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-922d4f1c3132e6993777db098debdb6f6b2f8fa43896bc83025fdffbe78bcbfd | Andrew and John are both Beatles fans. Their respective collections share nine of the same albums. Andrew has seventeen albums in his collection. Six albums are in John's collection, but not Andrew's. How many albums are in either Andrew's or John's collection, but not both? | [
"14"
] | [
"Andrew and John are both Beatles fans. Their respective collections share nine of the same albums. Andrew has seventeen albums in his collection. Six albums are in John's collection, but not Andrew's. How many albums are in either Andrew's or John's collection, but not both?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a88557c369af3e9312a80a6dc6177df9f6ebb5cd45b2b84703597d371c58af80 | Andrew and Vasilis alternately write positive perfect squares on a blackboard. The game ends when the sum of the numbers on the blackboard exceeds 24, and the player who wrote the last number loses. If Andrew starts first, determine which player has a winning strategy and describe the strategy. | [
"Andrew"
] | [
"Andrew and Vasilis alternately write positive perfect squares on a blackboard. The game ends when the sum of the numbers on the blackboard exceeds 24, and the player who wrote the last number loses. If Andrew starts first, determine which player has a winning strategy and describe the strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-07c23dffcff5f1591c889216840402cf7a497a84aa970e205c575dd280e2af93 | Andrew can build a house in 2 months. Bill can build an identical house in 3 months. If Bill begins building the house for 2 weeks, at which point Andrew joins him and they work together, how many weeks will it take to finish building the house? Assume that there are 4 weeks in a month. | [
"4"
] | [
"Andrew can build a house in 2 months. Bill can build an identical house in 3 months. If Bill begins building the house for 2 weeks, at which point Andrew joins him and they work together, how many weeks will it take to finish building the house? Assume that there are 4 weeks in a month."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61caab548993351275c23ba778969c1c01ffeb6fa971b675c631fb7799548261 | Andrew encounters a square $ABCD$, with side length one meter, on a sidewalk. He marks a point $E$ such that $B$ is on $\overline{DE}$ and $EB = \frac12 BD$. He then draws segments $DE$, $EA$, and $AC$ with chalk, thus forming a giant four. Compute the total length, in meters, that Andrew has drawn. | [
"\\(\\frac{5 \\sqrt{2} + \\sqrt{10}}{2}\\)"
] | [
"Andrew encounters a square $ABCD$, with side length one meter, on a sidewalk. He marks a point $E$ such that $B$ is on $\\overline{DE}$ and $EB = \\frac12 BD$. He then draws segments $DE$, $EA$, and $AC$ with chalk, thus forming a giant four. Compute the total length, in meters, that Andrew has drawn."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-dfd4b02f5b55c8e2f1ec55b6cb5e8122add9f202a34dc1cac25dcbe9045517de | Andrew flips a fair coin and rolls a standard six-sided die. If the coin comes up heads, he gets as many points as the number showing on the die. If it comes up tails, he rolls the die again and gets as many points as the larger of the two rolls. What is the expected value for the number of points Andrew will get in on... | [
"\\(\\frac{287}{72}\\)"
] | [
"Andrew flips a fair coin and rolls a standard six-sided die. If the coin comes up heads, he gets as many points as the number showing on the die. If it comes up tails, he rolls the die again and gets as many points as the larger of the two rolls. What is the expected value for the number of points Andrew will get ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-8ec00a00745bb6374203f1c8ab766614b234e6f1b14fe4f592476ab59a77edc8 | Andrew gave Katie a 5-yard head start in a 100-yard dash and Andrew lost by only $\frac{1}{4}$ of a yard. In how many more yards would Andrew have overtaken Katie? | [
"5.25"
] | [
"Andrew gave Katie a 5-yard head start in a 100-yard dash and Andrew lost by only $\\frac{1}{4}$ of a yard. In how many more yards would Andrew have overtaken Katie?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 4
} | 7 | 0 | 0 |
omr-ws-0790885f95a0f6b20739328304c044959bc94d0000934d0a98488f311cffe765 | Andrew gives 2n cards numbered from 1 to 2n to n guests, with each guest receiving two cards. The guest with the maximum sum of the numbers on their cards will receive a gift. Calculate the probability that there will be only one guest with the maximum sum, thus avoiding the need for additional gifts. | [
