The train covered half the distance at that speed and then stopped for 3 minutes; in order to arrive at $B$ on schedule it had to increase the speed by 10 km/h on the remaining half of the trip. Another t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72b68577380e4f987e97e2f34ebe70034b57676d6729c4c51d9a323ff0ad3695 | According to their plan, two plants had to turn out 360 machines during a month. The first plant fulfilled its plan by 112\%, and the second by 110\%, having produced collectively 400 machines during this month. How many machines were manufactured in excess of the plan by each plant? | [
"24 and 16"
] | [
"According to their plan, two plants had to turn out 360 machines during a month. The first plant fulfilled its plan by 112\\%, and the second by 110\\%, having produced collectively 400 machines during this month. How many machines were manufactured in excess of the plan by each plant?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63e95762c4929958b095dce30159cc22587ebed762dc1a3542aa91b85bc1c287 | Ace, Bea, and Cee are each given a positive integer. They are told that the sum of the three numbers is 15. Ace announces that he can deduce that the other two have different numbers, while Bea independently announces that she can deduce that no two of the three numbers are the same. Hearing both announcements, Cee ann... | [
"(2, 8, 5)"
] | [
"Ace, Bea, and Cee are each given a positive integer. They are told that the sum of the three numbers is 15. Ace announces that he can deduce that the other two have different numbers, while Bea independently announces that she can deduce that no two of the three numbers are the same. Hearing both announcements, Ce... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 5,
"tir": 21
} | 52 | 0 | 0 |
omr-ws-d9e62ec74606c3b0a74dc70304f41e6b81fdb9a790261e482f4ed77f716a8483 | Ackermann and Brahmagupta play a game with a fair two-sided coin. The coin is flipped until either the sequence $HTHHT$ or the sequence $TTHT$ occurs. Ackermann wins if the sequence $HTHHT$ occurs before $TTHT$, and Brahmagupta wins if $TTHT$ occurs before $HTHHT$. The probability that Ackermann wins the game can be ex... | [
"31"
] | [
"Ackermann and Brahmagupta play a game with a fair two-sided coin. The coin is flipped until either the sequence $HTHHT$ or the sequence $TTHT$ occurs. Ackermann wins if the sequence $HTHHT$ occurs before $TTHT$, and Brahmagupta wins if $TTHT$ occurs before $HTHHT$. The probability that Ackermann wins the game can ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 7,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-9cfb49726c1c1b8a9d02ee4995c7a83048d3fc48f03eb59445ea294646cf5b59 | Acme Corporation has released an alphabet soup in which each of the vowels (A, E, I, O, U) of the English alphabet appears five times (and the consonants do not appear at all). How many five-letter words can be formed from a bowl of Acme Vowel Soup? (Note: The words do not have to be actual words in English!) | [
"3125"
] | [
"Acme Corporation has released an alphabet soup in which each of the vowels (A, E, I, O, U) of the English alphabet appears five times (and the consonants do not appear at all). How many five-letter words can be formed from a bowl of Acme Vowel Soup? (Note: The words do not have to be actual words in English!)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5112c4faed71b7b13e11b135f929d2a4497ca572bc60d9a379642e9c6f1a3082 | Acme T-Shirt Company charges a $\$50$ set-up fee plus $\$9$ for each shirt printed. Beta T-shirt Company has no set up fee, but charges $\$14$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme? | [
"11"
] | [
"Acme T-Shirt Company charges a $\\$50$ set-up fee plus $\\$9$ for each shirt printed. Beta T-shirt Company has no set up fee, but charges $\\$14$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d6b5cb4ceccdc0270876067e83e9770eae79b70c26393b06b893cde58ad845f | Acorns nuts is putting out two new 1 lb nut mixes. Nutrageous will be made up of 30% almonds, 20% cashews, and 50% peanuts. Nutolicious will be made up of 20% almonds, 40% cashews, and 40% peanuts. Currently in supply, they have 100 lbs of peanuts, 60 lbs of almonds, and 80 lbs of cashews. Maximize profit if each bag o... | [
"1015.5"
] | [
