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-c0a147a238c833a71060d39ea2b6b54aed323d188c974af2449d32f6bcb54191 | You have 50 cards, each with numbers from 1 to 100 written on each side. The cards are placed on a table, and you can flip each card exactly once to maximize the sum of the numbers on the upper sides. What is the largest sum you can always ensure? | [
"2525"
] | [
"You have 50 cards, each with numbers from 1 to 100 written on each side. The cards are placed on a table, and you can flip each card exactly once to maximize the sum of the numbers on the upper sides. What is the largest sum you can always ensure?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ac8979771c0618ec1b1a974a6743317cba0d2a3d842833c51650817e2d00b729 | You have 50 dimes and 20 quarters. What percent of the value of your money is in quarters? | [
"50\\%"
] | [
"You have 50 dimes and 20 quarters. What percent of the value of your money is in quarters?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-95d5d62ed226b8b751dbc17ac301db07ce40896b4b23a20482e8a6a33e8dbb26 | You have 6 movies numbered 1-6. You roll a dice to determine which movie to watch next until all of them are watched. If you roll a number that corresponds to a movie you have already seen, you pick the nearest movie on the list. If there are two nearest movies you haven't seen, you pick the one with the highest number... | [
"\\dfrac{1}{32}"
] | [
"You have 6 movies numbered 1-6. You roll a dice to determine which movie to watch next until all of them are watched. If you roll a number that corresponds to a movie you have already seen, you pick the nearest movie on the list. If there are two nearest movies you haven't seen, you pick the one with the highest n... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-84ce6b9bda49d3fe272b57e0adbd08933307f17e7d6170199b6ae89bf7be1b65 | You have 6 points in a plane, with no three points collinear. Each segment connecting two points can be either green or yellow. Let \( N \) be the number of distinct sets of four points \( \{Q, R, S, T\} \) such that the segments \( QR, RS, ST, TQ \) are all the same color. Calculate the minimum possible value of \( N ... | [
"3"
] | [
"You have 6 points in a plane, with no three points collinear. Each segment connecting two points can be either green or yellow. Let \\( N \\) be the number of distinct sets of four points \\( \\{Q, R, S, T\\} \\) such that the segments \\( QR, RS, ST, TQ \\) are all the same color. Calculate the minimum possible v... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-ec93042ec7b19cd26aaee4b19ff340b09f5160f18c52d00d4972a6a3b78fe708 | You have 7 distinct rectangles, each with an area of \( \frac{1}{7} \). These rectangles can be arranged to form a 1 by 1 square. What are the dimensions of the rectangles? | [
"\\left(1, \\frac{1}{7}\\right), \\left(\\frac{1}{2}, \\frac{2}{7}\\right), \\left(\\frac{1}{3}, \\frac{3}{7}\\right), \\left(\\frac{1}{4}, \\frac{4}{7}\\right), \\left(\\frac{1}{5}, \\frac{5}{7}\\right), \\left(\\frac{1}{6}, \\frac{6}{7}\\right), \\left(\\frac{1}{7}, 1\\right)"
] | [
"You have 7 distinct rectangles, each with an area of \\( \\frac{1}{7} \\). These rectangles can be arranged to form a 1 by 1 square. What are the dimensions of the rectangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-f7ad28ff8d1e915aef1c3e63f90959cf3cc52929eca6904e924ebc37baf65f9d | You have 8 batteries, 4 of which are "good" and 4 are "bad." A device requires 6 batteries to operate, but only 4 of them need to be good. You can test the device by picking 6 batteries and attempting to turn it on, with no other way to check battery quality. In the worst-case scenario, how many tests are needed to ens... | [
"4"
] | [
"You have 8 batteries, 4 of which are \"good\" and 4 are \"bad.\" A device requires 6 batteries to operate, but only 4 of them need to be good. You can test the device by picking 6 batteries and attempting to turn it on, with no other way to check battery quality. In the worst-case scenario, how many tests are need... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f777c7abdc7624ed625263178e97309995b3e7384a22257bfdbe50cade207b43 | You have 8 different rings. Let \( n \) be the number of possible arrangements of 5 rings on the four fingers of one hand (each finger has zero or more rings, and the order matters). Find the three leftmost non-zero digits of \( n \). | [
"376"
] | [
"You have 8 different rings. Let \\( n \\) be the number of possible arrangements of 5 rings on the four fingers of one hand (each finger has zero or more rings, and the order matters). Find the three leftmost non-zero digits of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-6c097e858c4926a26fc3ebc03a174fde460b9882c29d05ea75686eea5eb24aa3 | You have 8 different rings. Let \(n\) be the number of possible arrangements of 5 rings on the four fingers of one hand (each finger has zero or more rings, and the order matters). Find the three leftmost non-zero digits of \(n\). | [
"376"
] | [
"You have 8 different rings. Let \\(n\\) be the number of possible arrangements of 5 rings on the four fingers of one hand (each finger has zero or more rings, and the order matters). Find the three leftmost non-zero digits of \\(n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-39c09ba18dac123ce8e0803e8dfcf40570926412d9fef68cad3c19f9c08c8968 | You have 80 coins, one of which is lighter than the others. Using a balance scale only four times, determine how to identify the lighter coin. | [
"4"
] | [
"You have 80 coins, one of which is lighter than the others. Using a balance scale only four times, determine how to identify the lighter coin."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 6
} | 10 | 0 | 0 |
omr-ws-bbce65d76f59a03b63e54624730270403b9ac2f2dbbcda5f15d65c12066906de | You have 850 eggs and 1650 cups of cream. Each quart of Creamy Vanilla ice cream requires 2 eggs and 3 cups of cream, and each quart of Continental Mocha ice cream requires 1 egg and 3 cups of cream. How many quarts of each flavor should you make to use up all the eggs and cream? | [
"300 quarts of Creamy Vanilla and 250 quarts of Continental Mocha."
] | [
"You have 850 eggs and 1650 cups of cream. Each quart of Creamy Vanilla ice cream requires 2 eggs and 3 cups of cream, and each quart of Continental Mocha ice cream requires 1 egg and 3 cups of cream. How many quarts of each flavor should you make to use up all the eggs and cream?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 26
} | 39 | 0 | 0 |
omr-ws-c40136018e892f1a8ca92f6bef1d643ab4a65a2ea88cd1350553921968f503a5 | You have Blue blocks, Green blocks, and Red blocks. Blue blocks and Green blocks are 2 inches thick, and Red blocks are 1 inch thick. In how many ways can you stack the blocks into a vertical column that is exactly 12 inches high? | [
"2731"
] | [
"You have Blue blocks, Green blocks, and Red blocks. Blue blocks and Green blocks are 2 inches thick, and Red blocks are 1 inch thick. In how many ways can you stack the blocks into a vertical column that is exactly 12 inches high?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-9c9e8750cb72ed04f5953c87920208fc3b84a71ff2b588049308a35ba63eaf5b | You have N beans. You divide your beans into 2 equal groups and give the remainder (N % 2) to your parrot. Then you divide the remaining beans (N - (N % 2)) into 6 groups and give the remainder to your parrot. You repeat this process with 30, 210, and 2310 groups. What is the maximum number of beans you can give to you... | [
"2309"
] | [
"You have N beans. You divide your beans into 2 equal groups and give the remainder (N % 2) to your parrot. Then you divide the remaining beans (N - (N % 2)) into 6 groups and give the remainder to your parrot. You repeat this process with 30, 210, and 2310 groups. What is the maximum number of beans you can give t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 5,
"tir": 1
} | 35 | 0 | 0 |
omr-ws-39b48a68c4d0d68eeb8e370a9bc76648cf77d35c2c47d20218064377c7b900f9 | You have \$100 and can play either of two gambling machines 20 times. Each machine costs \$10 to play. One machine has a \(\frac{1}{2}\) chance of winning \$24, and the other has a \(\frac{1}{3}\) chance of winning \$24. You do not know which machine is which. What is the optimal strategy to maximize your expected earn... | [
"3"
] | [
"You have \\$100 and can play either of two gambling machines 20 times. Each machine costs \\$10 to play. One machine has a \\(\\frac{1}{2}\\) chance of winning \\$24, and the other has a \\(\\frac{1}{3}\\) chance of winning \\$24. You do not know which machine is which. What is the optimal strategy to maximize you... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-b3c057c50832e99dd234a8e92e87a179cf8b9acd59d0fa746316404a5948e07c | You have \( n \) identically-looking coins, but one of them has a different weight from the others. Using a balance scale that can compare the weights of two groups of coins, determine which coin has the unique weight and whether it is heavier or lighter than the others, using the smallest number of weighings. | [
"\\lceil \\log_3(2n) \\rceil"
] | [
"You have \\( n \\) identically-looking coins, but one of them has a different weight from the others. Using a balance scale that can compare the weights of two groups of coins, determine which coin has the unique weight and whether it is heavier or lighter than the others, using the smallest number of weighings."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-f9111628dfbb9596dbfee5734250d0b74ef657f706d5e0d2bcc9a67a0c2ba6fe | You have a $10 \times 10$ board with black and white chips. Initially, all chips are black. You can change the color of one chip and also change the colors of all adjacent chips (sharing a side). Determine if it is possible to change the color of all chips to white. | [
"\\text{No}"
] | [
"You have a $10 \\times 10$ board with black and white chips. Initially, all chips are black. You can change the color of one chip and also change the colors of all adjacent chips (sharing a side). Determine if it is possible to change the color of all chips to white."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-e3462a6dff7265d7d40de5218aae327a70094d26f669268e5b752ac51ea69433 | You have a $12 \times 12$ room and a $16 \times 9$ carpet. How can you cut the carpet into 2 parts so that it covers the entire $12 \times 12$ room floor? | [
"The carpet can be cut into two parts using the zig-zag method described above to cover the entire $12 \\times 12$ room floor."