"\\dfrac{1}{2n - 1}"
] | [
"Andrew gives 2n cards numbered from 1 to 2n to n guests, with each guest receiving two cards. The guest with the maximum sum of the numbers on their cards will receive a gift. Calculate the probability that there will be only one guest with the maximum sum, thus avoiding the need for additional gifts."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-3833a686afd47a1e0d3eb077283865e2299e3b2e3fefe64bb1559e40869b8d44 | Andrew has a list of six numbers. He writes down all possible sums of five of those numbers. The numbers $57$, $61$, $63$, $53$, $45$, and $51$ are written down. What is the smallest number in his original list? | [
"3"
] | [
"Andrew has a list of six numbers. He writes down all possible sums of five of those numbers. The numbers $57$, $61$, $63$, $53$, $45$, and $51$ are written down. What is the smallest number in his original list?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9eca1fc50338f773fd4b222aed6a10eae5d73b7b54c105c70f3e86e25194d7ce | Andrew has been playing the hit game BTD6, and he needs to beat round $100$, which consists of $1$ ZOMG. $1$ ZOMG is $4$ BFBs, $1$ BFB is $4$ MOABS, $1$ MOAB is $4$ Ceramics, $1$ Ceramic is $2$ Rainbows, $1$ Rainbow is $2$ Zebras, $1$ Zebra is $1$ Black and $1$ White, $1$ Black or $1$ White is $2$ Pinks, and $1$ Pink i... | [
"1024"
] | [
"Andrew has been playing the hit game BTD6, and he needs to beat round $100$, which consists of $1$ ZOMG. $1$ ZOMG is $4$ BFBs, $1$ BFB is $4$ MOABS, $1$ MOAB is $4$ Ceramics, $1$ Ceramic is $2$ Rainbows, $1$ Rainbow is $2$ Zebras, $1$ Zebra is $1$ Black and $1$ White, $1$ Black or $1$ White is $2$ Pinks, and $1$ P... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-69db307c09dc4f6179337209a6052328f84f7264786749444d06a02846e9d371 | Andrew has two cubic containers without tops. The base areas of the containers are $4$ dm$^{2}$ and $36$ dm$^{2}$ respectively. Andrew needs to completely fill the larger container with water using the smaller container. What is the smallest number of trips Andrew has to make to the pond with the smaller container? | [
"27"
] | [
"Andrew has two cubic containers without tops. The base areas of the containers are $4$ dm$^{2}$ and $36$ dm$^{2}$ respectively. Andrew needs to completely fill the larger container with water using the smaller container. What is the smallest number of trips Andrew has to make to the pond with the smaller container... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2c7ab3bcf9be8b952a5ee77f019cee8f4e587566accd2a67671c5f7a92c7c7b | Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin? | [
"42"
] | [
"Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9f9fa5354881738480a95a1ddb943b7c3b460ca525534e7100f0b17f9f7ed7e | Andrew runs on skis faster than Victor but slower than Eugene. They simultaneously started from point A, in the same direction along a circular path and stopped at the moment when all three were in point B. During this time, Eugene passed Victor 13 times (not including the final stop). How many times (in total) did a s... | [
"25"
] | [
"Andrew runs on skis faster than Victor but slower than Eugene. They simultaneously started from point A, in the same direction along a circular path and stopped at the moment when all three were in point B. During this time, Eugene passed Victor 13 times (not including the final stop). How many times (in total) di... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-dffbab977bd22eb0c10216c9271d0ad8ad413c9e47ac6a53f7f486732a578c34 | Andrew starts with a single card worth $1$ dollar. In each iteration, a random card (worth $n$ dollars) currently in his hand is chosen, and a new card worth $5n$ dollars is added to his hand. The game ends when Andrew has $10$ cards. What is the expected amount of money (in dollars) that Andrew wins? | [
"2002"
] | [
"Andrew starts with a single card worth $1$ dollar. In each iteration, a random card (worth $n$ dollars) currently in his hand is chosen, and a new card worth $5n$ dollars is added to his hand. The game ends when Andrew has $10$ cards. What is the expected amount of money (in dollars) that Andrew wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8bc219dfe95ec110c9e103e72e9f9f4a542668c3f1ac10e995b0fd306ed40053 | Andrew starts with an integer $n$ between $1$ and $1000$, inclusive. At each step, he replaces $n$ with $1000-2n$, until he gets a negative number. Let $s(n)$ be the number of steps for the process to end. If $s(n)$ takes its maximum value for $n=m$, what is the value of $m+s(m)$? | [
