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-2b213ca01dbe2a73a606dbd70af3ddf75431ebfe5de5ab449beb0d59d45d69f6 | A bag contains 5 red, 6 green, 7 yellow, and 8 blue jelly beans. A jelly bean is selected at random. What is the probability that it is blue? | [
"\\frac{4}{13}"
] | [
"A bag contains 5 red, 6 green, 7 yellow, and 8 blue jelly beans. A jelly bean is selected at random. What is the probability that it is blue?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f17e1456de9bf738648d35c21e072da6324fb4048563176da966ba494f4bffd1 | A bag contains 5 white balls. In each trial, a ball is drawn uniformly at random. If the ball is white, it is colored black and returned to the bag. If the ball is black, it is returned without changing its color. What is the expected number of times the process must be repeated so that the bag contains only black ball... | [
"11.42"
] | [
"A bag contains 5 white balls. In each trial, a ball is drawn uniformly at random. If the ball is white, it is colored black and returned to the bag. If the ball is black, it is returned without changing its color. What is the expected number of times the process must be repeated so that the bag contains only black... | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-96bff15f3d57ec631e61cac0fa4270f76e2b5affcc8b317776cb81ba20d30781 | A bag contains 5 white, 7 red, and 3 black balls. If 3 balls are drawn one by one without replacement, find the probability that none of the balls drawn is red. | [
"\\(\\frac{8}{65}\\)"
] | [
"A bag contains 5 white, 7 red, and 3 black balls. If 3 balls are drawn one by one without replacement, find the probability that none of the balls drawn is red."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-00bef7665a2e7796cd540df7425a42aeeffc686b24e9795b57e020ea0c2dc13d | A bag contains 5 yellow, 6 red, 3 blue, and 2 white marbles. If two marbles are chosen with replacement, calculate the following probabilities: a. The probability of drawing a red marble followed by a blue marble. b. The probability of drawing a yellow marble followed by a white marble. c. The probability of drawing a ... | [
"a. \\( \\frac{9}{128} \\)\nb. \\( \\frac{5}{128} \\)\nc. \\( 0 \\)\nd. \\( \\frac{9}{64} \\)"
] | [
"A bag contains 5 yellow, 6 red, 3 blue, and 2 white marbles. If two marbles are chosen with replacement, calculate the following probabilities:\na. The probability of drawing a red marble followed by a blue marble.\nb. The probability of drawing a yellow marble followed by a white marble.\nc. The probability of dr... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-f1db4a3352af75ce05c3eee7c9be49607f3742a0ff760333f495c5ec2e445925 | A bag contains 50 tickets numbered from 1 to 50. Five tickets are drawn randomly and arranged in ascending order as \( x_1 < x_2 < x_3 < x_4 < x_5 \). Find the probability that \( x_3 = 30 \). | [
"\\(\\frac{3857}{105938}\\)"
] | [
"A bag contains 50 tickets numbered from 1 to 50. Five tickets are drawn randomly and arranged in ascending order as \\( x_1 < x_2 < x_3 < x_4 < x_5 \\). Find the probability that \\( x_3 = 30 \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-667746c3a8db05ccf28a36607210ead5ba4cf55473c37af0ba6fc22ed6bc0e0a | A bag contains 50 tickets, numbered from 1 to 50. Five tickets are drawn at random and arranged in ascending order of magnitude $(x_1 < x_2 < x_3 < x_4 < x_5)$. Find the probability that $x_3 = 30$. | [
"\\(\\frac{3857}{105938}\\)"
] | [
"A bag contains 50 tickets, numbered from 1 to 50. Five tickets are drawn at random and arranged in ascending order of magnitude $(x_1 < x_2 < x_3 < x_4 < x_5)$. Find the probability that $x_3 = 30$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-946bbfa55cd4dd9c696134b03a92b9f3dbe18b40e4acef43408dfa36e440c9a5 | A bag contains 6 different balls of 3 colors: white, green, and red. Find the probability that the bag contains 2 balls of each color. | [
"\\(\\frac{10}{81}\\)"
] | [
"A bag contains 6 different balls of 3 colors: white, green, and red. Find the probability that the bag contains 2 balls of each color."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-9d0424e74e4fd1c764ca7f5d2341297207ca4eb6b189f6757a9c17476dc52ec4 | A bag contains 6 different balls of 3 colors: white, green, and red. Three balls are picked up at random and found to be one of each color. Find the probability that the bag contained 4 red balls. | [
"\\(\\frac{1}{20}\\)"
] | [
"A bag contains 6 different balls of 3 colors: white, green, and red. Three balls are picked up at random and found to be one of each color. Find the probability that the bag contained 4 red balls."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-301703fd4376589f9e91f900e86a186baeb50c59ba761a01ee22057f680da49a | A bag contains 6 different balls of 3 colors: white, green, and red. Three balls are picked up at random and found to be one of each color. Find the probability that the bag contained an equal number of white and green balls. | [
"\\(\\frac{2}{5}\\)"
] | [
"A bag contains 6 different balls of 3 colors: white, green, and red. Three balls are picked up at random and found to be one of each color. Find the probability that the bag contained an equal number of white and green balls."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5025b38d5439c05e15112185d827221df76f72b31fd9a367838c1f0bb62ad4fa | A bag contains 7 red, 6 blue, and 5 green balls. If 5 balls are chosen at random, what is the probability that at least one ball of each color is selected? | [
"\\(\\frac{865}{1224}\\)"
] | [
"A bag contains 7 red, 6 blue, and 5 green balls. If 5 balls are chosen at random, what is the probability that at least one ball of each color is selected?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-0f8feb7e4c6482299638b142bbe0c593df392b9f0c66e08da726b1f86e24eeab | A bag contains 8 dice, where \( m \) dice are biased to always show a six and the rest are unbiased. To test the hypothesis \( H_0: m = 2 \) against the alternative hypothesis \( H_1: m = 1 \), the following procedure is used: throw the 8 dice and if the number of sixes is less than or equal to 3, reject \( H_0 \); oth... | [
"Type I error probability: \\( 0.7368 \\), Type II error probability: \\( 0.1095 \\)"
] | [
"A bag contains 8 dice, where \\( m \\) dice are biased to always show a six and the rest are unbiased. To test the hypothesis \\( H_0: m = 2 \\) against the alternative hypothesis \\( H_1: m = 1 \\), the following procedure is used: throw the 8 dice and if the number of sixes is less than or equal to 3, reject \\(... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ee6db669abed0985744a170409f0c1e53840096bec1ed6558e527c5fa8d5e0da | A bag contains 8 pennies and 1 quarter. Four coins are picked at random from the bag and placed into an urn that already contains 12 pennies. Then, three coins are picked from the urn and put back into the bag. What is the probability that the quarter ends up in the bag? | [
"\\(\\dfrac{23}{36}\\)"
] | [
"A bag contains 8 pennies and 1 quarter. Four coins are picked at random from the bag and placed into an urn that already contains 12 pennies. Then, three coins are picked from the urn and put back into the bag. What is the probability that the quarter ends up in the bag?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-b2b5b9f1dc14e631974c0fdd6bed5588738115e8acfd6d3df222b3d82c986ca0 | A bag contains 8 red and 6 green balls. Three balls are drawn one after another without replacement. Find the probability that at least 2 of the balls drawn are green. | [
"\\(\\frac{5}{13}\\)"
] | [
"A bag contains 8 red and 6 green balls. Three balls are drawn one after another without replacement. Find the probability that at least 2 of the balls drawn are green."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-845f0dbb0ca63828dbbebfbf6490e296f25d1b698a4068e35c10f72b61b2a40f | A bag contains 8 white balls and 2 red balls. One ball is drawn at random on each trial. If the ball is red, the trials end. If the ball is white, the ball is replaced and another trial is conducted. Let $X$ be the number of trials until a red ball is drawn. Find the expected value of $X^2$. | [
"45"
] | [
"A bag contains 8 white balls and 2 red balls. One ball is drawn at random on each trial. If the ball is red, the trials end. If the ball is white, the ball is replaced and another trial is conducted. Let $X$ be the number of trials until a red ball is drawn. Find the expected value of $X^2$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de2551133a28d362b71461a662575fb8876eb154b60f0e62e1ba0b3dc0d0e776 | A bag contains 9 blue marbles, 10 ugly marbles, and 1 special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. Given that none of the marbles he drew were ugly, what is the expected value of the number of total marbles he drew? | [