"Acorns nuts is putting out two new 1 lb nut mixes. Nutrageous will be made up of 30% almonds, 20% cashews, and 50% peanuts. Nutolicious will be made up of 20% almonds, 40% cashews, and 40% peanuts. Currently in supply, they have 100 lbs of peanuts, 60 lbs of almonds, and 80 lbs of cashews. Maximize profit if each ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 8
} | 11 | 0 | 0 |
omr-ws-d8e8b5b52e21f31176dcb6c7c4702f5e2e906e1db467328f37514cf94d712cb4 | Across any point K of the square ABCD, a straight line is drawn intersecting the sides AB and CD at points P and Q. Let L be the point, different from K, which lies on both the circle passing through K, B, P and the circle passing through K, D, Q. If the side length of the square ABCD is 10 units, calculate the length ... | [
"10"
] | [
"Across any point K of the square ABCD, a straight line is drawn intersecting the sides AB and CD at points P and Q. Let L be the point, different from K, which lies on both the circle passing through K, B, P and the circle passing through K, D, Q. If the side length of the square ABCD is 10 units, calculate the le... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-96d9c86e328e0357189e4e73456d9cdbc1ad2524fb2c8efde13137380af5e609 | Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Points $B_1, B_2, B_3$ are chosen on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3$. If the perim... | [
"12"
] | [
"Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Points $B_1, B_2, B_3$ are chosen on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3$. If the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0543fb7a74b5b2f4adf962eab733fd19aae20e075405b009f0ebf8789b9399d0 | Acute triangle \( ABC \) is inscribed in circle \( \omega \). Let \( H \) and \( O \) denote its orthocenter and circumcenter, respectively. Let \( M \) and \( N \) be the midpoints of sides \( AB \) and \( AC \), respectively. Rays \( MH \) and \( NH \) meet \( \omega \) at \( P \) and \( Q \), respectively. Lines \( ... | [
"10"
] | [
"Acute triangle \\( ABC \\) is inscribed in circle \\( \\omega \\). Let \\( H \\) and \\( O \\) denote its orthocenter and circumcenter, respectively. Let \\( M \\) and \\( N \\) be the midpoints of sides \\( AB \\) and \\( AC \\), respectively. Rays \\( MH \\) and \\( NH \\) meet \\( \\omega \\) at \\( P \\) and \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4fb09e64df2f6e8c754efbac67af3f193ca41622591b5c1108fbdaa1d4ab193a | Acute triangle \(ABC\) is inscribed in circle \(\omega\). The tangent lines to \(\omega\) at \(B\) and \(C\) meet at \(T\), and \(D\) is the intersection of \(AT\) with \(\omega\). Point \(S\) lies on ray \(BC\) such that \(AS \perp AT\). Points \(B_1\) and \(C_1\) lie on ray \(ST\) (with \(C_1\) in between \(B_1\) and... | [
"2\\sqrt{145}"
] | [
"Acute triangle \\(ABC\\) is inscribed in circle \\(\\omega\\). The tangent lines to \\(\\omega\\) at \\(B\\) and \\(C\\) meet at \\(T\\), and \\(D\\) is the intersection of \\(AT\\) with \\(\\omega\\). Point \\(S\\) lies on ray \\(BC\\) such that \\(AS \\perp AT\\). Points \\(B_1\\) and \\(C_1\\) lie on ray \\(ST\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9213425eb6d7abf0e61bb68100189c74a3dd316aad23b1076b05a4f246f484d2 | Acute, scalene $\triangle{ABC}$ is given with circumcircle $\Omega$ and orthocenter $H$; $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{PH}$ and $\overline{QH}$ are both tangent to the circumcircle of $\triangle{BHC}$. $X_1$ and $X_2$ lie on $\Omega$ such that the circumcircl... | [
"1"
] | [
"Acute, scalene $\\triangle{ABC}$ is given with circumcircle $\\Omega$ and orthocenter $H$; $P$ and $Q$ lie on $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $\\overline{PH}$ and $\\overline{QH}$ are both tangent to the circumcircle of $\\triangle{BHC}$. $X_1$ and $X_2$ lie on $\\Omega$ such that th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a958e6499dbd71258b63bab73752cc40bf1976ed45a27ad53d55876469a3f994 | Acute-angled triangle \(ABC\) is inscribed in circle \(\Omega\). Lines tangent to \(\Omega\) at \(B\) and \(C\) intersect at \(P\). Points \(D\) and \(E\) are on \(AB\) and \(AC\) such that \(PD\) and \(PE\) are perpendicular to \(AB\) and \(AC\) respectively. Let \(M\) be the midpoint of \(BC\). If the coordinates of ... | [
"(3, \\sqrt{3})"
] | [