] | [
"You have a $12 \\times 12$ room and a $16 \\times 9$ carpet. How can you cut the carpet into 2 parts so that it covers the entire $12 \\times 12$ room floor?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 5
} | 10 | 0 | 0 |
omr-ws-592693d2a3c5dedd43c28d6f5728c48397ca3b6b8932cf80171dfe6bad887678 | You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the re... | [
"1"
] | [
"You have a $3 \\times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-54f01392cebbf4c1d28b72e70bb7a13dd7450e9cce2de382b8e8b291f7ce4ff8 | You have a 1 by 2024 grid of squares in a column, with vertices labelled from $(0,0)$ to $(1,2024)$. Place a weed at $(0,0)$. A weed is placed at coordinates $(x,y)$ with a $50\%$ probability if and only if exactly one of the vertices $(x-1, y)$ or $(x, y-1)$ has a weed on it; otherwise, the attempt fails. Placement at... | [
"18207"
] | [
"You have a 1 by 2024 grid of squares in a column, with vertices labelled from $(0,0)$ to $(1,2024)$. Place a weed at $(0,0)$. A weed is placed at coordinates $(x,y)$ with a $50\\%$ probability if and only if exactly one of the vertices $(x-1, y)$ or $(x, y-1)$ has a weed on it; otherwise, the attempt fails. Placem... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-fd0f677226e12b765139644f6724b76913479f5fe0150d1f989afc57b3337d03 | You have a 10-sided die with numbers ranging from 1 to 10, each number coming up with equal probability. You roll the die repeatedly and sum the numbers until the sum exceeds 100. What is the expected value of your sum? | [
"105.5"
] | [
"You have a 10-sided die with numbers ranging from 1 to 10, each number coming up with equal probability. You roll the die repeatedly and sum the numbers until the sum exceeds 100. What is the expected value of your sum?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-0a80d5bf7a8731ed96fbefc30d6087d4ef86cfd50ad47f9a604c39792def47a7 | You have a 100 x 100 chessboard colored in the standard way with alternating colors going down and across. You can choose any rectangle within the dimensions of the chessboard and each individual square will flip to the opposite color. Find the minimum number of moves it takes to change the entire board to one color. | [
"100"
] | [
"You have a 100 x 100 chessboard colored in the standard way with alternating colors going down and across. You can choose any rectangle within the dimensions of the chessboard and each individual square will flip to the opposite color. Find the minimum number of moves it takes to change the entire board to one col... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-5458cf51ce425f25e6c8c5caf341f2b9f7159fd6f7652ab08b13cd17f629a5b1 | You have a 100 x 100 chessboard colored in the standard way with alternating colors. You can choose any rectangle within the dimensions of the chessboard and each individual square will flip to the opposite color. Find the minimum number of moves it takes to change the entire board to one color. | [
"100"
] | [
"You have a 100 x 100 chessboard colored in the standard way with alternating colors. You can choose any rectangle within the dimensions of the chessboard and each individual square will flip to the opposite color. Find the minimum number of moves it takes to change the entire board to one color."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-f06d13941353763a0aa8a64fc8324702e1ae70837362bba79d4ba04fffc1881d | You have a 17 x 17 chessboard with a coin on every square, initially all with their faces up. At each step, you can flip 5 consecutive coins in any row, column, or diagonal. Determine the minimum number of steps required to flip all the coins to their number sides up, if it is possible. If it is not possible, enter -1. | [
"-1"
] | [
"You have a 17 x 17 chessboard with a coin on every square, initially all with their faces up. At each step, you can flip 5 consecutive coins in any row, column, or diagonal. Determine the minimum number of steps required to flip all the coins to their number sides up, if it is possible. If it is not possible, ente... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-f2cd3eb1a0b818155d5ae012374b8dcb9ec349949a384d72bc1ff4f836a6241d | You have a 2-meter long string. You choose a point along the string uniformly at random and make a cut, discarding the shorter section. If you still have 0.5 meters or more of string, you repeat the process. You stop once you have less than 0.5 meters of string. On average, how many cuts will you make before stopping? | [
"2"
] | [
"You have a 2-meter long string. You choose a point along the string uniformly at random and make a cut, discarding the shorter section. If you still have 0.5 meters or more of string, you repeat the process. You stop once you have less than 0.5 meters of string. On average, how many cuts will you make before stopp... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12cdc682aa592c1f2316e2c9caa4b05596b373ea931963f7ed1a87985369117f | You have a 2-meter long string. You choose a point along the string uniformly at random and make a cut. You discard the shorter section. If you still have 0.5 meters or more of string, you repeat. You stop once you have less than 0.5 meters of string. On average, how many cuts will you make before stopping? | [
"8 - 4 \\ln 2"
] | [
"You have a 2-meter long string. You choose a point along the string uniformly at random and make a cut. You discard the shorter section. If you still have 0.5 meters or more of string, you repeat. You stop once you have less than 0.5 meters of string. On average, how many cuts will you make before stopping?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-199487e7f3145302c9245eb4c1e4bb637e1b78a1bdf0bb536059444be06a8139 | You have a 4-digit whole number that is a perfect square. Another number is formed by adding 1 to the unit's digit, subtracting 1 from the ten's digit, adding 1 to the hundred's digit, and subtracting 1 from the thousand's digit. If the resulting number is also a perfect square, find the original number. Is it unique? | [
"3025"
] | [
"You have a 4-digit whole number that is a perfect square. Another number is formed by adding 1 to the unit's digit, subtracting 1 from the ten's digit, adding 1 to the hundred's digit, and subtracting 1 from the thousand's digit. If the resulting number is also a perfect square, find the original number. Is it uni... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-33de01ba93b065862fd101cf4d35bfbd273c4923b457146f16a87dd7daa235d3 | You have a 4x4 grid (16 squares) that must be colored either red, blue, or green. The conditions are: 1. All the boxes in column 1 are the same color. 2. All the boxes in row 1 are red. 3. There must be an even number of blue squares. 4. There must be an even number of green squares. 5. Green squares cannot share a com... | [
"2"
] | [
"You have a 4x4 grid (16 squares) that must be colored either red, blue, or green. The conditions are:\n1. All the boxes in column 1 are the same color.\n2. All the boxes in row 1 are red.\n3. There must be an even number of blue squares.\n4. There must be an even number of green squares.\n5. Green squares cannot s... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-6b8ba5011db0b75cbd701407468147fb87b7571e50d2769c515cc1b8efd9fb2f | You have a Monopoly board, an $11 \times 11$ square grid with the $9 \times 9$ internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by rolling t... | [
"\\dfrac{1}{72}"
] | [