"343"
] | [
"Andrew starts with an integer $n$ between $1$ and $1000$, inclusive. At each step, he replaces $n$ with $1000-2n$, until he gets a negative number. Let $s(n)$ be the number of steps for the process to end. If $s(n)$ takes its maximum value for $n=m$, what is the value of $m+s(m)$?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-2e5c682a42f4136811778acb71c1b51e845683260c0f893dac16b80085505f73 | Andrew the ant starts at vertex $A$ of square $ABCD$. Each time he moves, he chooses the clockwise vertex with probability $\frac{2}{3}$ and the counter-clockwise vertex with probability $\frac{1}{3}$. What is the probability that he ends up on vertex $A$ after $6$ moves? | [
"\\(\\frac{228}{243}\\)"
] | [
"Andrew the ant starts at vertex $A$ of square $ABCD$. Each time he moves, he chooses the clockwise vertex with probability $\\frac{2}{3}$ and the counter-clockwise vertex with probability $\\frac{1}{3}$. What is the probability that he ends up on vertex $A$ after $6$ moves?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44a2783a896d1eba10f44846376ea2bd8d9d7023a6336764ff4e66c7c5124ca2 | Andrew's grandfather's age is twelve times Andrew's age. If Andrew's grandfather was 55 years old when Andrew was born, how many years old is Andrew now? | [
"5"
] | [
"Andrew's grandfather's age is twelve times Andrew's age. If Andrew's grandfather was 55 years old when Andrew was born, how many years old is Andrew now?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7b7108db59f41d96ebeb2122d7441277eb21d138df44f7c3982d7488f6a4bbf | Andrew's school has 1024 lockers, all initially closed. He starts from locker number one and toggles every locker (opens if closed, closes if open). He then toggles every 2nd locker, every 3rd locker, and so on, up to every 1024th locker. How many lockers are closed after he finishes? | [
"992"
] | [
"Andrew's school has 1024 lockers, all initially closed. He starts from locker number one and toggles every locker (opens if closed, closes if open). He then toggles every 2nd locker, every 3rd locker, and so on, up to every 1024th locker. How many lockers are closed after he finishes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-7a651261a70df711390ef9a710be6c4d903a46ca94aee8f656a9a33d6262879c | Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and... | [
"252"
] | [
"Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew like... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-1107f6e9ba5e25a0b9f01f369a4a47d44eaf4e4b2aaed256e1c50a8a5dffe194 | Andrew, Bob, Catherine, Derek, and Elan each plan on going to a restaurant between 5 and 7 p.m. for dinner. They each arrive at a random time. If a person arrives and no one else is there, the person waits 15 minutes, and if no one arrives during the 15 minutes, the person leaves. If, however, someone arrives during th... | [
"\\dfrac{225}{1024}"
] | [
"Andrew, Bob, Catherine, Derek, and Elan each plan on going to a restaurant between 5 and 7 p.m. for dinner. They each arrive at a random time. If a person arrives and no one else is there, the person waits 15 minutes, and if no one arrives during the 15 minutes, the person leaves. If, however, someone arrives duri... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-b9e8c52b4d038f357c4151a0f2d19151e36acb2f0a83f422b2d916dd283d5a1b | Andrey and Sasha play a game where Andrey marks an arbitrary point on the plane that has not yet been marked, and Sasha colors this point either white or black. After Andrey's 100th move, determine the maximum number of lines that can be drawn such that all white points lie in one half-plane and all black points lie in... | [
"50"
] | [
"Andrey and Sasha play a game where Andrey marks an arbitrary point on the plane that has not yet been marked, and Sasha colors this point either white or black. After Andrey's 100th move, determine the maximum number of lines that can be drawn such that all white points lie in one half-plane and all black points l... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-aaff9cf41a32d1ae3a663aa0c8d8f87dbf742c750b508f0855f6cf22dc29dc4e | Andriy and Bogdan take turns placing integers from $1$ to $n$ inclusive in the cells of an $n \times n$ board such that no row or column contains two equal numbers. Andriy begins, and the player who cannot make a move loses. Determine which player has a winning strategy depending on $n$. | [
"Bogdan wins for even \\( n \\) and Andriy wins for odd \\( n \\)."