"Acute-angled triangle \\(ABC\\) is inscribed in circle \\(\\Omega\\). Lines tangent to \\(\\Omega\\) at \\(B\\) and \\(C\\) intersect at \\(P\\). Points \\(D\\) and \\(E\\) are on \\(AB\\) and \\(AC\\) such that \\(PD\\) and \\(PE\\) are perpendicular to \\(AB\\) and \\(AC\\) respectively. Let \\(M\\) be the midpo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-790d6d6759e4395e64bf23240038a692f0ae6940a549d46f2d0f992759f70178 | Ada and Byron play a game. First, Ada chooses a non-zero real number $a$ and announces it. Then Byron chooses a non-zero real number $b$ and announces it. Then Ada chooses a non-zero real number $c$ and announces it. Finally, Byron chooses a quadratic polynomial whose three coefficients are $a, b, c$ in some order. (a)... | [
"Byron has a winning strategy for part (a) and Ada has a winning strategy for part (b)."
] | [
"Ada and Byron play a game. First, Ada chooses a non-zero real number $a$ and announces it. Then Byron chooses a non-zero real number $b$ and announces it. Then Ada chooses a non-zero real number $c$ and announces it. Finally, Byron chooses a quadratic polynomial whose three coefficients are $a, b, c$ in some order... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-614acd3a178bd2e5483bb382aec17d1831e5594567abce4a0e2f4eeda24f37c6 | Ada and Charles play a game starting with an integer \( n > 1 \) on the blackboard. On each turn, a player can either replace the number \( k \) with a positive divisor of \( k \) different from 1 and \( k \) (gaining one point) or replace \( k \) with \( k+1 \) (losing one point). Each player starts with 1000 points, ... | [
"\\( n \\in \\{ \\text{primes except } 2, 7, 13 \\} \\cup \\{ 8, 14, 26, 49, 91, 169 \\} \\)"
] | [
"Ada and Charles play a game starting with an integer \\( n > 1 \\) on the blackboard. On each turn, a player can either replace the number \\( k \\) with a positive divisor of \\( k \\) different from 1 and \\( k \\) (gaining one point) or replace \\( k \\) with \\( k+1 \\) (losing one point). Each player starts w... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a4a019d1c22590fd7d63d07df00ea046a9bfb11442790f2bbbd4767b418880bd | Ada and Otto are playing a game on a figure with six dots (A, B, C, D, E, F) and nine sticks connecting pairs of dots. Ada starts by placing a pebble on any dot. Players then take turns moving the pebble along an adjacent stick, removing that stick from the figure. The game ends when there are no sticks adjacent to the... | [
"Ada"
] | [
"Ada and Otto are playing a game on a figure with six dots (A, B, C, D, E, F) and nine sticks connecting pairs of dots. Ada starts by placing a pebble on any dot. Players then take turns moving the pebble along an adjacent stick, removing that stick from the figure. The game ends when there are no sticks adjacent t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-3fc91a28b437eed201a629f566c55e906ee5b0959e01470bc8033c60819463a6 | Ada drew a triangle, chose a point on each side, and chose a point \( P \) on the inside of the triangle. She then drew segments that join \( P \) with the three vertices and the three points on the sides. Show, by example, how Ada could have achieved dividing the initial triangle into six isosceles triangles. | [
"\\text{Ada used an equilateral triangle with centroid and midpoints, connecting them to form six isosceles triangles.}"
] | [
"Ada drew a triangle, chose a point on each side, and chose a point \\( P \\) on the inside of the triangle. She then drew segments that join \\( P \\) with the three vertices and the three points on the sides. Show, by example, how Ada could have achieved dividing the initial triangle into six isosceles triangles.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-dc43d55bc0be37136730649ef0c503cc96e6309e5349b107c98fdf345d9e1e61 | Ada the ant starts at a point O on a plane. At the start of each minute, she chooses North, South, East, or West, and marches 1 meter in that direction. After 2018 minutes, she returns to point O. Let \( n \) be the number of possible journeys she could have made. What is the highest power of 10 that divides \( n \)? | [
"4"
] | [