"10"
] | [
"A bag contains 9 blue marbles, 10 ugly marbles, and 1 special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. Given that none of the marbles he drew were ugly, what is the expected value of the number of total marbles he drew?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-ee176346d33e8dee28e2452788034e300dd2e2d60ba71e155a82933cc3135f19 | A bag contains 99 black beans and 1 white bean. Beans are drawn one at a time from the bag until the white bean is drawn. Find the expected number of black beans drawn before the white bean is drawn. | [
"49.5"
] | [
"A bag contains 99 black beans and 1 white bean. Beans are drawn one at a time from the bag until the white bean is drawn. Find the expected number of black beans drawn before the white bean is drawn."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b536f090d167478ca5cf2070c85c743815c68ec2b4f4c36b07d00f3bab508fb0 | A bag contains 99 red marbles and 99 blue marbles. You repeatedly draw two marbles from the bag and replace them with one marble according to the following rules: - If the two marbles are the same color, you put a red marble back in the bag. - If the two marbles are different colors, you put a blue marble back in the b... | [
"blue"
] | [
"A bag contains 99 red marbles and 99 blue marbles. You repeatedly draw two marbles from the bag and replace them with one marble according to the following rules:\n- If the two marbles are the same color, you put a red marble back in the bag.\n- If the two marbles are different colors, you put a blue marble back i... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-1a016278adb2b83421b5e594ff175069f84dcf016e292862a564e18b0814caa5 | A bag contains \( N \) balls, each labelled with a different integer from \(\{1, 2, \ldots, N\}\). \( n \leq N \) balls are selected one by one without replacement. Given \( 1 \leq M \leq N \) and \( 1 \leq m \leq n \), find the probability that exactly \( m \) of the \( n \) balls chosen are labelled by integers which... | [
"\\(\\dfrac{\\dbinom{M}{m}\\cdot \\dbinom{N-M}{n-m}}{\\dbinom{N}{n}}\\)"
] | [
"A bag contains \\( N \\) balls, each labelled with a different integer from \\(\\{1, 2, \\ldots, N\\}\\). \\( n \\leq N \\) balls are selected one by one without replacement. Given \\( 1 \\leq M \\leq N \\) and \\( 1 \\leq m \\leq n \\), find the probability that exactly \\( m \\) of the \\( n \\) balls chosen are... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-b62996bf5ba0060616b4d9e5571046d30e85be063ccecb68f43f2fb020f010ae | A bag contains \(a\) white and \(b\) black balls. Balls are chosen according to the following method: 1. A ball is chosen at random and discarded. 2. A second ball is chosen. If its color is different from that of the preceding ball, it is replaced in the bag and the process is repeated from the beginning. Let \(P_{a,b... | [
"1"
] | [
"A bag contains \\(a\\) white and \\(b\\) black balls. Balls are chosen according to the following method:\n1. A ball is chosen at random and discarded.\n2. A second ball is chosen. If its color is different from that of the preceding ball, it is replaced in the bag and the process is repeated from the beginning.\n... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-9b7dbd8ff23a43c2dfdfe40083bacdbc475c4ad7391d73668daa7c3df5d95c21 | A bag contains a coin of value \( M \) and other coins whose total value is \( m \). A person draws coins one at a time until they draw the coin of value \( M \). Find the expected value of the coins drawn before drawing the coin of value \( M \). | [
"\\( \\frac{m}{2} \\)"
] | [
"A bag contains a coin of value \\( M \\) and other coins whose total value is \\( m \\). A person draws coins one at a time until they draw the coin of value \\( M \\). Find the expected value of the coins drawn before drawing the coin of value \\( M \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-82e605563c2c060bfaa7c24721dc10833c4522eba9e243f3d7f6e8b768f787b3 | A bag contains a number of colored balls, with equal numbers of each color. Adding 20 balls of a new color to the bag would not change the probability of drawing (without replacement) two balls of the same color. How many balls are in the bag before the 20 balls were added? | [
"40"
] | [
"A bag contains a number of colored balls, with equal numbers of each color. Adding 20 balls of a new color to the bag would not change the probability of drawing (without replacement) two balls of the same color. How many balls are in the bag before the 20 balls were added?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e158ce6fa503de0538ae895cd16ffbd88f161edb5c69ba5eed0ac387814c41d7 | A bag contains balls of 2021 types, with $i$ balls of the $i$th type. We randomly draw a ball, note its type, and return it along with an additional $2020^{2021}$ balls of the same type. This process is repeated $2021^{2020}$ times. What is the probability that the ball drawn after these repetitions is of the 2019th ty... | [
"\\(\\frac{2019}{2021 \\cdot 1011}\\)"
] | [
"A bag contains balls of 2021 types, with $i$ balls of the $i$th type. We randomly draw a ball, note its type, and return it along with an additional $2020^{2021}$ balls of the same type. This process is repeated $2021^{2020}$ times. What is the probability that the ball drawn after these repetitions is of the 2019... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 1,
"tir": 1
} | 36 | 0 | 0 |
omr-ws-d61fd50f2f8f5a01c18e8486ee315b187442636cb4f804ada1a3f3beae516658 | A bag contains eight yellow marbles, seven red marbles, and five black marbles. Igor removes N marbles all at once. What is the maximum possible value of N such that there are at least four marbles of one color and at least three marbles of another color left in the bag? | [
"7"
] | [
"A bag contains eight yellow marbles, seven red marbles, and five black marbles. Igor removes N marbles all at once. What is the maximum possible value of N such that there are at least four marbles of one color and at least three marbles of another color left in the bag?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e89c4f2de128e55dff6ffb1ae6dd691c4c7742a2d27d1609f438d3f8459b8a9c | A bag contains five ping pong balls numbered from one to five. If two balls are drawn at random with replacement, given that the total of the two numbers on the selected balls is at most four, what is the probability that the total is an odd number? | [
"\\(\\frac{2}{7}\\)"
] | [
"A bag contains five ping pong balls numbered from one to five. If two balls are drawn at random with replacement, given that the total of the two numbers on the selected balls is at most four, what is the probability that the total is an odd number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f60e835fe2a63a159daf1976f055fda93f78a97a090b0b00bff565e2a461f53e | A bag contains four balls. Two balls are drawn and found to be white. What is the probability that the two balls left in the bag are also white? | [
"\\(\\frac{3}{5}\\)"
] | [
"A bag contains four balls. Two balls are drawn and found to be white. What is the probability that the two balls left in the bag are also white?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-8261b2b19b01bb3f1f2191e0d8b0e1583787f8a94d677acac4346f9a0f37f824 | A bag contains nine balls of which three are red, three are green, and three are blue. If the nine balls are randomly arranged in a row, what is the probability that the arrangement will have all three balls of each color together in groups? | [
"\\(\\frac{2}{7}\\)"
] | [
"A bag contains nine balls of which three are red, three are green, and three are blue. If the nine balls are randomly arranged in a row, what is the probability that the arrangement will have all three balls of each color together in groups?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80daef000b3c6d79979704a90c0a91a1ee6155a771d99e0aa1f9ee6199405df1 | A bag contains nine balls of which three are red, three are green, and three are blue. If the nine balls are randomly arranged in a row, what is the probability that the row begins and ends with balls of the same color? | [
"\\(\\frac{2}{7}\\)"
] | [
"A bag contains nine balls of which three are red, three are green, and three are blue. If the nine balls are randomly arranged in a row, what is the probability that the row begins and ends with balls of the same color?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-182446186ef44f35abd53ca93f428b4b8eea486bd09cdb4b557c36b6afdbf7cf | A bag contains nine balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. How many of the nine balls were of one co... | [
"3 white balls and 6 black balls or 6 white balls and 3 black balls."