"You have a Monopoly board, an $11 \\times 11$ square grid with the $9 \\times 9$ internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by ro... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-6e4f43b1f328a206002db31e4f3c3070434ac6924d00499e38b907add8f06996 | You have a Monte Carlo algorithm that outputs either YES or NO, with a probability of 2/3 of being correct. You run this algorithm \( k \) times, where \( k \) is a large odd number, and return the most frequent output (YES or NO) from these \( k \) runs. Calculate the value of \( k \) such that the probability of the ... | [
"45"
] | [
"You have a Monte Carlo algorithm that outputs either YES or NO, with a probability of 2/3 of being correct. You run this algorithm \\( k \\) times, where \\( k \\) is a large odd number, and return the most frequent output (YES or NO) from these \\( k \\) runs. Calculate the value of \\( k \\) such that the probab... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-e8644400a0436a7ff040c0b947b30dbda89014e6d8bb44d25355ff4e82f18bd2 | You have a \(300 \times 300\) board and pieces consisting of three unit squares forming an L-shape. Determine the minimum number \(n\) of painted unit squares such that no three painted squares form an L-piece, and adding any additional painted square will create an L-piece. | [
"30000"
] | [
"You have a \\(300 \\times 300\\) board and pieces consisting of three unit squares forming an L-shape. Determine the minimum number \\(n\\) of painted unit squares such that no three painted squares form an L-piece, and adding any additional painted square will create an L-piece."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-d3edc6f0402edcf600a853ca338424e7990d65905e930076ac1cb13aaff05143 | You have a bag containing 1024 balls numbered from 1 to 1024, with 17 of them being prize-winning balls. You can choose a set of at least 17 balls and ask a judge if the set contains all 17 prize-winning balls, who will reply with a yes or no. Determine the minimum number of attempts needed to guarantee that you can id... | [
"10"
] | [
"You have a bag containing 1024 balls numbered from 1 to 1024, with 17 of them being prize-winning balls. You can choose a set of at least 17 balls and ask a judge if the set contains all 17 prize-winning balls, who will reply with a yes or no. Determine the minimum number of attempts needed to guarantee that you c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-335d8c31671a9871df26015d7d21d2cc6ff94bfa37836e4df2fae08b99a24d39 | You have a bag with 2 red balls and \( n \) green balls. On each turn, you remove a random ball from the bag, then remove a green ball. The process stops when there are no red balls left or the number of red balls is greater than or equal to the number of green balls. Let \( P(n) \) be the probability that the process ... | [
"1"
] | [
"You have a bag with 2 red balls and \\( n \\) green balls. On each turn, you remove a random ball from the bag, then remove a green ball. The process stops when there are no red balls left or the number of red balls is greater than or equal to the number of green balls. Let \\( P(n) \\) be the probability that the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-a4412b3580622b6a2747b8ca873ac4879de26186e9ef5945344ad7be59219506 | You have a birthday cake and you want to divide it into several parts so that it can be distributed equally among either $m$ or $n$ guests. What is the minimal number of parts you need to divide the cake into to ensure that it can be distributed equally regardless of whether $m$ or $n$ guests attend? | [
"\\( m + n - \\gcd(m, n) \\)"
] | [
"You have a birthday cake and you want to divide it into several parts so that it can be distributed equally among either $m$ or $n$ guests. What is the minimal number of parts you need to divide the cake into to ensure that it can be distributed equally regardless of whether $m$ or $n$ guests attend?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 7,
"tir": 23
} | 76 | 0 | 0 |
omr-ws-4f199c5d9a97c63e3e2b1c2a1e98c5892a3762ec74a81348a5dc30d29497f492 | You have a cake that you split into $n$ pieces. $2 \leq k \leq m$ guests come to your party. Find the minimum $n$ such that for all $k,$ the guests can choose pieces so that everyone gets an equal share of the cake, and all pieces are taken. Each piece is taken by exactly one person. | [
"\\(\\text{lcm}(2, 3, \\ldots, m)\\)"
] | [
"You have a cake that you split into $n$ pieces. $2 \\leq k \\leq m$ guests come to your party. Find the minimum $n$ such that for all $k,$ the guests can choose pieces so that everyone gets an equal share of the cake, and all pieces are taken. Each piece is taken by exactly one person."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 23
} | 37 | 0 | 0 |
omr-ws-9c31d726659d3a89c67d3ad7a210c849d1778563b3b680e6e282b42780a7fb8b | You have a calculator with 3 operations: (a) determine if two numbers are equal, (b) add two numbers together, and (c) find the roots of \(x^2 + ax + b\), if they exist. The calculator can only use values stored in memory. If you start with a number \(k\) in memory, how do you determine if \(k\) is 1? | [
"1"
] | [
"You have a calculator with 3 operations: (a) determine if two numbers are equal, (b) add two numbers together, and (c) find the roots of \\(x^2 + ax + b\\), if they exist. The calculator can only use values stored in memory. If you start with a number \\(k\\) in memory, how do you determine if \\(k\\) is 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-80966144d678e431b62c9567a87f92cfdf9e23f74cea2590bab8d265ee86718a | You have a candy bag initially containing $n$ chocolates, of which exactly 1 is a yellow chocolate and $n-1$ are dark chocolates. You repeatedly take out a random chocolate from the bag. If it is yellow, you put a dark chocolate back in the bag. If it is dark, you put a yellow chocolate back instead. What is the expect... | [
"\\(-1 + \\sum_{k=0}^{n-1} \\frac{2^{n-1}}{\\binom{n-1}{k}}\\)"
] | [
"You have a candy bag initially containing $n$ chocolates, of which exactly 1 is a yellow chocolate and $n-1$ are dark chocolates. You repeatedly take out a random chocolate from the bag. If it is yellow, you put a dark chocolate back in the bag. If it is dark, you put a yellow chocolate back instead. What is the e... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2d3b33fb0b974560dc891c5ec874e97d1f0f7aa91b7b2d2b4ce833d0ef03a2f | You have a choice to play one of two games. In both games, a coin is flipped four times. In game 1, if at least two flips land heads, you win. In game 2, if at least two consecutive flips land heads, you win. Let $N$ be the number of the game that gives you a better chance of winning, and let $p$ be the absolute differ... | [
"\\(\\frac{21}{16}\\)"
] | [
"You have a choice to play one of two games. In both games, a coin is flipped four times. In game 1, if at least two flips land heads, you win. In game 2, if at least two consecutive flips land heads, you win. Let $N$ be the number of the game that gives you a better chance of winning, and let $p$ be the absolute d... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-64a1d898271a8769c6d1357051d4b05a4f36553f4e15299f4579c269996e9f47 | You have a circle with two points A and B that are diametrically opposite to each other. You need to travel from A to B. If you travel along the diameter, your speed is \( v_2 \), and if you travel along the circle, your speed is \( v_1 \), where \( v_1 > v_2 \). You can also mix the two paths. Which path should you ch... | [
"- If \\( \\frac{v_1}{v_2} \\leq \\frac{\\pi}{2} \\), choose the diameter.\n- If \\( \\frac{v_1}{v_2} > \\frac{\\pi}{2} \\), choose the semicircle."