"Ada the ant starts at a point O on a plane. At the start of each minute, she chooses North, South, East, or West, and marches 1 meter in that direction. After 2018 minutes, she returns to point O. Let \\( n \\) be the number of possible journeys she could have made. What is the highest power of 10 that divides \\(... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-d25a9e8bd6ab15ea6ce85ef37135ae85d3e2d192063fd2bfda6f19b3fc26fefd | Ada walks along a number line with the integers from -2 to 3, starting from 0. How many 12-step paths can she take so that she ends on 0? | [
"638"
] | [
"Ada walks along a number line with the integers from -2 to 3, starting from 0. How many 12-step paths can she take so that she ends on 0?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bada6aa72ea502c23394cb718c7363d5513d5b39be2db94414429362dbde0a3a | Adam and Bardon take turns writing a natural number with distinct 6 digits, with each player adding one digit at a time to the right of the last digit written by the other. Bardon wins if the resulting 6-digit number is prime; otherwise, Adam wins. Determine which player has a winning strategy. | [
"Adam has a winning strategy."
] | [
"Adam and Bardon take turns writing a natural number with distinct 6 digits, with each player adding one digit at a time to the right of the last digit written by the other. Bardon wins if the resulting 6-digit number is prime; otherwise, Adam wins. Determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-70c679da1531c45724ed9faabf46fd32712d48a29a026f4e0840df7c541dc7ff | Adam and Ben start their new jobs on the same day. Adam's schedule is 3 workdays followed by 1 rest day. Ben's schedule is 7 workdays followed by 3 rest days. On how many of their first 1000 days do both have rest-days on the same day? | [
"50"
] | [
"Adam and Ben start their new jobs on the same day. Adam's schedule is 3 workdays followed by 1 rest day. Ben's schedule is 7 workdays followed by 3 rest days. On how many of their first 1000 days do both have rest-days on the same day?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6d53259578b92e602679c09159d9123c0a01ccf87c956cd804e1c171d5fddd1 | Adam and Bettie are playing a game where they take turns generating a random number between $0$ and $127$ inclusive. The scoring rules are: - If the number is zero, it receives no points. - If the number is odd, it receives one more point than the number one less than it. - If the number is even, it receives the same s... | [
"429"
] | [
"Adam and Bettie are playing a game where they take turns generating a random number between $0$ and $127$ inclusive. The scoring rules are:\n- If the number is zero, it receives no points.\n- If the number is odd, it receives one more point than the number one less than it.\n- If the number is even, it receives th... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-a35b22ed2e1ba504a3d94abf73fdb01bde9bc88f35fcfdf7bc952e57cc85be9c | Adam and Bob are playing a game where they roll a fair $20$-sided die, obtaining values $a$ and $b$, respectively. They calculate their cumulative score as $\frac{1}{2^a3^b}$. Let $\mu$ denote the expected value of this score. Compute the value of the greatest integer less than or equal to $\frac{1}{\mu}$. | [
"800"
] | [
"Adam and Bob are playing a game where they roll a fair $20$-sided die, obtaining values $a$ and $b$, respectively. They calculate their cumulative score as $\\frac{1}{2^a3^b}$. Let $\\mu$ denote the expected value of this score. Compute the value of the greatest integer less than or equal to $\\frac{1}{\\mu}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-4b0fecaf3aa2de19f0d276d1feb0a13719879b202a0f2245f08bc76338fff37c | Adam and Bob are playing a number game. They first agree on a positive integral constant $N$. Then, starting with zero, Adam and Bob alternate turns with Adam adding a composite number to the running total and Bob adding a prime number to the running total. The first person to make the running total $\ge N$ loses. Adam... | [
"24"
] | [
"Adam and Bob are playing a number game. They first agree on a positive integral constant $N$. Then, starting with zero, Adam and Bob alternate turns with Adam adding a composite number to the running total and Bob adding a prime number to the running total. The first person to make the running total $\\ge N$ loses... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c76cca159771471fd541e94bc081d1351207badc0353d3759439f3f143715a20 | Adam and Eve decide to produce offspring and continue until they have more boys than girls (they stop if their first child is a boy). How many offspring do they expect to produce? | [