] | [
"A bag contains nine balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. How many of the nine balls were of o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-13c7c8ff4f45b47625097b464d5f6e7c96e06cf0e6b1af1b9905337f1bdbd6c6 | A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. Given that none of the marbles he drew were ugly, what is the expected value of the number of total marbles he drew? | [
"\\(\\frac{20}{11}\\)"
] | [
"A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. Given that none of the marbles he drew were ugly, what is the expected value of the number of total marbles he drew?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-1d75176110afed469151d235653ef25c51b8f90e223d560b84f7a9e365c6f484 | A bag contains one marble known to be either red or green. A green marble is added to the bag, and the bag is shaken well. Two random draws are made, both resulting in green marbles, which are then returned to the bag. What is the probability that the original marble in the bag is also green? | [
"\\(\\frac{4}{5}\\)"
] | [
"A bag contains one marble known to be either red or green. A green marble is added to the bag, and the bag is shaken well. Two random draws are made, both resulting in green marbles, which are then returned to the bag. What is the probability that the original marble in the bag is also green?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-d55e9b6287c8ba76e559faab28db7e082c83661862095133ea8922943c2585c4 | A bag contains one marble, either green or yellow. A yellow marble is added to the bag, and one marble is randomly chosen. The chosen marble is yellow. What is the probability that the marble left in the bag is yellow? Express your answer as a common fraction. | [
"$\\frac{2}{3}$"
] | [
"A bag contains one marble, either green or yellow. A yellow marble is added to the bag, and one marble is randomly chosen. The chosen marble is yellow. What is the probability that the marble left in the bag is yellow? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4132abc6c2b72074840ec7bd5326745a41822ceea0a83573badf989fb9cc33b5 | A bag contains red and blue marbles. When two marbles are drawn, the probability that they are both red is equal to the probability that they are both blue. The probability that one of each color is drawn is 4/7. What is the least number of marbles in the bag? | [
"8"
] | [
"A bag contains red and blue marbles. When two marbles are drawn, the probability that they are both red is equal to the probability that they are both blue. The probability that one of each color is drawn is 4/7. What is the least number of marbles in the bag?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-583ab0757f919f1d88808b49d13291ddcedb0a7c9c34e8ba5a4a16ccfcf37d0a | A bag contains red balls and white balls. If eight balls are to be pulled from the bag, one at a time with replacement, the probability of getting exactly four red balls is 40 times the probability of getting exactly two red balls. What percent of the balls originally in the bag are red? | [
"80 percent"
] | [
"A bag contains red balls and white balls. If eight balls are to be pulled from the bag, one at a time with replacement, the probability of getting exactly four red balls is 40 times the probability of getting exactly two red balls. What percent of the balls originally in the bag are red?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-16acbb351689d1b8e05e44e4c3007408c54edf4b3dc38b818287496e1df941ad | A bag contains several chips which are red, white, or blue. The odds in favor of selecting a blue chip are 1 to 2, a red chip are 1 to 4, and a white chip are 1 to 14. What is the least number of chips the bag must contain? | [
"15"
] | [
"A bag contains several chips which are red, white, or blue. The odds in favor of selecting a blue chip are 1 to 2, a red chip are 1 to 4, and a white chip are 1 to 14. What is the least number of chips the bag must contain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-a5b7d722b9c4c576f11cdf3e46c885765faa417c3d0630bb7ec6534d560b40b4 | A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | [
"3"
] | [
"A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\\frac{1}{15}$. How many balls in the bag are red?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b75777a5adae8330616da11ca757fcd266922d9726ff45b80310a93db3e225ad | A bag contains ten identical balls, each numbered with a different number from the integers 1 to 10. Lucky Pix draws three balls from the bag. She wins if the smallest numbered ball in her hand is odd and the next smallest numbered ball in her hand is even. Determine the probability that Lucky Pix wins. | [
"\\(\\frac{1}{3}\\)"
] | [
"A bag contains ten identical balls, each numbered with a different number from the integers 1 to 10. Lucky Pix draws three balls from the bag. She wins if the smallest numbered ball in her hand is odd and the next smallest numbered ball in her hand is even. Determine the probability that Lucky Pix wins."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-4be6519b55754ae95a1a82e8819f723a4c40b57510f8a0e206276464e58b6dc8 | A bag contains three red, five green, and two blue marbles. When two marbles are chosen, what is the probability that they are of different colors? | [
"\\(\\frac{31}{45}\\)"
] | [
"A bag contains three red, five green, and two blue marbles. When two marbles are chosen, what is the probability that they are of different colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7a9e8911df519779962da0dd57151fceb05fcd69f2a4b8eb4fc4b59d6c67abca | A bag contains three tiles marked $A$, $L$, and $I$. Ali wishes to pick the letters in $ALI$, in that order. He randomly draws one tile from the bag; if the tile drawn is $A$, he discards the tile, otherwise, he puts the tile back in the bag. Ali repeats this process until he draws the $A$ tile, then similarly draws ti... | [
"721"
] | [
"A bag contains three tiles marked $A$, $L$, and $I$. Ali wishes to pick the letters in $ALI$, in that order. He randomly draws one tile from the bag; if the tile drawn is $A$, he discards the tile, otherwise, he puts the tile back in the bag. Ali repeats this process until he draws the $A$ tile, then similarly dra... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-c5a585e9f389cdb1a231778298e63923782e8edc20a973a961b7ea19a4789ec4 | A bag has 3 red marbles and 5 white marbles. Two marbles are drawn from the bag and not replaced. What is the probability that the first marble is red and the second marble is white? | [
"\\dfrac{15}{56}"
] | [
"A bag has 3 red marbles and 5 white marbles. Two marbles are drawn from the bag and not replaced. What is the probability that the first marble is red and the second marble is white?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87a95f49da67e9f713170ea4d5f4e8986ca9d65c3880411f6e6cb3738974b4c8 | A bag has 4 red marbles, 5 white marbles, and 6 blue marbles. Three marbles are drawn from the bag (without replacement). What is the probability that they are all the same color? | [
"\\frac{34}{455}"
] | [
"A bag has 4 red marbles, 5 white marbles, and 6 blue marbles. Three marbles are drawn from the bag (without replacement). What is the probability that they are all the same color?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ab6871c21a136938e3460c673097a597a86dc657d4adc733a43cf5e21858e99 | A bag has 6 red marbles and 4 blue marbles. What are the chances of pulling out a red and a blue marble? | [