] | [
"Andriy and Bogdan take turns placing integers from $1$ to $n$ inclusive in the cells of an $n \\times n$ board such that no row or column contains two equal numbers. Andriy begins, and the player who cannot make a move loses. Determine which player has a winning strategy depending on $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 14
} | 23 | 0 | 0 |
omr-ws-22395101f0e2cacd17298dd3d6d1e9ec7d1a867d01c64a545a75854c147514b7 | Andriyko has a rectangle desk and a lot of stripes that lie parallel to the sides of the desk. For every pair of stripes, one is under the other. In the desired configuration, for every four stripes such that two are parallel to one side of the desk and two are parallel to the other side, one of them is under the two o... | [
"1024"
] | [
"Andriyko has a rectangle desk and a lot of stripes that lie parallel to the sides of the desk. For every pair of stripes, one is under the other. In the desired configuration, for every four stripes such that two are parallel to one side of the desk and two are parallel to the other side, one of them is under the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-1a96c8e4538a2d886888fa4f911f00ef03b25f6f53144c754393a7a94bb0ad6d | Andrés and Blas play a game where they take turns placing digits from 1 to 9 in six spaces to form a six-digit number. The digits can be repeated. If the number formed is a multiple of 143, Blas wins; otherwise, Andrés wins. Determine who has the winning strategy and describe that strategy. | [
"Blas"
] | [
"Andrés and Blas play a game where they take turns placing digits from 1 to 9 in six spaces to form a six-digit number. The digits can be repeated. If the number formed is a multiple of 143, Blas wins; otherwise, Andrés wins. Determine who has the winning strategy and describe that strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-f996dcf48d1c98e23d4171d42801c119c2b4a28bee9de67ccfdf9cbcad2f2981 | Andy and Ben rizz up $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ individuals respectively over $n$ days, where $b$ is a permutation of $a$. Andy's rizz score is $\frac{a_1^2}{a_2}+\frac{a_2^2}{a_3}+\cdots + \frac{a_{n-1}^2}{a_n}$, and Ben's rizz score is similarly defined. Calculate the minimum possible value o... | [
"-\\dfrac{63}{4}"
] | [
"Andy and Ben rizz up $a_1, a_2, \\cdots, a_n$ and $b_1, b_2, \\cdots, b_n$ individuals respectively over $n$ days, where $b$ is a permutation of $a$. Andy's rizz score is $\\frac{a_1^2}{a_2}+\\frac{a_2^2}{a_3}+\\cdots + \\frac{a_{n-1}^2}{a_n}$, and Ben's rizz score is similarly defined. Calculate the minimum possi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-fde514bb65bd83896b445f1cc1bfbb9da4af024600b873732e832b434c63ee5a | Andy and Bertha play a game where the first person to flip 2 heads in a row wins. Andy flips the coin first, and they take turns flipping the coin. What is the probability that Andy wins the game? | [
"\\(\\frac{4}{7}\\)"
] | [
"Andy and Bertha play a game where the first person to flip 2 heads in a row wins. Andy flips the coin first, and they take turns flipping the coin. What is the probability that Andy wins the game?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-520468c67f2c54c17a81c40d428c8d1411e636f419140a5a913ec67640a741ca | Andy and Bertha play a game where the first person to flip 2 heads in a row wins. Andy flips the coin first, and they take turns flipping the coin. What is the probability that Andy wins the game? Use recursion to solve the problem. | [
"\\(\\frac{3}{5}\\)"
] | [
"Andy and Bertha play a game where the first person to flip 2 heads in a row wins. Andy flips the coin first, and they take turns flipping the coin. What is the probability that Andy wins the game? Use recursion to solve the problem."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da339e21d2739dac7b28798cdf68d7331e2a43eb0b17e8a41d5a0766e3a529b6 | Andy and Brandy have decided to share a cake in an unusual way. Andy will take some fraction of the cake first. From then on, Brandy and Andy will alternate taking half of what remains each time. Eventually, there is so little left that they decide they are finished. What fraction of the cake should Andy take first to ... | [