"\\(\\infty\\)"
] | [
"Adam and Eve decide to produce offspring and continue until they have more boys than girls (they stop if their first child is a boy). How many offspring do they expect to produce?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 25
} | 46 | 0 | 0 |
omr-ws-e5881862ae86066727510d5f3a9abffe9de8c6c2bb7b74e243253db5bb4a2bad | Adam and Kris play a game with bananas. Initially, Adam has 201 bananas and Kris has 156 bananas. Each day, they each eat one banana from their pile if possible, and then simultaneously give the other half of their remaining bananas (rounded down) to the other player. After all bananas are eaten, how many bananas did K... | [
"179"
] | [
"Adam and Kris play a game with bananas. Initially, Adam has 201 bananas and Kris has 156 bananas. Each day, they each eat one banana from their pile if possible, and then simultaneously give the other half of their remaining bananas (rounded down) to the other player. After all bananas are eaten, how many bananas ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 21
} | 31 | 0 | 0 |
omr-ws-83ceaa78dd4190b4527a48f702b45cc19887d9846bb50ffa097465348aecda4e | Adam and Simon start on bicycle trips from the same point at the same time. Adam travels east at 8mph and Simon travels south at 6mph. After how many hours are they 60 miles apart? | [
"6"
] | [
"Adam and Simon start on bicycle trips from the same point at the same time. Adam travels east at 8mph and Simon travels south at 6mph. After how many hours are they 60 miles apart?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1c5d96226207383e1745909f36b55104f9e166a75a1c725d06e068e872e241e1 | Adam can eat one muffin every 20 seconds without dunking it in milk and one muffin every 10 seconds if he dunks it in milk. Ben can eat one muffin every 15 seconds without dunking it in milk and one muffin every 5 seconds if he dunks it in milk. If Adam dunks half of his muffins in milk and Ben dunks one-third of his m... | [
"51"
] | [
"Adam can eat one muffin every 20 seconds without dunking it in milk and one muffin every 10 seconds if he dunks it in milk. Ben can eat one muffin every 15 seconds without dunking it in milk and one muffin every 5 seconds if he dunks it in milk. If Adam dunks half of his muffins in milk and Ben dunks one-third of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 4
} | 24 | 0 | 0 |
omr-ws-92dbd36f983b9528c351470fc7ae7a39ae3946bdd217b589be53e8f8f999cbfe | Adam draws 7 circles on a paper, with radii 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm, and 7 cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm²) between the total area of the red circles and the total ... | [
"0"
] | [
"Adam draws 7 circles on a paper, with radii 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm, and 7 cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm²) between the total area of the red circles and the t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-8023071c557b373ededab6b4df03957a02b75ecca8cf07d67de3b662d20a6331 | Adam has $15$ of a certain type of rare coin and is interested in knowing how much this collection is worth. He discovers that $5$ of these coins are worth $12$ dollars in total. Assuming that the value of each coin is the same, how many dollars is his entire collection worth? | [
"36"
] | [
"Adam has $15$ of a certain type of rare coin and is interested in knowing how much this collection is worth. He discovers that $5$ of these coins are worth $12$ dollars in total. Assuming that the value of each coin is the same, how many dollars is his entire collection worth?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1a1c9ceaeaff65789f3af6b562de65696f84ccb581a97282d5506c3bbbccb3a | Adam has $n$ boxes numbered from $1$ to $n$ arranged in their original positions. He wants to rearrange the boxes such that no box is in its original position and the boxes numbered $n-3$, $n-2$, $n-1$, and $n$ are placed in the last 4 positions. If there are 2304 such arrangements, what is the value of $n$? | [
"10"
] | [