"\\(\\frac{8}{15}\\)"
] | [
"A bag has 6 red marbles and 4 blue marbles. What are the chances of pulling out a red and a blue marble?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-afaf972d40fa87934807f865f21e08c77213f8b0124bfcb9df25a29feaa18be9 | A bag has cards in it, numbered from \( 1 \) to \( 6 \). Cards are removed one at a time in random order. Find the probability that the \( n \)-th card removed, where \( 3 < n < 7 \), has a number less than \( n \) on it. | [
"\\dfrac{n-1}{6}"
] | [
"A bag has cards in it, numbered from \\( 1 \\) to \\( 6 \\). Cards are removed one at a time in random order. Find the probability that the \\( n \\)-th card removed, where \\( 3 < n < 7 \\), has a number less than \\( n \\) on it."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-418893492d63e00f52f2e4b50cdfa74b4c350d0fcee349575db33059e45f0298 | A bag initially contains one marble, which is either green or red. A green marble is added to the bag, and the bag is shaken. Two random draws are made, both resulting in green marbles, which are then returned to the bag. What is the probability that the original marble in the bag is also green? | [
"\\dfrac{4}{5}"
] | [
"A bag initially contains one marble, which is either green or red. A green marble is added to the bag, and the bag is shaken. Two random draws are made, both resulting in green marbles, which are then returned to the bag. What is the probability that the original marble in the bag is also green?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee5c7220c86656ea652c9a61f0405388ff3573ac48df259e734f932ec9e4d9b4 | A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only $1/3$ of the marbles in the bag are blue. Then yellow marbles are added to the bag until only $1/5$ of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubl... | [
"\\(\\frac{1}{3}\\)"
] | [
"A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only $1/3$ of the marbles in the bag are blue. Then yellow marbles are added to the bag until only $1/5$ of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-5be8ff47e34bdb6bbb971e1a7e9f505b45d758d52129ceb15661133b2215343a | A bag of grapes is to be distributed evenly to 5 kids in a class, and the grapes that are left over will be thrown out. If each student receives the greatest possible number of grapes, what is the greatest possible number of grapes that could be thrown out? | [
"4"
] | [
"A bag of grapes is to be distributed evenly to 5 kids in a class, and the grapes that are left over will be thrown out. If each student receives the greatest possible number of grapes, what is the greatest possible number of grapes that could be thrown out?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c351aceeb5bd76f47b39bb6074da4f4443141bc1c122059208ab235506b92843 | A bag weighing 11111 grams is placed in the right pan of a balance. A person sequentially places weights of 1, 2, 4, 8, ... grams in either pan, starting with the 1 gram weight. At a certain point, the pans are in equilibrium. Calculate the sum of the weights in grams that are placed in the left pan, including the 16 g... | [
"13747"
] | [
"A bag weighing 11111 grams is placed in the right pan of a balance. A person sequentially places weights of 1, 2, 4, 8, ... grams in either pan, starting with the 1 gram weight. At a certain point, the pans are in equilibrium. Calculate the sum of the weights in grams that are placed in the left pan, including the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ae6e01bea0ba64a4281ef4e65a29e805dcd033e7afe817f77036ec0a6a976606 | A bagel store sells six different kinds of bagels, one of which is Sesame. If you choose 15 bagels at random, what is the probability that your choice contains at least three Sesame bagels? | [
"\\dfrac{91}{228}"
] | [
"A bagel store sells six different kinds of bagels, one of which is Sesame. If you choose 15 bagels at random, what is the probability that your choice contains at least three Sesame bagels?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-a9ac2fc53bc3e4294869ede05f9a9849ff6d16f6de739d280af4067ceb26fb9f | A bagel store sells six different kinds of bagels. If you choose 15 bagels at random, what is the probability that your choice contains at least one bagel of each kind? | [
"\\dfrac{1001}{7752}"
] | [
"A bagel store sells six different kinds of bagels. If you choose 15 bagels at random, what is the probability that your choice contains at least one bagel of each kind?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-49d6788e7f5641799a285ce642d7f095902df9ba93fc252573f8b84d42e94dd8 | A baker adds 30 cups of water. How many ounces did he add? He takes away 21 cups of water. How many ounces does he have at the end? | [
"72"
] | [
"A baker adds 30 cups of water. How many ounces did he add? He takes away 21 cups of water. How many ounces does he have at the end?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d58803ded6d4839a8f81f52466fa61a4a51879f5d587cc26fc0d41b8f6065d0 | A baker can serve 5 different types of bread. A family of 5 is sitting at a round table such that no two adjacent people have the same type of bread. If the person in seat 4 gets Rye bread, in how many ways can the baker put down the bread? | [
"204"
] | [
"A baker can serve 5 different types of bread. A family of 5 is sitting at a round table such that no two adjacent people have the same type of bread. If the person in seat 4 gets Rye bread, in how many ways can the baker put down the bread?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 13,
"tir": 16
} | 73 | 0 | 0 |
omr-ws-a0da1babd7803105b7b0692672922275457f9a34f7b638fab48cf3010bbd11f3 | A baker offers two stuffed cakes, each of which can contain up to five different fillings. A customer claims there are $1,048,576$ different ways to choose the two stuffed cakes. How many different fillings does the pastry chef have? | [
"10"
] | [
"A baker offers two stuffed cakes, each of which can contain up to five different fillings. A customer claims there are $1,048,576$ different ways to choose the two stuffed cakes. How many different fillings does the pastry chef have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-a543e65b4e3421e5302c8b9cedadf5f7a3d2d13f143959a8b5f159df5b2706ce | A baker uses \( n \) different spices. He bakes 10 loaves, each with a different combination of spices, and each loaf uses more than \( n/2 \) spices. Let \( S \) be the set of all spices used. Determine the minimum value of \( n \) such that there are three spices in \( S \) such that every loaf contains at least one ... | [
"5"
] | [
"A baker uses \\( n \\) different spices. He bakes 10 loaves, each with a different combination of spices, and each loaf uses more than \\( n/2 \\) spices. Let \\( S \\) be the set of all spices used. Determine the minimum value of \\( n \\) such that there are three spices in \\( S \\) such that every loaf contain... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-f159915018980bcb18f2ffe7d5738dfcaab94c0330c340972f991aa289f71141 | A baker wants to grow wheat on a $2$ km$^2$ field in the form of a "slice of bread" (a rectangle surmounted by a semicircle). What dimensions should the field have so that the fence delimiting it is as short as possible? | [