] | [
"You have a circle with two points A and B that are diametrically opposite to each other. You need to travel from A to B. If you travel along the diameter, your speed is \\( v_2 \\), and if you travel along the circle, your speed is \\( v_1 \\), where \\( v_1 > v_2 \\). You can also mix the two paths. Which path sh... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 19
} | 42 | 0 | 0 |
omr-ws-e4f7f99d221c6ee0919dbce4d64723212131dcff78051a7e1ba3a79600d417a7 | You have a coin that comes up heads with probability \( p \in (0, 1) \). You need to simulate an event that happens with probability \( q \in [0, 1] \). Devise a strategy (an algorithm) using only this coin to simulate the event such that the strategy has a finite expected number of tosses. | [
"\\text{Use the recursive algorithm outlined above.}"
] | [
"You have a coin that comes up heads with probability \\( p \\in (0, 1) \\). You need to simulate an event that happens with probability \\( q \\in [0, 1] \\). Devise a strategy (an algorithm) using only this coin to simulate the event such that the strategy has a finite expected number of tosses."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-b73fd8b974abda8dbe6f0c11320a570a59b5f4c56d3b8d7f56ff4099a935bd40 | You have a deck of 15 cards numbered from 1 to 15. You write down the number "1" and then flip cards over until you reach the card numbered "1". Next, you pick any number that has not been flipped over yet, write it down, and continue flipping cards until you reach that number. You repeat this process until all cards h... | [
"\\( H_{15} \\)"
] | [
"You have a deck of 15 cards numbered from 1 to 15. You write down the number \"1\" and then flip cards over until you reach the card numbered \"1\". Next, you pick any number that has not been flipped over yet, write it down, and continue flipping cards until you reach that number. You repeat this process until al... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 8
} | 9 | 0 | 0 |
omr-ws-67402494b65dbe0ec2a7a6f12c5998f162faeaa0d29fd2c87164e37c4ec578d9 | You have a deck of 20 cards, with 8 blue, 5 red, and the rest green. If you randomly pick 5 cards from the deck, what is the probability that you will have at least one card of each color? | [
"\\dfrac{875}{1292}"
] | [
"You have a deck of 20 cards, with 8 blue, 5 red, and the rest green. If you randomly pick 5 cards from the deck, what is the probability that you will have at least one card of each color?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b634fa0d353ba0f6c147cd599763c46ce9d68fbf3de6e21aabfa843bd59164f | You have a deck of 52 cards numbered from 1 to 52. You write down the number "1" and then flip cards over until you reach the card numbered "1". Next, you pick any number that has not been flipped over yet, write it down, and continue flipping cards until you reach that number. You repeat this process until all cards h... | [
"\\( H_{52} \\) or approximately \\( 4.5380439506974491902904868759210285188182639657472426682712704563 \\)"
] | [
"You have a deck of 52 cards numbered from 1 to 52. You write down the number \"1\" and then flip cards over until you reach the card numbered \"1\". Next, you pick any number that has not been flipped over yet, write it down, and continue flipping cards until you reach that number. You repeat this process until al... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-2053143e4f00465ebc543451899e774983cfe9cc37493ace72e82d7666b79cfd | You have a deck of 6 cards labeled $ABCDEF$. To shuffle, you split the deck in half: $ABCDEF \to ABC~~~DEF$. Then, you interleave the cards such that the first half ($ABC$) is placed after the corresponding card in the second half ($DEF$): $ABC~~~DEF \to D^AE^BF^C \to DAEBFC$. How many shuffles are required to return t... | [
"3"
] | [
"You have a deck of 6 cards labeled $ABCDEF$. To shuffle, you split the deck in half: $ABCDEF \\to ABC~~~DEF$. Then, you interleave the cards such that the first half ($ABC$) is placed after the corresponding card in the second half ($DEF$): $ABC~~~DEF \\to D^AE^BF^C \\to DAEBFC$. How many shuffles are required to ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-791110cab94b0cad1c12246ed246710fd5cebfc023616539d85536dede41e7bc | You have a deck of 60 cards, where 9 cards are numbered from 1 to 9 and the other 51 cards are blank. You randomly pick 7 cards from the deck. If a blank card is drawn, it is discarded and replaced with another card from the deck until only numbered cards remain in your hand. You then attempt to arrange these numbered ... | [
"\\dfrac{1}{1378}"
] | [
"You have a deck of 60 cards, where 9 cards are numbered from 1 to 9 and the other 51 cards are blank. You randomly pick 7 cards from the deck. If a blank card is drawn, it is discarded and replaced with another card from the deck until only numbered cards remain in your hand. You then attempt to arrange these numb... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-094f333485d2baaae0c891186ad73f26d6ce3a1a035f3cdf187b8f1c465e9b02 | You have a fair 12-sided die with sides numbered 0 through 11. After rolling the die 4 times, you construct a base 12 number where the nth roll corresponds to the nth place value starting from the units digit. What is the probability that the resulting number is divisible by $11 \cdot 9 \cdot 8$ (in base 10)? If the pr... | [
"793"
] | [
"You have a fair 12-sided die with sides numbered 0 through 11. After rolling the die 4 times, you construct a base 12 number where the nth roll corresponds to the nth place value starting from the units digit. What is the probability that the resulting number is divisible by $11 \\cdot 9 \\cdot 8$ (in base 10)? If... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb2847dade20c0dd9a1de172fd1f22cc1ba15c0e85091c10d588af3a44ac9b40 | You have a fair, 6-sided die with faces labeled 1 through 6. You roll the die three times and multiply together the numbers you get. The expected value of your result can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | [
"351"
] | [
"You have a fair, 6-sided die with faces labeled 1 through 6. You roll the die three times and multiply together the numbers you get. The expected value of your result can be written as $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-9c05ada22b82139b79a30bbf49ee6f7e1c29854e6b76f9593028fff8914d8e87 | You have a flight on Air China from Beijing to New York that will depart any time between 1 p.m. and 6 p.m., uniformly at random. Your friend, Henry, is flying American Airlines, also from Beijing to New York, and his flight will depart any time between 3 p.m. and 5 p.m., uniformly at random. What is the probability th... | [
"\\(\\frac{2}{5}\\)"
] | [
"You have a flight on Air China from Beijing to New York that will depart any time between 1 p.m. and 6 p.m., uniformly at random. Your friend, Henry, is flying American Airlines, also from Beijing to New York, and his flight will depart any time between 3 p.m. and 5 p.m., uniformly at random. What is the probabili... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-05a19ef83a243c9acdb01a221d9034070896a3aca4b46a9cada4d328ff98bbc5 | You have a group of \( 2n \) people (\( n > 2 \)). Every person hates exactly one other person (if \( A \) hates \( B \), then \( B \) hates \( A \)). In how many ways can you divide this group into 3 groups such that no two people who hate each other are in the same group? | [
"6^n"
] | [
"You have a group of \\( 2n \\) people (\\( n > 2 \\)). Every person hates exactly one other person (if \\( A \\) hates \\( B \\), then \\( B \\) hates \\( A \\)). In how many ways can you divide this group into 3 groups such that no two people who hate each other are in the same group?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-a45e01b5c92481c89f3eade62b5e16d9d3dec3df5e6e8d2f760bf5b051797ce7 | You have a krook at the bottom-left corner of a chessboard. The krook can move either one square up or down, or one square left or right. How many different ways can the krook reach the top-right corner of the chessboard in exactly $17$ moves? | [
"0"
] | [
"You have a krook at the bottom-left corner of a chessboard. The krook can move either one square up or down, or one square left or right. How many different ways can the krook reach the top-right corner of the chessboard in exactly $17$ moves?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74fce1a3fa3f2f9615efe7a34e94337551e8a7b3837aa23ec11e1d528e059f0e | You have a lock with four dials, each set to a number between 0 and 9 (inclusive). The digits of the code are not prime, 0, or 1, and the average value of the digits is 5. How many combinations will you have to try? | [