"Adam has $n$ boxes numbered from $1$ to $n$ arranged in their original positions. He wants to rearrange the boxes such that no box is in its original position and the boxes numbered $n-3$, $n-2$, $n-1$, and $n$ are placed in the last 4 positions. If there are 2304 such arrangements, what is the value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-4c2ce9801acf82c3910f1f5bda1c7f833aeb2b26313d0f2aed06a5a890d02529 | Adam has $n$ matches lying on a table, forming $n$ one-match piles. He wants to combine them into a single pile of $n$ matches using $n-1$ operations, each of which consists of combining two piles into one. Every time he combines a pile of $a$ matches with a pile of $b$ matches, he receives $a \cdot b$ candies. Calcula... | [
"45"
] | [
"Adam has $n$ matches lying on a table, forming $n$ one-match piles. He wants to combine them into a single pile of $n$ matches using $n-1$ operations, each of which consists of combining two piles into one. Every time he combines a pile of $a$ matches with a pile of $b$ matches, he receives $a \\cdot b$ candies. C... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-704d3f7da4d093dda97bee8b966b681dc57d382378ff1179c3bc9dab6a672387 | Adam has RM2010 in his bank account. He donates RM10 to charity every day, starting on Monday. On what day will he donate his last RM10? | [
"Friday"
] | [
"Adam has RM2010 in his bank account. He donates RM10 to charity every day, starting on Monday. On what day will he donate his last RM10?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 7,
"tir": 24
} | 76 | 0 | 0 |
omr-ws-b4e6ed7fc90a55d8084c528af8c1cd6c39e67e6e561835bce9eac5a94e7401a0 | Adam has a secret natural number \( x \). Eve can ask questions of the form "is \( x + n \) a prime number?" for any natural number \( n \) of her choice. What is the minimum number of questions Eve needs to ask to guarantee she can determine \( x \) if \( x \) is less than or equal to 100? | [
"7"
] | [
"Adam has a secret natural number \\( x \\). Eve can ask questions of the form \"is \\( x + n \\) a prime number?\" for any natural number \\( n \\) of her choice. What is the minimum number of questions Eve needs to ask to guarantee she can determine \\( x \\) if \\( x \\) is less than or equal to 100?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eede6e7910764cc5528ee83edf9803f5a9bb6879b17d94421e0be3e35ac96b55 | Adam has a single stack of $3 \cdot 2^n$ rocks, where $n$ is a nonnegative integer. Each move, Adam can either split an existing stack into two new stacks whose sizes differ by $0$ or $1$, or he can combine two existing stacks into one new stack. Adam keeps performing such moves until he eventually gets at least one st... | [
"2^{n-1} - 1"
] | [
"Adam has a single stack of $3 \\cdot 2^n$ rocks, where $n$ is a nonnegative integer. Each move, Adam can either split an existing stack into two new stacks whose sizes differ by $0$ or $1$, or he can combine two existing stacks into one new stack. Adam keeps performing such moves until he eventually gets at least ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-ab54abeeb779a4426c79bef7a94e107c6bba57352b6d6f7ecc29e519df5fcceb | Adam has buried the treasure at the point \((x, y)\) with integer coordinates. He has indicated the values of \(x^2 + y\) and \(x + y^2\) on the map, where these values are distinct. If \(x^2 + y = 100\) and \(x + y^2 = 50\), find the sum of all possible values of \(x + y\). | [
"0"
] | [
"Adam has buried the treasure at the point \\((x, y)\\) with integer coordinates. He has indicated the values of \\(x^2 + y\\) and \\(x + y^2\\) on the map, where these values are distinct. If \\(x^2 + y = 100\\) and \\(x + y^2 = 50\\), find the sum of all possible values of \\(x + y\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bf3feb4062465e7bd9271f542288159692c37050a06526739da0b86d639f41d0 | Adam is bored, and has written the string “EMCC” on a piece of paper. For fun, he decides to erase every letter “C”, and replace it with another instance of “EMCC”. For example, after one step, he will have the string “EMEMCCEMCC”. How long will his string be after $8$ of these steps? | [
"1534"
] | [