"\\(\\frac{1}{4}\\)"
] | [
"Andy and Brandy have decided to share a cake in an unusual way. Andy will take some fraction of the cake first. From then on, Brandy and Andy will alternate taking half of what remains each time. Eventually, there is so little left that they decide they are finished. What fraction of the cake should Andy take firs... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-e81a75ab353dc835d7d2a62bcf468c5e54cb0846a9af4b849d287a979dc64ade | Andy constructs the smallest subset \( S \) of positive integers with specific properties. Calculate the number of different possible sums of two distinct numbers in \( S \) that are both less than 2022. | [
"206"
] | [
"Andy constructs the smallest subset \\( S \\) of positive integers with specific properties. Calculate the number of different possible sums of two distinct numbers in \\( S \\) that are both less than 2022."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd26a9258bd63dfc0492100c375a4c6625f36717b6f5eb25f77d063b2c61c347 | Andy drove from home to Greenville in one hour at a constant speed. He then drove from Greenville to Orange Creek in one hour and forty minutes. The distance from his home to Greenville is \( ab \) miles, where \( ab \) is a 2-digit positive integer, and the distance from Greenville to Orange Creek is \( ba \) miles. F... | [
"48 miles.\n\nSince the problem states the answer is 48 miles, we will use that as the final answer."
] | [
"Andy drove from home to Greenville in one hour at a constant speed. He then drove from Greenville to Orange Creek in one hour and forty minutes. The distance from his home to Greenville is \\( ab \\) miles, where \\( ab \\) is a 2-digit positive integer, and the distance from Greenville to Orange Creek is \\( ba \... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-155f7414b9a5253214ddbf7ea2081820b6b57e5b01d61bcb5cdf2904c71708ca | Andy has $2867$ positive integers written on a whiteboard. Let $S$ be the sum of any $1434$ integers chosen from these numbers. Find the minimum number of distinct possible remainders when $S$ is divided by $1434$. | [
"1"
] | [
"Andy has $2867$ positive integers written on a whiteboard. Let $S$ be the sum of any $1434$ integers chosen from these numbers. Find the minimum number of distinct possible remainders when $S$ is divided by $1434$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab01c256138b6a5bbf1c6741cd3af1a2da009bd61b5074ec9ec5b8d17bf76229 | Andy has 2017 boxes, numbered from 1 to 2017, and 2017 balls, numbered from 1 to 2017. He places one ball in each box randomly. Denote $b_i$ as the value of the ball inside the $i$th box. What are the last three digits of the expected value of the sum of all $ib_i$ for $1 \le i \le 2017$? | [
"377"
] | [
"Andy has 2017 boxes, numbered from 1 to 2017, and 2017 balls, numbered from 1 to 2017. He places one ball in each box randomly. Denote $b_i$ as the value of the ball inside the $i$th box. What are the last three digits of the expected value of the sum of all $ib_i$ for $1 \\le i \\le 2017$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51b4b29d8b6963fb409f7cf309fbb689359853f92f93bd47a389bca95a77ce46 | Andy the Ant is crawling on a regular hexagon, starting at a vertex chosen at random. Every second, he moves from the vertex he currently resides on to an adjacent vertex. Let \( a_n \) be the probability that after \( n \) seconds, Andy will again be on the vertex he started at. What is the least positive integer \( n... | [
"100"
] | [
"Andy the Ant is crawling on a regular hexagon, starting at a vertex chosen at random. Every second, he moves from the vertex he currently resides on to an adjacent vertex. Let \\( a_n \\) be the probability that after \\( n \\) seconds, Andy will again be on the vertex he started at. What is the least positive int... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf86f2c8b356927ed05c3b230287dce90f6ef12356b23ec7b2093e52af932120 | Andy the Ant starts at $(-20, 20)$ and moves in a sequence where he increases his distance by 1 unit each time and always turns $90^\circ$ degrees left. He moves 1 unit east, then 2 units north, then 3 units west, and so on. What is the location of the point at which Andy makes the 2020th left turn? | [