"\\dfrac{4}{\\sqrt{4 + \\pi}} \\text{ km}, \\dfrac{2}{\\sqrt{4 + \\pi}} \\text{ km}"
] | [
"A baker wants to grow wheat on a $2$ km$^2$ field in the form of a \"slice of bread\" (a rectangle surmounted by a semicircle). What dimensions should the field have so that the fence delimiting it is as short as possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-5a9f57b6849f88834bd678d8fa1baa8a5b5dc9f45a0212c91019d63ad4e2deab | A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM, the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? | [
"4:30 PM"
] | [
"A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM, the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-58b48062b5b4025efb2f0fec680c9b675342bbc7dbedf01f4a5a4b7507642f71 | A bakery sells chocolate, cinnamon, and plain doughnuts and has 6 chocolate, 6 cinnamon, and 3 plain doughnuts available. If a box contains 12 doughnuts, how many different combinations of doughnuts can be made for a box? | [
"10"
] | [
"A bakery sells chocolate, cinnamon, and plain doughnuts and has 6 chocolate, 6 cinnamon, and 3 plain doughnuts available. If a box contains 12 doughnuts, how many different combinations of doughnuts can be made for a box?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-859abffad58bdeb9e9eb787809458314fb1d0aa0d5990067411d802f438ba90d | A bakery sells three kinds of rolls. How many different combinations of rolls could Jack purchase if he buys a total of six rolls and includes at least one of each kind? | [
"10"
] | [
"A bakery sells three kinds of rolls. How many different combinations of rolls could Jack purchase if he buys a total of six rolls and includes at least one of each kind?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef155bcbb67e47985d9753781b77b19b1138bee25144c97b0f02dc522ef08367 | A balance scale sits on a teacher's table, currently tipped to the right. There is a set of weights on the scale, and each weight has the name of at least one pupil. When each pupil enters the classroom, they move all the weights carrying their name to the opposite side of the scale. Let $W$ be the set of all weights, ... | [
"2^{n-1} (R - L)"
] | [
"A balance scale sits on a teacher's table, currently tipped to the right. There is a set of weights on the scale, and each weight has the name of at least one pupil. When each pupil enters the classroom, they move all the weights carrying their name to the opposite side of the scale. Let $W$ be the set of all weig... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-b92e863533ea662aafe1f529fb5abea5f4bc268d6bf62d2a166e5461f025f8ff | A balanced die is rolled \( n \) times independently where \( n \geq 2 \). Let \( X \) denote the number of times 6 dots show up and \( Y \) denote the number of times 5 dots show up in these \( n \) rolls. Find the probability distribution of \( Z = X + Y \). | [
"Z \\sim \\text{Binomial}\\left(n, \\frac{1}{3}\\right)"
] | [
"A balanced die is rolled \\( n \\) times independently where \\( n \\geq 2 \\). Let \\( X \\) denote the number of times 6 dots show up and \\( Y \\) denote the number of times 5 dots show up in these \\( n \\) rolls. Find the probability distribution of \\( Z = X + Y \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-02c8f0a57bdf3ff01b6fbb6e3220c014e893426b43c8d41af2c59f319b0211b1 | A balanced die is rolled \( n \) times independently where \( n \geq 2 \). Let \( X \) denote the number of times 6 dots show up and \( Y \) denote the number of times 5 dots show up in these \( n \) rolls. What is the joint probability distribution of \( (X, Y) \)? | [
"P(X = x, Y = y) = \\frac{n!}{x! \\, y! \\, (n - x - y)!} \\left(\\frac{1}{6}\\right)^{x+y} \\left(\\frac{2}{3}\\right)^{n - x - y} \\text{ for } x, y \\geq 0 \\text{ and } x + y \\leq n"
] | [
"A balanced die is rolled \\( n \\) times independently where \\( n \\geq 2 \\). Let \\( X \\) denote the number of times 6 dots show up and \\( Y \\) denote the number of times 5 dots show up in these \\( n \\) rolls. What is the joint probability distribution of \\( (X, Y) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9db20485d48a8db9dac3f307063adbe918cfad6c36f45fb8a409cdef7eda54fe | A balanced six-sided die is rolled ten times. Add the total of all the numbers which were rolled immediately preceding those times a six is rolled. Call this total the score. Find the mean and variance of the score. | [
"\\dfrac{3395}{144}"
] | [
"A balanced six-sided die is rolled ten times. Add the total of all the numbers which were rolled immediately preceding those times a six is rolled. Call this total the score. Find the mean and variance of the score."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2a6efa7b3b27cd2aa972760b19100cfb289a8550365b1d0ff099bf4fce8070d4 | A ball $K$ of radius $r$ is touched from the outside by mutually equal balls of radius $R$. Two of these balls are tangent to each other. Additionally, for two balls $K_1$ and $K_2$ that are tangent to $K$ and tangent to each other, there exist two other balls that are tangent to $K_1$, $K_2$, and also to $K$. Determin... | [
"\\( R = r \\) and the number of balls tangent to \\( K \\) is 12."
] | [
"A ball $K$ of radius $r$ is touched from the outside by mutually equal balls of radius $R$. Two of these balls are tangent to each other. Additionally, for two balls $K_1$ and $K_2$ that are tangent to $K$ and tangent to each other, there exist two other balls that are tangent to $K_1$, $K_2$, and also to $K$. Det... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72a5b1fdcca04e75b5ee887494fe6e92912b5d97436af7071976dfdf6ccb236c | A ball bounces back up $\frac{2}{3}$ of the height from which it falls. If the ball is dropped from a height of $243$ cm, after how many bounces does the ball first rise less than $30$ cm? | [
"6"
] | [
"A ball bounces back up $\\frac{2}{3}$ of the height from which it falls. If the ball is dropped from a height of $243$ cm, after how many bounces does the ball first rise less than $30$ cm?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1cbaba27bad57faf54368d740de9416d9a4990a127ebd4f45e137825ed13669b | A ball bounces between two parallel walls while traversing the zigzag $ABCDEF$. The distance $AA'$ between the two walls is $5$ meters. The length of the zigzag is exactly equal to the sum of the distances $AA'$ and $A'F$. What is the length of the zigzag traversed by the ball? Note: The ball is considered as a geometr... | [
"65"
] | [
"A ball bounces between two parallel walls while traversing the zigzag $ABCDEF$. The distance $AA'$ between the two walls is $5$ meters. The length of the zigzag is exactly equal to the sum of the distances $AA'$ and $A'F$. What is the length of the zigzag traversed by the ball? Note: The ball is considered as a ge... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-c20da27410e4c52533c57106da571cdd8a9af4303ea350fcbfe4c1c71c50fc33 | A ball bounces up and down, reaching 80% of its previous height after each bounce. If it is dropped from a height of 2 feet, what is the total distance the ball will travel before coming to rest on the ground? | [
"20\n\nExplanation:\nPost 2 provides a solution for problem 2 (P2), which matches the problem we are looking at. The solution calculates the total distance the ball travels as an infinite geometric series and arrives at the answer of 20 feet."