"10"
] | [
"You have a lock with four dials, each set to a number between 0 and 9 (inclusive). The digits of the code are not prime, 0, or 1, and the average value of the digits is 5. How many combinations will you have to try?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2997bd45a883f4b487ffcdadf7237b7fd05e4e9f4665e5aec4a866372c42097b | You have a number of gold coins that you were going to distribute equally among your 11 best friends. However, after dividing your gold coins into 11 equal piles, you realize that if you give away all your gold coins, 2 people will receive an extra gold coin. You have less than 100 gold coins. What is the largest numbe... | [
"90"
] | [
"You have a number of gold coins that you were going to distribute equally among your 11 best friends. However, after dividing your gold coins into 11 equal piles, you realize that if you give away all your gold coins, 2 people will receive an extra gold coin. You have less than 100 gold coins. What is the largest ... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-4d5b0bb2746ae5fa3fe01d59109d9331b6d66ab7d04c87bf1fc75275c9975acd | You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules? | [
"258"
] | [
"You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-c3c3cafdbab87922199c12b160de68420b8260dd340f4bb1b4831bd24a8ed0ff | You have a piece of paper with a length of 12 inches and a height of 6 inches. The lower right-hand corner is folded over so it touches the left long edge. Find the minimum length of the resulting crease. | [
"\\(6\\sqrt{2}\\)"
] | [
"You have a piece of paper with a length of 12 inches and a height of 6 inches. The lower right-hand corner is folded over so it touches the left long edge. Find the minimum length of the resulting crease."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95a80c589a285a7623eb81cd39ffca640ba9e5de9c171173466397a548806556 | You have a rectangle of paper. The shortest side of the rectangle is $6$ and the diagonal is $12$. The paper is folded along a diagonal, thus obtaining a triangle in which the two parts overlap and two triangles without overlapping. Calculate the area of the overlapping triangle. | [
"\\(12 \\sqrt{3}\\)"
] | [
"You have a rectangle of paper. The shortest side of the rectangle is $6$ and the diagonal is $12$. The paper is folded along a diagonal, thus obtaining a triangle in which the two parts overlap and two triangles without overlapping. Calculate the area of the overlapping triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 19
} | 42 | 0 | 0 |
omr-ws-15eaff9b39444d32957fbf84081220cd82a820dd8f652e7f2be9cfce8f95f8d0 | You have a rectangular prism box with length $x+5$ units, width $x-5$ units, and height $x^{2}+25$ units. For how many positive integer values of $x$ is the volume of the box less than 700 units? | [
"1"
] | [
"You have a rectangular prism box with length $x+5$ units, width $x-5$ units, and height $x^{2}+25$ units. For how many positive integer values of $x$ is the volume of the box less than 700 units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7afc050c8c556f4f8e4eeff6abbb4e1715a2368a52300b6a21e4fd6170fefa37 | You have a robot that each morning performs one of four actions with equal probability (1/4): - Do nothing. - Self-destruct. - Create one clone. - Create two clones. Compute the probability that you eventually have no robots. | [
"\\(\\sqrt{2} - 1\\)"
] | [
"You have a robot that each morning performs one of four actions with equal probability (1/4): \n- Do nothing.\n- Self-destruct.\n- Create one clone.\n- Create two clones.\n\nCompute the probability that you eventually have no robots."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7ebaf9c5baa5cddafa0b0eb6b1510b0db197ae176ddc77645e0c145f5321f5f7 | You have a robot. Each morning the robot performs one of four actions, each with probability \(1/4\): - Nothing. - Self-destruct. - Create one clone. - Create two clones. Compute the probability that you eventually have no robots. | [
"\\sqrt{2} - 1"
] | [
"You have a robot. Each morning the robot performs one of four actions, each with probability \\(1/4\\):\n- Nothing.\n- Self-destruct.\n- Create one clone.\n- Create two clones.\nCompute the probability that you eventually have no robots."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-599c83de3c7d161d7e0044a659986baf50dde301cc8d1ba9eeb73063478bf8fd | You have a sequence of 100 Othello bricks, some facing black and some white. In a move, you can flip 7 bricks that are equally spaced from one another. Determine the minimum number of moves required to get every brick to face black, starting from a configuration where all bricks are facing white. | [
"16"
] | [
"You have a sequence of 100 Othello bricks, some facing black and some white. In a move, you can flip 7 bricks that are equally spaced from one another. Determine the minimum number of moves required to get every brick to face black, starting from a configuration where all bricks are facing white."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-35efc61274b6ab10a596d2cfebeb51f82493f414628a845ceaae1188e6a2ad8c | You have a set of 10 different kinds of cards. Each turn, you collect 3 cards from the set. If you currently have exactly 8 different kinds of cards, what is the probability that in the previous turn you had exactly 6 different kinds of cards? | [
"\\dfrac{12}{55}"
] | [
"You have a set of 10 different kinds of cards. Each turn, you collect 3 cards from the set. If you currently have exactly 8 different kinds of cards, what is the probability that in the previous turn you had exactly 6 different kinds of cards?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-1fae4ee051b6d0c483835855790a97c32c487b873fc7acf9ba2caa51e5467083 | You have a set of 90 different trading cards and are missing the final scene card. You buy 10 packs, each containing 8 different cards. Each card is equally likely to be in a given pack. Find the probability that you will get at least one copy of the final scene card. | [
"\\(1 - \\left(\\frac{41}{45}\\right)^{10}\\)"
] | [
"You have a set of 90 different trading cards and are missing the final scene card. You buy 10 packs, each containing 8 different cards. Each card is equally likely to be in a given pack. Find the probability that you will get at least one copy of the final scene card."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-df7c47c1c426b1496380b0232a1620a9cbaa01b37d1ac4054fbcb73f0941c97a | You have a set of \( n \) biased coins. The \( m \)-th coin has a probability of \( \frac{1}{2m+1} \) of landing heads (for \( m = 1, 2, \ldots, n \)), and the results for each coin are independent. What is the probability that if each coin is tossed once, you get an odd number of heads? | [
"\\(\\frac{n}{2n+1}\\)"
] | [
"You have a set of \\( n \\) biased coins. The \\( m \\)-th coin has a probability of \\( \\frac{1}{2m+1} \\) of landing heads (for \\( m = 1, 2, \\ldots, n \\)), and the results for each coin are independent. What is the probability that if each coin is tossed once, you get an odd number of heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-aea9d51f9e635a677837c3b73cdfaba3710c89e2dd713233ece8a626a5b72d4a | You have a sheet of paper and a pair of scissors. Turn 1: You cut the paper into either 8 pieces or 12 pieces. Turn 2: You take one of these pieces and cut it into 8 pieces or 12 pieces. Hence after two turns you have 15, 19, or 23 pieces of paper sitting in front of you. There is no way to end up with 20 pieces of pap... | [
"60"
] | [
"You have a sheet of paper and a pair of scissors. Turn 1: You cut the paper into either 8 pieces or 12 pieces. Turn 2: You take one of these pieces and cut it into 8 pieces or 12 pieces. Hence after two turns you have 15, 19, or 23 pieces of paper sitting in front of you. There is no way to end up with 20 pieces o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-4157a071e388526eb5d0f367293fb2ea39e2b60323c20623350d2571d16403c5 | You have a sheet of paper and a pair of scissors. You cut the paper into either 8 or 12 pieces. Then you take one of these pieces and cut it into either 8 or 12 pieces, and so on. After 2 cuts, you may have 15, 19, or 23 pieces (by making two '8-cuts', one '8-cut' and one '12-cut', or two '12-cuts'). There is an intege... | [
"60"
] | [