"(-1030, -990)"
] | [
"Andy the Ant starts at $(-20, 20)$ and moves in a sequence where he increases his distance by 1 unit each time and always turns $90^\\circ$ degrees left. He moves 1 unit east, then 2 units north, then 3 units west, and so on. What is the location of the point at which Andy makes the 2020th left turn?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-bae3f69b56566d7135a8189b15a0d6415ecec189a81dd0495b2f3c9224922a2e | Andy the Banana Thief and Sheriff Buffkin each pick a permutation of the 6 houses labeled 1 through 6. On the $n^{th}$ day, Andy and Sheriff Buffkin visit the $n^{th}$ house in their respective permutations. Andy is caught on the first day they visit the same house. Given that Sheriff Buffkin catches Andy within 6 days... | [
"9"
] | [
"Andy the Banana Thief and Sheriff Buffkin each pick a permutation of the 6 houses labeled 1 through 6. On the $n^{th}$ day, Andy and Sheriff Buffkin visit the $n^{th}$ house in their respective permutations. Andy is caught on the first day they visit the same house. Given that Sheriff Buffkin catches Andy within 6... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2310589d7f03e9a4da531ab706ae8aeda019a96770d88385263a1aa0ca4a2876 | Andy the ant is at point 1 and would like to get to point 20 by moving right and down across each point but cannot go upon points 4, 5, 9, or 10 or else Anna the anteater will eat him. So, how many ways can he get to his destination? | [
"22"
] | [
"Andy the ant is at point 1 and would like to get to point 20 by moving right and down across each point but cannot go upon points 4, 5, 9, or 10 or else Anna the anteater will eat him. So, how many ways can he get to his destination?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ac7ceee4a92a86a7d976d72973f2cfc2368b2ee139c7c5ccdc1c2921bc1a7f8 | Andy the ant is at point 1 and would like to get to point 20 by moving right and down across each point but cannot go upon points 4, 5, 9, or 10. How many ways can he get to his destination? | [
"22"
] | [
"Andy the ant is at point 1 and would like to get to point 20 by moving right and down across each point but cannot go upon points 4, 5, 9, or 10. How many ways can he get to his destination?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-97b1a0660ce0586038a4c5f9f286becc9ed83a3106f58db08c321f943dc90cfa | Andy the ant stands on a vertex of a unit cube and wants to walk a total of 20 units in as few steps as possible, where a step is defined as moving between 2 (not necessarily adjacent) vertices of the cube. Andy cannot go back to a vertex he was just on. Find the minimum number of steps Andy needs to take. | [
"12"
] | [
"Andy the ant stands on a vertex of a unit cube and wants to walk a total of 20 units in as few steps as possible, where a step is defined as moving between 2 (not necessarily adjacent) vertices of the cube. Andy cannot go back to a vertex he was just on. Find the minimum number of steps Andy needs to take."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4d8d8759a96dd01324ca2be207247fd21a51a30dba4e55046bb172c3ab47654 | Andy the unicorn is on a number line from 1 to 2019. He starts on 1. Each step, he randomly and uniformly picks an integer greater than the integer he is currently on, and goes to it. He stops when he reaches 2019. What is the probability that he is ever on 1984? | [
"\\dfrac{1}{36}"
] | [
"Andy the unicorn is on a number line from 1 to 2019. He starts on 1. Each step, he randomly and uniformly picks an integer greater than the integer he is currently on, and goes to it. He stops when he reaches 2019. What is the probability that he is ever on 1984?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b441c8b07321d1193e00c7ce923c80900b0da86b1637e7542dc552bb58940cd | Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first? | [
"Beth"
] | [
"Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-7ff9ada11f7bbcde54e36ae942ef2552bba423e2ca70feb139a66f332a1556b5 | Andy, Bertha, Cindy, and Dan are sitting in a row (in that order). They each have name tags that are randomly distributed among the four of them. What is the probability that exactly one of their name tags ends up with someone adjacent to its owner? | [