] | [
"A ball bounces up and down, reaching 80% of its previous height after each bounce. If it is dropped from a height of 2 feet, what is the total distance the ball will travel before coming to rest on the ground?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4167264ea4a183eb261229a7b532017bd0346b24c935df80536ac075055f2571 | A ball is being heated up on a special stove where the rate of temperature change of the ball is proportional to the difference in the fourth powers of the temperatures of the ball and the stove. The stove is held at a constant 450K. If the ball is currently 270K and heating up at a rate of 17K/minute, how long will it... | [
"6.22"
] | [
"A ball is being heated up on a special stove where the rate of temperature change of the ball is proportional to the difference in the fourth powers of the temperatures of the ball and the stove. The stove is held at a constant 450K. If the ball is currently 270K and heating up at a rate of 17K/minute, how long wi... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5a2ceabeb4cadd77eeba85f3603f7e7594df79ec571a762c8f5484113e6d23d4 | A ball is dropped from 10 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot? | [
"4"
] | [
"A ball is dropped from 10 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61ff7db15f555c6f0460cccfe78471beecb43bdc0a52b33ef45d369f603e4404 | A ball is dropped from 1000 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot? | [
"10"
] | [
"A ball is dropped from 1000 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21acf002d03e09553fcc6aa11d55c813d4aacc353fbf4e91256d51fb7bf12804 | A ball is dropped from a height of 16 feet. Each time it drops, it rebounds 80% of the height from which it is falling. Find the total distance traveled after $x$ bounces. | [
"\\( 16 \\left(-5\\left( \\left(\\frac{4}{5} \\right)^{x+1} -1\\right)-1 \\right) \\)"
] | [
"A ball is dropped from a height of 16 feet. Each time it drops, it rebounds 80% of the height from which it is falling. Find the total distance traveled after $x$ bounces."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 8
} | 15 | 0 | 0 |
omr-ws-466515753d5fea8ede8d27254a9520ee5c0eaa7dfb878e4900c49dcd26d25995 | A ball is dropped from a height of 16 feet. Each time it drops, it rebounds 80% of the height from which it is falling. Find the total distance traveled in 15 bounces. | [
"\\( 16 \\left(-5\\left( \\left(\\frac{4}{5} \\right)^{16} -1\\right)-1 \\right) \\)"
] | [
"A ball is dropped from a height of 16 feet. Each time it drops, it rebounds 80% of the height from which it is falling. Find the total distance traveled in 15 bounces."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 10
} | 14 | 0 | 0 |
omr-ws-b5d7e67a666edd74f901cd6ebbb2af783931580c572e97e361f88e740c438d09 | A ball is dropped from a height of 20 feet onto a perfect flat surface. Each time the ball hits the surface after falling \(h\) feet it bounces a distance of \(\frac{2}{5}h\) feet. Find the total distance the ball travels up and down if it is allowed to continue bouncing indefinitely. | [
"\\(\\frac{140}{3}\\)"
] | [
"A ball is dropped from a height of 20 feet onto a perfect flat surface. Each time the ball hits the surface after falling \\(h\\) feet it bounces a distance of \\(\\frac{2}{5}h\\) feet. Find the total distance the ball travels up and down if it is allowed to continue bouncing indefinitely."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-2d22cc1ed87cc1ff3a67328e185f8612e6086dd8574bda5ec98544d8a4d6c0a0 | A ball is dropped from a height of 20 meters, 12 meters away from the top of a 20-meter lamppost. The ball's shadow, caused by the light at the top of the lamppost, is moving along the level ground. How fast is the shadow moving 1 second after the ball is released? | [
"97.96 meters per second."
] | [
"A ball is dropped from a height of 20 meters, 12 meters away from the top of a 20-meter lamppost. The ball's shadow, caused by the light at the top of the lamppost, is moving along the level ground. How fast is the shadow moving 1 second after the ball is released?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 22
} | 35 | 0 | 0 |
omr-ws-624795a828ffeb0d56cc01d9a173ff231d966d496456a0b3cace3c4289157ffd | A ball is dropped from a height of 250 feet. Each time it hits the ground, it bounces back up to half the height it fell. Find the total distance traveled by the ball when it hits the ground for the fourth time. | [
"687.5"
] | [
"A ball is dropped from a height of 250 feet. Each time it hits the ground, it bounces back up to half the height it fell. Find the total distance traveled by the ball when it hits the ground for the fourth time."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-5a837098aa14c79ea4b447766bac13e02ef67fc9f5a65e79b52e6d2b5448410e | A ball is dropped from a height of 4 meters. Each time it strikes the pavement after falling from a height of \(h\) meters, it rebounds to a height of \(0.6h\) meters. Find the total number of seconds that the ball travels. (Hint: A freely falling ball travels \(4.9t^2\) meters in \(t\) seconds, so it will fall \(h\) m... | [
"5.33"
] | [
"A ball is dropped from a height of 4 meters. Each time it strikes the pavement after falling from a height of \\(h\\) meters, it rebounds to a height of \\(0.6h\\) meters. Find the total number of seconds that the ball travels. (Hint: A freely falling ball travels \\(4.9t^2\\) meters in \\(t\\) seconds, so it will... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-386dcab944f14dda72c0475b1ce174f126e9ec8543578fb40c0d38762479b8e5 | A ball is dropped from a height of 4096 ft. Each time it bounces, it rebounds to 1/16 of its previous height. Find the distance the ball has traveled in feet if it is caught at the highest point after the third bounce. | [
"4641"
] | [
"A ball is dropped from a height of 4096 ft. Each time it bounces, it rebounds to 1/16 of its previous height. Find the distance the ball has traveled in feet if it is caught at the highest point after the third bounce."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-e1ddb6c579e2c011c3a40535d03f33df833380a7dbb9800416d232816b1f5fe5 | A ball is dropped from a height of 5 ft on a surface that causes it to bounce up 10% of the distance that it fell. If the probability that the ball is on its way up at any point in time is represented by the fraction $ \frac{m}{n}$ in lowest terms, where $ m$ and $ n$ are positive integers, what is $ m+n$? | [
"12"
] | [
"A ball is dropped from a height of 5 ft on a surface that causes it to bounce up 10% of the distance that it fell. If the probability that the ball is on its way up at any point in time is represented by the fraction $ \\frac{m}{n}$ in lowest terms, where $ m$ and $ n$ are positive integers, what is $ m+n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 5
} | 25 | 0 | 0 |
omr-ws-3046ee8d64369019fbbb3331d355039a0fb4e07b5dfb6d2575ac6b10a785ba70 | A ball is dropped from a height of 5 ft on a surface that causes it to bounce up 10% of the distance that it fell. If the probability that the ball is on its way up at any point in time is represented by the fraction m/n in lowest terms, where m and n are positive integers, what is m+n? | [
"12"
] | [
"A ball is dropped from a height of 5 ft on a surface that causes it to bounce up 10% of the distance that it fell. If the probability that the ball is on its way up at any point in time is represented by the fraction m/n in lowest terms, where m and n are positive integers, what is m+n?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-3138e17c09a5d12ddc97fa297f9681afa88b9d9954f7c55d88a6ba4aada0c3c0 | A ball is dropped from a height of 5 ft on a surface that causes it to bounce up to 10% of the distance it fell. If the probability that the ball is on its way up at any point in time is represented by the fraction \( \frac{m}{n} \) in lowest terms, where \( m \) and \( n \) are positive integers, what is \( m+n \)? | [
"12"
] | [
"A ball is dropped from a height of 5 ft on a surface that causes it to bounce up to 10% of the distance it fell. If the probability that the ball is on its way up at any point in time is represented by the fraction \\( \\frac{m}{n} \\) in lowest terms, where \\( m \\) and \\( n \\) are positive integers, what is \... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-b244da065286ecb22b3284545d9223deda6fa2400e0294793a2b9dcc18bf66ea | A ball is launched on a flat, square table with a slope of $\frac{11}{12}$ from the bottom left corner into the interior. When the ball reaches the edge of the table, it bounces back at an angle that is symmetric with the incidence angle. What is the total distance it travels before it reaches a corner for the first ti... | [