"You have a sheet of paper and a pair of scissors. You cut the paper into either 8 or 12 pieces. Then you take one of these pieces and cut it into either 8 or 12 pieces, and so on. After 2 cuts, you may have 15, 19, or 23 pieces (by making two '8-cuts', one '8-cut' and one '12-cut', or two '12-cuts'). There is an i... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-20a023fae290a74023b0b13a843e0c3ce248a9f6d2a14625e86333acb0dd38a0 | You have a sheet of paper, which you lay on the xy plane so that its vertices are at \( (-1, 0) \), \( (1, 0) \), \( (1, 100) \), \( (-1, 100) \). You remove a section of the bottom of the paper by cutting along the function \( y = f(x) \), where \( f \) satisfies \( f(1) = f(-1) = 0 \). (In other words, you keep the b... | [
"6"
] | [
"You have a sheet of paper, which you lay on the xy plane so that its vertices are at \\( (-1, 0) \\), \\( (1, 0) \\), \\( (1, 100) \\), \\( (-1, 100) \\). You remove a section of the bottom of the paper by cutting along the function \\( y = f(x) \\), where \\( f \\) satisfies \\( f(1) = f(-1) = 0 \\). (In other wo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-9b2516b4f584f1447ae65cc49b8fb2b156d2d858986c3d6f8496210537f210b7 | You have a six-sided die labeled 1-6 and your opponent has a seven-sided die labeled 1-7. What is the probability of you winning if both of you roll your dice? | [
"$\\frac{5}{18}$"
] | [
"You have a six-sided die labeled 1-6 and your opponent has a seven-sided die labeled 1-7. What is the probability of you winning if both of you roll your dice?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fdbeb51ed6abeb18a09d19ecef781de1beada3af79661da83f7c49d9a1d38ca8 | You have a six-sided die labeled 1-6 and your opponent has a six-sided die labeled 2-7. What is the probability of you winning if both of you roll your dice? | [
"$\\frac{5}{18}$"
] | [
"You have a six-sided die labeled 1-6 and your opponent has a six-sided die labeled 2-7. What is the probability of you winning if both of you roll your dice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9cc4f0a2cfe0c84c1e284c833913141d1ab62d48fa85a3b1b336bd4ae73692fd | You have a six-sided die labeled 1-6 and your opponent has an eight-sided die labeled 1-8. What is the probability of you winning if both of you roll your dice? | [
"\\(\\frac{5}{16}\\)"
] | [
"You have a six-sided die labeled 1-6 and your opponent has an eight-sided die labeled 1-8. What is the probability of you winning if both of you roll your dice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6c876d675d0b0bc645dd273c8171e89bd3ed9c542fe050507a5405beed1efdca | You have a six-sided die labeled 1-6 and your opponent has an n-sided die labeled 1-n. What is the probability of you winning if both of you roll your dice? | [
"\\(\\frac{15}{6(n-5)}\\)"
] | [
"You have a six-sided die labeled 1-6 and your opponent has an n-sided die labeled 1-n. What is the probability of you winning if both of you roll your dice?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b478ad24fc035456819dc03460b3b8624b06991fe084b1c001fc89113d34bfa3 | You have a square divided into $n^2$ smaller squares. A snake moves from one square to another, visiting each square exactly once and placing the number of the move in the square. After the snake's path, the numbers $1, 4, 9, \ldots, n^2$ are found in one column. Determine the value of $n$. | [
"3"
] | [
"You have a square divided into $n^2$ smaller squares. A snake moves from one square to another, visiting each square exactly once and placing the number of the move in the square. After the snake's path, the numbers $1, 4, 9, \\ldots, n^2$ are found in one column. Determine the value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a68718764bcbeaa580f66b729bf8e6e5d13bebd4e51f516c3b377f4a9d53f91 | You have a stack of $2n+1$ cards. You can shuffle the stack using two operations: 1) Cut: Remove any number of cards from the top of the pile and put them on the bottom. 2) Perfect riffle shuffle: Remove the top $n$ cards from the deck and place them in order in the spaces between the other $n+1$ cards. Calculate the m... | [
"110"
] | [
"You have a stack of $2n+1$ cards. You can shuffle the stack using two operations:\n1) Cut: Remove any number of cards from the top of the pile and put them on the bottom.\n2) Perfect riffle shuffle: Remove the top $n$ cards from the deck and place them in order in the spaces between the other $n+1$ cards.\nCalcula... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1198007e448259bc589595750a411578b47224e175dc94d8ea9f5d39a43da66 | You have a stack of \(2n+1\) cards. You can shuffle the stack using the following operations: - Cut: Remove any number of cards from the top of the pile and put them on the bottom. - Perfect riffle shuffle: Remove the top \(n\) cards from the deck and place them in order in the spaces between the other \(n+1\) cards. L... | [
"110"
] | [
"You have a stack of \\(2n+1\\) cards. You can shuffle the stack using the following operations:\n- Cut: Remove any number of cards from the top of the pile and put them on the bottom.\n- Perfect riffle shuffle: Remove the top \\(n\\) cards from the deck and place them in order in the spaces between the other \\(n+... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30141d7bca24d144f156f32d8682099e65e2e3c8a04b7813b70a87cd0d1bfa14 | You have a stack of coins numbered from 1 to 2006, with coin 1 at the bottom and coin 2006 at the top, all initially facing heads up. A $\text{k-flip}$ is a procedure where you take the top $k$ coins and flip them over as a whole. How many $\text{k-flips}$ are needed to rearrange these coins so that their order does no... | [
"6016"
] | [
"You have a stack of coins numbered from 1 to 2006, with coin 1 at the bottom and coin 2006 at the top, all initially facing heads up. A $\\text{k-flip}$ is a procedure where you take the top $k$ coins and flip them over as a whole. How many $\\text{k-flips}$ are needed to rearrange these coins so that their order ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87318bc6daba4d5f4b0ecc1cc88694a1161d8205c596b65967274a1e4b78a7d3 | You have a steel beam that is 25 meters long. It needs to pass through a corridor that is 13 meters wide and then through a second corridor that is perpendicular to the first. How wide must the second corridor be to allow the steel beam to pass through? | [
"\\( x \\approx 5.251080936 \\)"
] | [
"You have a steel beam that is 25 meters long. It needs to pass through a corridor that is 13 meters wide and then through a second corridor that is perpendicular to the first. How wide must the second corridor be to allow the steel beam to pass through?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 25
} | 44 | 0 | 0 |
omr-ws-51c6be43cac69ccec195af17fad254202a7a75aa7ab5722d126c6e523e9350da | You have a stick of length $L$ and you divide it into four parts to make a quadrilateral with area $A$. If $L=16$, find the maximum possible value of $A$ and give the ratio of the sides when this maximum is achieved. | [
"1:1:1:1"
] | [
"You have a stick of length $L$ and you divide it into four parts to make a quadrilateral with area $A$. If $L=16$, find the maximum possible value of $A$ and give the ratio of the sides when this maximum is achieved."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-caad669f7aab6b63768bc8212c1fc0d431afc8c24deea72975b7f1431e2c04f8 | You have a stick of length $n > 1$. In how many ways can you break it into $n$ pieces of unit length? (At each step, you can break one of the pieces with length $>1$ into two pieces of integer lengths.) | [
"\\(\\frac{1}{n}\\binom{2(n-1)}{n-1}\\)"
] | [
"You have a stick of length $n > 1$. In how many ways can you break it into $n$ pieces of unit length? (At each step, you can break one of the pieces with length $>1$ into two pieces of integer lengths.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 1,
"tir": 3
} | 35 | 0 | 0 |
omr-ws-d390b3a2dedaa7930513aa056c4b6e408572ad840c2f6ea5bb395a966e1ff5d4 | You have a table with N rows and M columns. For each row i, you know the number A_i. You can color either the first A_i cells or the last M - A_i cells in each row. The "boredness" of a column is defined as the product of the number of colored cells and the number of empty cells in that column. The "boredness" of the t... | [
""
] | [
"You have a table with N rows and M columns. For each row i, you know the number A_i. You can color either the first A_i cells or the last M - A_i cells in each row. The \"boredness\" of a column is defined as the product of the number of colored cells and the number of empty cells in that column. The \"boredness\"... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 1 |
omr-ws-9319f1cb78d75c17f8135e67de2d5f7f93d53bf28cded986ad4640197e87e270 | You have a twig of length $1$. You repeatedly select two points on the twig independently and uniformly at random, make cuts at these points, and keep only the largest piece. After $2012$ repetitions, what is the expected length of the remaining piece? | [