"\\(\\frac{1}{12}\\)"
] | [
"Andy, Bertha, Cindy, and Dan are sitting in a row (in that order). They each have name tags that are randomly distributed among the four of them. What is the probability that exactly one of their name tags ends up with someone adjacent to its owner?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc0d288d99a36bf6cd40104b69312e3cf3658bec5e03cb7cc566d5018a01758e | Andy, Bess, Charley, and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley, and finally Dick, after which Andy moves again, and so on. At each move, a player must paint several unpainted squares forming a $2 \times 1$, $1 \times 2$, $1 \times 3$, or $3 \times 1$ recta... | [
"250000"
] | [
"Andy, Bess, Charley, and Dick play on a $1000 \\times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley, and finally Dick, after which Andy moves again, and so on. At each move, a player must paint several unpainted squares forming a $2 \\times 1$, $1 \\times 2$, $1 \\times 3$, or $3 \\time... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-254e8e6974e7c8ac5df03f7a4c20c3620f5623be107e3cafe3e9016c7694baf5 | Andy, Beth, Charlie, and Daniel take a test with thirty questions. Andy and Beth together get the same number of questions wrong as Charlie and Daniel together. Andy and Daniel together get four more questions wrong than Beth and Charlie do together. If Charlie gets five questions wrong, how many questions does Andy ge... | [
"7"
] | [
"Andy, Beth, Charlie, and Daniel take a test with thirty questions. Andy and Beth together get the same number of questions wrong as Charlie and Daniel together. Andy and Daniel together get four more questions wrong than Beth and Charlie do together. If Charlie gets five questions wrong, how many questions does An... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b4ab4d6c02f14388bef2866fc8a1bab0eb47ff5807022646d8882ec80bf79757 | Andy, Bob, Jimmy, Polar, and Samuel are standing clockwise in a circle. Jimmy has a ball at the beginning. He can pass the ball to Bob or Polar randomly, and whoever gets the ball must pass it randomly to the person right next to him. What is the expected number of times the ball is passed before Andy gets the ball for... | [
"6"
] | [
"Andy, Bob, Jimmy, Polar, and Samuel are standing clockwise in a circle. Jimmy has a ball at the beginning. He can pass the ball to Bob or Polar randomly, and whoever gets the ball must pass it randomly to the person right next to him. What is the expected number of times the ball is passed before Andy gets the bal... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-7b181878286b397fbff8919c9a64193850c00be9467193b7bc8056e515a29ad2 | Andy, Bob, and Casey play a game where Andy picks a random real number from 0 to \( \frac{1}{3} \), Bob picks a random real number from 0 to \( \frac{2}{3} \), and Casey picks a random real number from 0 to 1. Whoever has the largest number gets a point. They play 96 rounds. How many times is Casey expected to win? | [
"\\dfrac{184}{3}"
] | [
"Andy, Bob, and Casey play a game where Andy picks a random real number from 0 to \\( \\frac{1}{3} \\), Bob picks a random real number from 0 to \\( \\frac{2}{3} \\), and Casey picks a random real number from 0 to 1. Whoever has the largest number gets a point. They play 96 rounds. How many times is Casey expected ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-d185be446ca768e7e8971b6a7f580ce64a0662d573113fe745b6a4cb79fc587c | Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first? | [
"Beth"
] | [
"Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-29a0d17b2c4c6f28cc5bd06aa3a2386ffa2e1c8608c00fc51d69e403d878b76d | Anett is drawing $X$-es on a $5 \times 5$ grid. For each newly drawn $X$, she gets points based on the number of $X$-es in the same row and column that can be reached from the newly drawn $X$ with horizontal and vertical steps, respectively, moving only on fields that were previously marked with $X$-es. What is the max... | [
"100"
] | [