"Adam is bored, and has written the string “EMCC” on a piece of paper. For fun, he decides to erase every letter “C”, and replace it with another instance of “EMCC”. For example, after one step, he will have the string “EMEMCCEMCC”. How long will his string be after $8$ of these steps?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21ac6dc40e02ad283932268eab25c39f6deab459fedbd08edd95f441c715f77c | Adam is playing a game: he rolls a regular die until he rolls a $6$. The probability that he rolls the die at most twice can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime. Find $m+n$. | [
"47"
] | [
"Adam is playing a game: he rolls a regular die until he rolls a $6$. The probability that he rolls the die at most twice can be expressed in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-ce6ffac759074a0da0dacd0d335dfc91560a8fe089f377346c87cb34b9e5d8ef | Adam is playing the same game again, except that the die is weighted. After rolling the die many times, he finds that $25\%$ more of his game results in him rolling the die at most twice compared to rolling a regular die. The probability that rolling the dice will result in a $6$ can be expressed in the form $\frac{m}{... | [
"4"
] | [
"Adam is playing the same game again, except that the die is weighted. After rolling the die many times, he finds that $25\\%$ more of his game results in him rolling the die at most twice compared to rolling a regular die. The probability that rolling the dice will result in a $6$ can be expressed in the form $\\f... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-16bbf72329f92b727b4f17caee6a64bfc9b1f172ca32ba235520d4dfb0684a60 | Adam owns a square $S$ with side length $42$. First, he places rectangle $A$, which is $6$ times as long as it is wide, inside the square, so that all four vertices of $A$ lie on sides of $S$, but none of the sides of $A$ are parallel to any side of $S$. He then places another rectangle $B$, which is $7$ times as long ... | [
"\\(\\frac{29}{4}\\)"
] | [
"Adam owns a square $S$ with side length $42$. First, he places rectangle $A$, which is $6$ times as long as it is wide, inside the square, so that all four vertices of $A$ lie on sides of $S$, but none of the sides of $A$ are parallel to any side of $S$. He then places another rectangle $B$, which is $7$ times as ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-2da6bc3f255c9eb6db9612eea7b7b6a0ef2df06e922f101f4e2495a6260ae7c8 | Adam picks three non-zero real numbers, and Bob arranges these numbers as the coefficients of a quadratic equation \(ax^2 + bx + c = 0\). Adam wins if the resulting equation has two distinct rational roots. Who has a winning strategy? | [
"Adam"
] | [
"Adam picks three non-zero real numbers, and Bob arranges these numbers as the coefficients of a quadratic equation \\(ax^2 + bx + c = 0\\). Adam wins if the resulting equation has two distinct rational roots. Who has a winning strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-67ae3a8dce91be8cef8bb41a60d9706fda13f855ca2473264116036ee5110757 | Adam starts at the bottom of a 2018-step staircase. He licks a lollipop after walking up \( x_1 \) steps, then after walking up \( x_2 \) steps, and so on, until he reaches the top. Each \( x_i \) satisfies \( 0 < x_i \leq 2018 \) and no two \( x_i \) are necessarily equal. If there are \( N \) ways for Adam to reach t... | [
"2"
] | [
"Adam starts at the bottom of a 2018-step staircase. He licks a lollipop after walking up \\( x_1 \\) steps, then after walking up \\( x_2 \\) steps, and so on, until he reaches the top. Each \\( x_i \\) satisfies \\( 0 < x_i \\leq 2018 \\) and no two \\( x_i \\) are necessarily equal. If there are \\( N \\) ways f... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b6fdbd229f515a0e00d4db71b91820fb2bd508b4ad8c7f231ff90386bcbb6fd | Adam the Ant is on point $A$ of a cylinder with height $4$ and radius $3$. He wishes to walk to point $B$, where $\overline{BC}$ is a height of the cylinder and $C$ is on the top base such that $\overline{AC}$ is a diameter. What is the length of the shortest path Adam can take if he is willing to walk across a base of... | [
"\\(2\\sqrt{13}\\)"
] | [