"\\(\\left(\\frac{11}{18}\\right)^{2012}\\)"
] | [
"You have a twig of length $1$. You repeatedly select two points on the twig independently and uniformly at random, make cuts at these points, and keep only the largest piece. After $2012$ repetitions, what is the expected length of the remaining piece?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-8cbf5d4ca001df703763e075e6655080a395c22cfac09cee759b51e337b23efb | You have a whole pizza in the refrigerator. On your first trip to the refrigerator, you eat half the pizza. On each successive trip, you eat half of the remaining pizza. After five trips to the refrigerator, what fractional part of the pizza have you eaten? | [
"\\frac{31}{32}"
] | [
"You have a whole pizza in the refrigerator. On your first trip to the refrigerator, you eat half the pizza. On each successive trip, you eat half of the remaining pizza. After five trips to the refrigerator, what fractional part of the pizza have you eaten?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1b5204a0eb50f41fac946c62275ea7cd532f3169841dc6f21cb08b555ba0930 | You have already invested $450 in a stock with an annual return of 9.00%. How much of an additional $900 should be invested at 10.50% and how much at 6% so that the total return on the entire $1350 is 8.67%? | [
"$494.33 at 10.50%, $405.67 at 6.00%",
"\\( x = 501 \\), \\( y = 399 \\)"
] | [
"You have already invested $450 in a stock with an annual return of 9.00%. How much of an additional $900 should be invested at 10.50% and how much at 6% so that the total return on the entire $1350 is 8.67%?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 33
} | 45 | 0 | 0 |
omr-ws-a02814b3689c302fce5f1bbe4b490437704a42d88b74d149e9af38a703cb5d17 | You have an $8 \times 8$ square board, with some squares filled with red and some with white. An opponent will flip exactly one square’s color while you are not looking. What is the minimum number of bits of memory that you need to guarantee identifying the flipped square when you look back? | [
"6"
] | [
"You have an $8 \\times 8$ square board, with some squares filled with red and some with white. An opponent will flip exactly one square’s color while you are not looking. What is the minimum number of bits of memory that you need to guarantee identifying the flipped square when you look back?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-790812ca9898a28db8cd6c2e7696b5e35b4a2c230d9734237cb847fa97305334 | You have an $a \times b \times c$ lattice of originally off unit cube lights. You can select any orthogonal plane cross section and toggle all the cubes in that cross section. How many different lightings are possible? | [
"2^{a + b + c - 2}"
] | [
"You have an $a \\times b \\times c$ lattice of originally off unit cube lights. You can select any orthogonal plane cross section and toggle all the cubes in that cross section. How many different lightings are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75166ec74a85912431b3375fbd7db97a545d499e08c91a34fec22cfb6f93fd4b | You have an $a \times b \times c$ lattice of originally off unit cube lights. You can select any orthogonal row (a $1 \times 1 \times n$ segment) and toggle all the cubes in that row. How many different lightings are possible? | [
"\\( 2^{a \\times b + a \\times c + b \\times c} \\)"
] | [
"You have an $a \\times b \\times c$ lattice of originally off unit cube lights. You can select any orthogonal row (a $1 \\times 1 \\times n$ segment) and toggle all the cubes in that row. How many different lightings are possible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afbfbba1a22c36abab54d371cc2e877d24e417ac7c700025f03a4b835f5269d2 | You have an $n \times n$ grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive $i+j$ points if there were $i$ crosses in the same row and $j$ crosses in the same column before you placed the new cross. What are the possible total scores you ca... | [
"\\( n^2(n-1) \\)"
] | [
"You have an $n \\times n$ grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive $i+j$ points if there were $i$ crosses in the same row and $j$ crosses in the same column before you placed the new cross. What are the possible total scores ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-3b6bcb5425c0f4ba0f47464e5f3a462b925f09bd66d657b65b9f222f6c252ca0 | You have an 8 x 8 square and a 6 x 6 square. You must cut each into 2 pieces and then form a 10 x 10 square. The cuts must be horizontal or vertical (or a combination, like a zig-zag). How can this be done? | [
"10 \\times 10"
] | [
"You have an 8 x 8 square and a 6 x 6 square. You must cut each into 2 pieces and then form a 10 x 10 square. The cuts must be horizontal or vertical (or a combination, like a zig-zag). How can this be done?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-0fc45dcedbe3ea914e7752fea5d987b7db6dcb012557d8d933171930348ebaca | You have an \( n \times m \) array of 0s and 1s. Let \( a \) be the maximum number of 1s you can pick such that no two are in the same row or column. Pick a subrectangle (not necessarily non-empty) of zeros with dimensions \( b \times c \); that is, pick \( b \) special rows and \( c \) special columns so that any entr... | [
"25"
] | [
"You have an \\( n \\times m \\) array of 0s and 1s. Let \\( a \\) be the maximum number of 1s you can pick such that no two are in the same row or column. Pick a subrectangle (not necessarily non-empty) of zeros with dimensions \\( b \\times c \\); that is, pick \\( b \\) special rows and \\( c \\) special columns... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1a77d0275b7c1c7afca0434a264f36b23eb6700c9f45940be4d468e84fa147d | You have an ample supply of fair coins. You flip all of them, keeping the ones that are heads and discarding any tails. This process is then repeated until no coins remain. What is the probability that at some point, there is exactly $1$ coin remaining? | [
"\\( \\frac{1}{\\log 4} \\)"
] | [
"You have an ample supply of fair coins. You flip all of them, keeping the ones that are heads and discarding any tails. This process is then repeated until no coins remain. What is the probability that at some point, there is exactly $1$ coin remaining?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-c9073768ec04ab95fdd7e29f651b6e7a15dba601403ebdbbbfd9e24ee374e46c | You have an equilateral paper triangle of area $9$. Fold it in half along a straight line that passes through the center of the triangle and does not contain any vertex. This creates a quadrilateral where the two pieces overlap and three triangles without overlaps. Determine the smallest possible area of the overlappin... | [
"3"
] | [
"You have an equilateral paper triangle of area $9$. Fold it in half along a straight line that passes through the center of the triangle and does not contain any vertex. This creates a quadrilateral where the two pieces overlap and three triangles without overlaps. Determine the smallest possible area of the overl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-39012eff0616ed247b662cb6a869bfaf1008fa5c023ff4e7b21dd0b50b44ecd6 | You have an equilateral triangle and a regular hexagon. The perimeter of the regular hexagon is equal to \(30\) times the side of the equilateral triangle. If we divide the area of the hexagon by the area of the triangle, what number do we get? | [
"1350"
] | [
"You have an equilateral triangle and a regular hexagon. The perimeter of the regular hexagon is equal to \\(30\\) times the side of the equilateral triangle. If we divide the area of the hexagon by the area of the triangle, what number do we get?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d78a46d3cad706b8b296dd07cc07cf27d53f714b97f8e5e4ef648cc202d229fc | You have an equilateral triangular array of points with \( n \) points on the bottom row. Every point is initially colored blue. You can perform moves by choosing a line parallel to the bottom row or intersecting it at a \( 60^\circ \) angle, and then coloring every red point on that line blue and every blue point on t... | [
"\\(2^{3n-2}\\)"
] | [
"You have an equilateral triangular array of points with \\( n \\) points on the bottom row. Every point is initially colored blue. You can perform moves by choosing a line parallel to the bottom row or intersecting it at a \\( 60^\\circ \\) angle, and then coloring every red point on that line blue and every blue ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f89835efd2429a6a512ba212413d96a8c234fcf851b13345cd9190aeafac713d | You have an identification number that is equally likely to be any one of the integers $1, 2, \cdots, n$. You punch in integers in order until you get the right one at a rate of $r$ integers per minute. The police will be alerted as soon as you punch in the first wrong number, and the probability that they will arrive ... | [