"\\( s\\sqrt{265} \\)"
] | [
"A ball is launched on a flat, square table with a slope of $\\frac{11}{12}$ from the bottom left corner into the interior. When the ball reaches the edge of the table, it bounces back at an angle that is symmetric with the incidence angle. What is the total distance it travels before it reaches a corner for the fi... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d090a64f7a32d12835b12727390c99f0ac57b1efe51f93a4350861cd23c6c850 | A ball is moving in 11 positions, marked as \( 0, 1, 2, \ldots, 10 \) in that order. The probabilities that it moves forward or backward are both \( \frac{1}{2} \). When the ball is at 0, it can only move forward to 1. If the ball initially starts from 0, find the expected number of steps it takes to arrive at 10 for t... | [
"100"
] | [
"A ball is moving in 11 positions, marked as \\( 0, 1, 2, \\ldots, 10 \\) in that order. The probabilities that it moves forward or backward are both \\( \\frac{1}{2} \\). When the ball is at 0, it can only move forward to 1. If the ball initially starts from 0, find the expected number of steps it takes to arrive ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c9d68fb78cc3a9cc6844f9c2fd7c7c4fe8c899c43f7563e4353bcbde9972bcf7 | A ball is placed on flat ground with a marked point. The ball is rotated along a closed polygon such that it returns to its original position. Let $N$ be the total number of distinct positions the marked point can have relative to the ball after the rotation, considering all possible closed polygonal paths. Find the re... | [
"1"
] | [
"A ball is placed on flat ground with a marked point. The ball is rotated along a closed polygon such that it returns to its original position. Let $N$ be the total number of distinct positions the marked point can have relative to the ball after the rotation, considering all possible closed polygonal paths. Find t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-5e34f5b6f4f710002ce56c1c2947c862c2451175933c3408b2a30fe8342e7f84 | A ball is propelled from corner \( A \) of a square snooker table with side length 2 metres. The ball bounces off the cushions \( BC \), \( CD \), and \( DA \) before going into the pocket at \( B \). The total distance travelled by the ball is \( \sqrt{k} \) metres. What is the value of \( k \)? | [
"52"
] | [
"A ball is propelled from corner \\( A \\) of a square snooker table with side length 2 metres. The ball bounces off the cushions \\( BC \\), \\( CD \\), and \\( DA \\) before going into the pocket at \\( B \\). The total distance travelled by the ball is \\( \\sqrt{k} \\) metres. What is the value of \\( k \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-6a4e37cab2b532156417dd8b7b06ac562b4692742e4eeedccfc50af801782053 | A ball is released from a height of 1024 meters. Each time it touches the ground, it bounces back to half of its previous height. How many meters will the ball cover before it reaches a height of 4 meters or less? | [
"3056"
] | [
"A ball is released from a height of 1024 meters. Each time it touches the ground, it bounces back to half of its previous height. How many meters will the ball cover before it reaches a height of 4 meters or less?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-2bd5f1d6be6148d170085efb5a2df271fae0b9369a88caff1201bcdfacef43a6 | A ball is sent at a 45° angle from a corner of a rectangular billiard table with sides of integer lengths \( m \) and \( n \) (where \( m \) and \( n \) are relatively prime). Calculate the total number of bounces the ball will make before it is pocketed in another corner. | [
"m + n - 2"
] | [
"A ball is sent at a 45° angle from a corner of a rectangular billiard table with sides of integer lengths \\( m \\) and \\( n \\) (where \\( m \\) and \\( n \\) are relatively prime). Calculate the total number of bounces the ball will make before it is pocketed in another corner."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad0f92c7d92775258a0c741e9ade294b07173fb8386d89d683cbe4cb36af8193 | A ball is sent in a straight line on a $2 \times 8$ billiards table from its center and stops at a corner after traveling 29 meters. How many times did the ball bounce off the borders of the table? The ball bounces according to the law of reflection. | [
"12"
] | [
"A ball is sent in a straight line on a $2 \\times 8$ billiards table from its center and stops at a corner after traveling 29 meters. How many times did the ball bounce off the borders of the table? The ball bounces according to the law of reflection."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 8,
"tir": 8
} | 51 | 0 | 0 |
omr-ws-d46b6378dd9a2550ff4824b7b81c724ec5fe6a17b91648b43098cb080334ab0f | A ball is shot at an angle from the corner of a square-shaped pool table and bounces off exactly 99 edges before hitting a corner for the first time. For how many unique angles will this be true? | [
"38"
] | [
"A ball is shot at an angle from the corner of a square-shaped pool table and bounces off exactly 99 edges before hitting a corner for the first time. For how many unique angles will this be true?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5bd6fe4d6e2e07add7485aaafb5907c9c74d760272ab23e37ceceb4f4b11a71f | A ball is shot from one corner of a 6 foot by 10 foot pool table at a 45 degree angle. How many times will it bounce off the sides before landing in a corner pocket? | [
"6"
] | [
"A ball is shot from one corner of a 6 foot by 10 foot pool table at a 45 degree angle. How many times will it bounce off the sides before landing in a corner pocket?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-71bf2b277a124a748575a9615b79b3e7781c5ea1abc2f6c4b5831a62b857ea9d | A ball is thrown downward at 10 feet per second from the top of a 100-foot building. Calculate the exact time in seconds when the ball hits the ground. Express your answer as a simplified radical expression. | [
"\\dfrac{5(\\sqrt{65} - 1)}{16}"
] | [
"A ball is thrown downward at 10 feet per second from the top of a 100-foot building. Calculate the exact time in seconds when the ball hits the ground. Express your answer as a simplified radical expression."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-359ab6b9a755aacda0d9ef114218802a2abd60b4dab45398950f7ed666236ab9 | A ball is thrown from the center of a pool table that is 8 meters long and 2 meters wide. The ball travels in a straight line for 29 meters and stops at a corner of the table. How many times did the ball hit the edges of the table? Note: The ball rebounds off the edge such that the angles of incidence and reflection ar... | [
"7"
] | [
"A ball is thrown from the center of a pool table that is 8 meters long and 2 meters wide. The ball travels in a straight line for 29 meters and stops at a corner of the table. How many times did the ball hit the edges of the table? Note: The ball rebounds off the edge such that the angles of incidence and reflecti... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4461d12ebcdef92958357d5f3b1616699d2ece7f519fb319a38d9ce6b596f96b | A ball is thrown into the air with an initial velocity of 30 feet per second from a height of 5 feet. After how many seconds will the ball hit the ground? | [
"2"
] | [
"A ball is thrown into the air with an initial velocity of 30 feet per second from a height of 5 feet. After how many seconds will the ball hit the ground?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-b4230f75627f3f79fa901c8f48014c31e02bcd74df854d74ea9c4146afb1cd97 | A ball is thrown up a hill that makes an angle of 45 degrees with the horizontal. The ball lands 100 feet up the hill, and its trajectory is described by the parabola \( y = -x^2 + ax \). Find the value of \( a \). | [
"100"
] | [
"A ball is thrown up a hill that makes an angle of 45 degrees with the horizontal. The ball lands 100 feet up the hill, and its trajectory is described by the parabola \\( y = -x^2 + ax \\). Find the value of \\( a \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f6efbb94e2d3010ab54c546885a2568978add54add88e018d7ab17007569e84 | A ball is thrown upward from a second floor window of a building. It takes 3 seconds for it to reach the ground below. It strikes the ground with a velocity of -57 ft/sec. Find the ball's initial velocity and the height from which it was thrown. Acceleration due to gravity is -32 ft/sec^2. | [
"Initial velocity = 39 ft/sec, Height = 27 ft."