"Adam the Ant is on point $A$ of a cylinder with height $4$ and radius $3$. He wishes to walk to point $B$, where $\\overline{BC}$ is a height of the cylinder and $C$ is on the top base such that $\\overline{AC}$ is a diameter. What is the length of the shortest path Adam can take if he is willing to walk across a ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48ecdc46f4aa8672310d8c5b717d290bc021f50a7b76f2435d6b433acc1ef6a0 | Adam the spider is sitting at the bottom left of a 4 × 4 coordinate grid and wants to crawl to the top right corner. He starts with 9 "crumb's" worth of energy, and each unit of movement costs 1 crumb. There is a crumb located one unit to the right and one unit up from his starting position, which gives him 1 additiona... | [
"20"
] | [
"Adam the spider is sitting at the bottom left of a 4 × 4 coordinate grid and wants to crawl to the top right corner. He starts with 9 \"crumb's\" worth of energy, and each unit of movement costs 1 crumb. There is a crumb located one unit to the right and one unit up from his starting position, which gives him 1 ad... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-d79582986a0bc771b06ed1216328f8fba21a5e8c100ce343c0a7ae81a2953586 | Adam went to a store to buy 2 souvenirs. After buying the first souvenir, he was left with half the money he started with. After buying the second item, which cost $10, he was left with only $5. How much money did he have before he bought the souvenirs? | [
"30"
] | [
"Adam went to a store to buy 2 souvenirs. After buying the first souvenir, he was left with half the money he started with. After buying the second item, which cost $10, he was left with only $5. How much money did he have before he bought the souvenirs?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-58f27e01828bf77415cb5a536441e0d875ef695e30ac2700382ee0c5a64f9aa7 | Adam wrote a number in each of the empty circles so that the sum of all the numbers in the circles equals 25. Whenever three circles are such that every two of them are connected with a single line, the numbers in these three circles add up to 12. What number did he put in the central circle? Find all possible answers. | [
"-1"
] | [
"Adam wrote a number in each of the empty circles so that the sum of all the numbers in the circles equals 25. Whenever three circles are such that every two of them are connected with a single line, the numbers in these three circles add up to 12. What number did he put in the central circle? Find all possible ans... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-2f5e23a1e6f4e009b66fad40af1bd2ecce35d8649f92d707929bbb241f45d790 | Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? | [
"170"
] | [
"Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 15,
"tir": 2
} | 41 | 0 | 0 |
omr-ws-b1cf7aeaee5bfdb46a58194b4b55040acca3ceeac29679d909a5ff159e15206f | Adam, Bob, and Charlie each flip a coin every day, starting from Day 1, until all three of them have flipped heads at least once. The last of them to flip heads for the first time does so on Day \( X \). The probability that \( X \) is even can be expressed in the form \( \frac{m}{n} \), where \( m \) and \( n \) are r... | [
"68"
] | [
"Adam, Bob, and Charlie each flip a coin every day, starting from Day 1, until all three of them have flipped heads at least once. The last of them to flip heads for the first time does so on Day \\( X \\). The probability that \\( X \\) is even can be expressed in the form \\( \\frac{m}{n} \\), where \\( m \\) and... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a24dbffed6b6a4348d8e2f3e53dc3d210e4b23b6b468cefadb70462108d0f8c7 | Adam, Steve, Greg, and John want to cross an old bridge at night. To be safe, they decide to cross at most two people at a time. They only have one working flashlight, and since everyone crossing the bridge needs light, they can only cross at the speed of the slower person in a pair. If Adam, Steve, Greg, and John can ... | [
"35.5"
] | [
"Adam, Steve, Greg, and John want to cross an old bridge at night. To be safe, they decide to cross at most two people at a time. They only have one working flashlight, and since everyone crossing the bridge needs light, they can only cross at the speed of the slower person in a pair. If Adam, Steve, Greg, and John... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-77c7038f01a1dd4af71a4389a90caf880cd5dec83bfaf4a668f80015ddf56328 | Add $101_2 + 11_2 + 1100_2 + 11101_2.$ Express your answer in base $2.$ | [
"110001_2"
] | [
"Add $101_2 + 11_2 + 1100_2 + 11101_2.$ Express your answer in base $2.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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