"0.585"
] | [
"You have an identification number that is equally likely to be any one of the integers $1, 2, \\cdots, n$. You punch in integers in order until you get the right one at a rate of $r$ integers per minute. The police will be alerted as soon as you punch in the first wrong number, and the probability that they will a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98384233983cf1c8af9f6ca821500a7a620883a3ac6aaaa88fee921585b5d7e6 | You have an infinite number of machines. The first one rounds a number you give it to the nearest integer, the second rounds to the nearest tenth, the third one rounds to the nearest hundredth, and so on. Let \( a \) be the largest number between 0 and 1 that cannot be rounded up to 1 in any way. Let \( b \) be the sma... | [
"\\dfrac{1}{9}"
] | [
"You have an infinite number of machines. The first one rounds a number you give it to the nearest integer, the second rounds to the nearest tenth, the third one rounds to the nearest hundredth, and so on. Let \\( a \\) be the largest number between 0 and 1 that cannot be rounded up to 1 in any way. Let \\( b \\) b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-dfda6692658f349987656a6c4ab6348adce2a14841e4f6ff1af94c8802a57922 | You have an infinite number of pencils, an ample amount of erasers (in multiples of 4), and a pair of glued compasses. You need to bring at least one pencil and can bring any number of items from 1 to 2022. The number of erasers must be a multiple of 4, and you can bring either 0 or 2 compasses. Determine the number of... | [
"\\( 0 \\) if \\( k = 1 \\), \\( \\frac{k-4}{4} + 1 \\) if \\( k \\equiv 0 \\pmod{4} \\) and \\( k > 1 \\), \\( \\frac{k-5}{4} + 1 \\) if \\( k \\equiv 1 \\pmod{4} \\) and \\( k > 1 \\), \\( \\frac{k-2}{4} + 1 \\) if \\( k \\equiv 2 \\pmod{4} \\) and \\( k > 1 \\), \\( \\frac{k-3}{4} + 1 \\) if \\( k \\equiv 3 \\pm... | [
"You have an infinite number of pencils, an ample amount of erasers (in multiples of 4), and a pair of glued compasses. You need to bring at least one pencil and can bring any number of items from 1 to 2022. The number of erasers must be a multiple of 4, and you can bring either 0 or 2 compasses. Determine the numb... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-1e09dab694dafe553da69bc279573ce7300fec4f6ca7dafde9e8eba87ce9b9af | You have an m-sided die and your opponent has an n-sided die. What is your strategy to maximize your chances of winning when both of you roll your dice? | [
"\\text{Choose the die with more sides}"
] | [
"You have an m-sided die and your opponent has an n-sided die. What is your strategy to maximize your chances of winning when both of you roll your dice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-cb210cafbd05055c5e43ec29a7af335732174d17e373305350d7d2da59849bce | You have an unknown amount of sugar, a balance scale, and a weight of 1 g. What is the fastest way (in terms of the minimal number of balancing actions) to obtain a 1 kg portion of sugar using the balance scale? | [
"10"
] | [
"You have an unknown amount of sugar, a balance scale, and a weight of 1 g. What is the fastest way (in terms of the minimal number of balancing actions) to obtain a 1 kg portion of sugar using the balance scale?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-be14b5b41403f07c71812e17fd6d705ba9a9dfac8bc371cbe9e8c7b06c5442e3 | You have an urn containing 75 blue balls, 25 red balls, and 1 yellow ball. You receive $1 for each red ball drawn, but if you draw the yellow ball, you lose everything. After each draw, you can choose to stop or draw again. What is the optimal strategy for maximizing your expected earnings? | [
"Stop drawing after winning \\$13."
] | [
"You have an urn containing 75 blue balls, 25 red balls, and 1 yellow ball. You receive $1 for each red ball drawn, but if you draw the yellow ball, you lose everything. After each draw, you can choose to stop or draw again. What is the optimal strategy for maximizing your expected earnings?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-c26abaefc459cd3c521ca3fb5880aa300b2a8e18f510d7c3ee41bb7136613274 | You have been given the task to transport 3,000 apples 1,000 miles from Appleland to Bananaville. Your truck can carry 1,000 apples at a time. Every time you travel a mile towards Bananaville, you must pay the tax of 1 apple, but you pay nothing when going in the other direction (towards Appleland). What is the highest... | [
"833 apples"
] | [
"You have been given the task to transport 3,000 apples 1,000 miles from Appleland to Bananaville. Your truck can carry 1,000 apples at a time. Every time you travel a mile towards Bananaville, you must pay the tax of 1 apple, but you pay nothing when going in the other direction (towards Appleland). What is the hi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9a5aa00a5c1ccbc1b558204433fa4a57e4874900ee88fd998bef685e33a366e5 | You have bought \( n \) chocolates from a shop. For every 3 wrappers returned, the shopkeeper gives you 1 more chocolate. How many chocolates can you eat in total? | [
"\\(n + \\left\\lfloor \\frac{n}{2} \\right\\rfloor\\)"
] | [
"You have bought \\( n \\) chocolates from a shop. For every 3 wrappers returned, the shopkeeper gives you 1 more chocolate. How many chocolates can you eat in total?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06ceeb54dd6285e12062f865634515633e5c63de94ffae721349a087923b28a0 | You have developed a model using truncated data and received a benchmark sample of 100 random observations of \( X \). Using this benchmark data, calculate the mean squared error (MSE) between the predicted values from your model and the actual values in the benchmark sample. Assume the truncation occurs at \( X = k \)... | [
"10"
] | [
"You have developed a model using truncated data and received a benchmark sample of 100 random observations of \\( X \\). Using this benchmark data, calculate the mean squared error (MSE) between the predicted values from your model and the actual values in the benchmark sample. Assume the truncation occurs at \\( ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d2fa16f856894a7ce91175e3c410e8ec19c133d0893ff1d939a8d5e3cdd2e7d | You have drawn a five-card hand from a standard 52-card deck and have a pair and three "worthless" cards. You discard the three "worthless" cards and draw three new ones. What is the probability that your hand will improve to a three of a kind, another pair, or four of a kind? Find and describe all the cases. | [
"\\dfrac{1497}{5405}"
] | [
"You have drawn a five-card hand from a standard 52-card deck and have a pair and three \"worthless\" cards. You discard the three \"worthless\" cards and draw three new ones. What is the probability that your hand will improve to a three of a kind, another pair, or four of a kind? Find and describe all the cases."... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-9db3eb6c902c8361529908c82d6b8cf92d8334dc089f408e07829e749e1af42c | You have five circles that are all tangent to each other and to two lines. The radius of the smallest circle is 8, and the radius of the largest circle is 18. What is the radius of the third (middle) circle? | [
"12"
] | [
"You have five circles that are all tangent to each other and to two lines. The radius of the smallest circle is 8, and the radius of the largest circle is 18. What is the radius of the third (middle) circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-7f07d3d3dd247d6f1a8b597f8621db84bb90d59ff5f51f0df952855ad65e2510 | You have five pieces of paper. You pick one or more of them and cut each of them into five smaller pieces. Now you take one or more of the pieces from this lot and cut each of these into five smaller pieces. And so on. After a series of such operations, you find that you have exactly $N$ pieces of paper. Determine the ... | [
"2001"
] | [
"You have five pieces of paper. You pick one or more of them and cut each of them into five smaller pieces. Now you take one or more of the pieces from this lot and cut each of these into five smaller pieces. And so on. After a series of such operations, you find that you have exactly $N$ pieces of paper. Determine... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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