] | [
"A ball is thrown upward from a second floor window of a building. It takes 3 seconds for it to reach the ground below. It strikes the ground with a velocity of -57 ft/sec. Find the ball's initial velocity and the height from which it was thrown. Acceleration due to gravity is -32 ft/sec^2."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 32
} | 37 | 0 | 0 |
omr-ws-618765bec04231495d6b86a78e26038a6edb7f12849fa0b88542ed37b9bb78e1 | A ball is thrown upward with an initial velocity of 64 ft/sec from a height of 80 ft. Determine the time at which the ball reaches its maximum height and the maximum height above the base it reaches. | [
"\\( t = 2 \\text{ seconds}, \\text{ maximum height} = 144 \\text{ feet} \\)"
] | [
"A ball is thrown upward with an initial velocity of 64 ft/sec from a height of 80 ft. Determine the time at which the ball reaches its maximum height and the maximum height above the base it reaches."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 18
} | 19 | 0 | 0 |
omr-ws-9db97e961fdb3c73bf60e72880754433c0a2916581bce67690d4cce3493e677e | A ball is thrown upward with an initial velocity of 64 ft/sec from a height of 80 ft. Determine when the ball strikes the ground and the velocity of the ball at that instant. | [
"\\( t = 5 \\) seconds, \\( v = -96 \\) ft/sec"
] | [
"A ball is thrown upward with an initial velocity of 64 ft/sec from a height of 80 ft. Determine when the ball strikes the ground and the velocity of the ball at that instant."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 23
} | 25 | 0 | 0 |
omr-ws-2320f88c6f0f8d515f3b98afecab6adb9a34bc94bf81c557d492e494b9b2d35a | A ball is thrown upward with an initial velocity of 64 ft/sec from a height of 80 ft. Find the velocity and height of the ball \( t \) seconds after it is released. | [
"\\( v(t) = -32t + 64 \\) and \\( y(t) = -16t^2 + 64t + 80 \\)"
] | [
"A ball is thrown upward with an initial velocity of 64 ft/sec from a height of 80 ft. Find the velocity and height of the ball \\( t \\) seconds after it is released."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 21
} | 23 | 0 | 0 |
omr-ws-4ce3dc86728ce17fd6b9ec6967caca3b3d72689a4d3c5d39167d461aade87763 | A ball of ice melts in such a way that the rate of melting is proportional to the amount of ice at that instant. If half of the quantity of ice melts in 30 minutes, calculate the fraction of the original volume of the ice ball that remains after 90 minutes from the start of melting. | [
"\\dfrac{1}{8}"
] | [
"A ball of ice melts in such a way that the rate of melting is proportional to the amount of ice at that instant. If half of the quantity of ice melts in 30 minutes, calculate the fraction of the original volume of the ice ball that remains after 90 minutes from the start of melting."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3a832a1427f5c0888b53077b00bf479936dcd755fdd088c1eaca3158e965d283 | A ball of radius $3$ is dropped from a random point from the ceiling of a room shaped like a cube of side length $18$. A right circular cone of radius $2\sqrt{3}$ and height $6$ is placed at the center of the room. Find the probability that the ball hits the cone before hitting the ground. | [
"\\(\\frac{\\left(2\\sqrt{3} + \\frac{3}{2}\\right)^2 \\pi}{144}\\)"
] | [
"A ball of radius $3$ is dropped from a random point from the ceiling of a room shaped like a cube of side length $18$. A right circular cone of radius $2\\sqrt{3}$ and height $6$ is placed at the center of the room. Find the probability that the ball hits the cone before hitting the ground."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-6111d645bf96c2048b2a2b4fe32a49e47e28dec5a9f38b82757367ec53f301c4 | A ball of radius 1 is in the corner of a room, tangent to two walls and the floor. A smaller ball is also in the corner, tangent to both walls, the floor, and the larger ball. What is the radius of the smaller ball? | [
"\\(2 - \\sqrt{3}\\)"
] | [
"A ball of radius 1 is in the corner of a room, tangent to two walls and the floor. A smaller ball is also in the corner, tangent to both walls, the floor, and the larger ball. What is the radius of the smaller ball?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-dd68f0c9b1d65969747d5a3c7ac24858145d572049e986b5aeb2f1ed34e8a675 | A ball of radius 1 is placed in the corner of a room, tangent to the floor and walls. A sequence of additional balls is placed such that each ball is tangent to the walls, the floor, and the previous ball. What is the sum of the radii of each of the resulting spheres? | [
"\\(\\frac{1+\\sqrt{3}}{2}\\)"
] | [
"A ball of radius 1 is placed in the corner of a room, tangent to the floor and walls. A sequence of additional balls is placed such that each ball is tangent to the walls, the floor, and the previous ball. What is the sum of the radii of each of the resulting spheres?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 30
} | 77 | 0 | 0 |
omr-ws-e403fbf3808fe0078d3ede7ac807949b7b6fae5c9f1b94c5175bb4939e6f77c7 | A ball of radius 1 is placed in the corner of a room, tangent to the floor and walls. A sequence of additional balls is placed such that each ball is tangent to the walls, the floor, and the previous ball. What is the sum of the surface areas of each of the resulting spheres? | [
"\\(\\left(2 + \\frac{4}{\\sqrt{3}}\\right)\\pi\\)"
] | [
"A ball of radius 1 is placed in the corner of a room, tangent to the floor and walls. A sequence of additional balls is placed such that each ball is tangent to the walls, the floor, and the previous ball. What is the sum of the surface areas of each of the resulting spheres?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 30
} | 77 | 0 | 0 |
omr-ws-9f6fbabcd9f891e128616a349e5dda3d5f892f2324b6bc2f8484a74580d3570e | A ball of radius 1 is placed in the corner of a room, tangent to the floor and walls. A sequence of additional balls is placed such that each ball is tangent to the walls, the floor, and the previous ball. What is the sum of the volumes of each of the resulting spheres? | [
"\\(\\frac{2}{15}(3+\\sqrt{3})\\pi\\)"
] | [
"A ball of radius 1 is placed in the corner of a room, tangent to the floor and walls. A sequence of additional balls is placed such that each ball is tangent to the walls, the floor, and the previous ball. What is the sum of the volumes of each of the resulting spheres?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 6
} | 12 | 0 | 0 |
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