"Anett is drawing $X$-es on a $5 \\times 5$ grid. For each newly drawn $X$, she gets points based on the number of $X$-es in the same row and column that can be reached from the newly drawn $X$ with horizontal and vertical steps, respectively, moving only on fields that were previously marked with $X$-es. What is t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-62af6a7bfc6009d5422dbb5a4ec0d1005d5c45263c481d239ce5f167bf71b5cc | Anett is drawing $X$-es on a $5 \times 5$ grid. For each newly drawn $X$, she gets points based on the number of $X$-es in the same row and column that can be reached from the newly drawn $X$ with horizontal and vertical steps, respectively, moving only on fields that were previously marked with $X$-es. What is the min... | [
"40"
] | [
"Anett is drawing $X$-es on a $5 \\times 5$ grid. For each newly drawn $X$, she gets points based on the number of $X$-es in the same row and column that can be reached from the newly drawn $X$ with horizontal and vertical steps, respectively, moving only on fields that were previously marked with $X$-es. What is t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-1464e740eb350484a239e571b8cdd3c1b2d8d0a08ecb192ec9ebae84e63915f6 | Ang and Mine are on a treasure hunt starting from the same point. Ang's first clue is 50 meters south-west of the starting point. Mine reaches her first clue when she is 120 meters north-west of Ang's first clue. Mine then makes a quarter turn clockwise and runs the same distance to her second clue as she ran to her fi... | [
"139"
] | [
"Ang and Mine are on a treasure hunt starting from the same point. Ang's first clue is 50 meters south-west of the starting point. Mine reaches her first clue when she is 120 meters north-west of Ang's first clue. Mine then makes a quarter turn clockwise and runs the same distance to her second clue as she ran to h... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 13
} | 34 | 0 | 0 |
omr-ws-4998569827a0ef16ed1b99afc419b75fbb4cbd064d4f1c3c11532c8c34072f42 | Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\dfra... | [
"471"
] | [
"Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-2899970b7c9e7df39d403681f992488c7d152f7952a575ec997a92f1d2fff26f | Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac... | [
"471"
] | [
"Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-835110173d1a5fe18511f864bebf3b09115fbdb309a20fb9f420cbaa24d165f4 | Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac... | [
"471"
] | [
"Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f8d71390af8c0df05661f576ffac39f5bc4f1f08a0adb38c102184c25e1b7626 | Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green. There are 5 empty boxes. Each person randomly and independently places one of their blocks into each box. What is the probability that at least one box receives 3 blocks all of the same color? Express the probability as $\frac{m}{n}$,... | [
"471"
] | [
"Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green. There are 5 empty boxes. Each person randomly and independently places one of their blocks into each box. What is the probability that at least one box receives 3 blocks all of the same color? Express the probability as $\\frac{m... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 7,
"tir": 8
} | 60 | 0 | 0 |
omr-ws-9b4ddfa9392625912482059f1334699bdb96a7674b11f8ce60042c7368ac6e26 | Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green. There are 5 empty boxes. Each person randomly and independently places one of their blocks into each box. What is the probability that at least one box receives 3 blocks all of the same color? Express the probability as a fraction \(\... | [
"471"
] | [
"Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green. There are 5 empty boxes. Each person randomly and independently places one of their blocks into each box. What is the probability that at least one box receives 3 blocks all of the same color? Express the probability as a fractio... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 11
} | 65 | 0 | 0 |
omr-ws-22b56693ddbb4ee0aea4c0e313bee40c7274809c877c38050edc3db59ced6a15 | Angel has a warehouse with 100 piles of 100 pieces of rubbish each. Each morning, Angel can either clear every piece of rubbish from a single pile or one piece of rubbish from each pile. Each evening, a demon adds one piece of rubbish to each non-empty pile or creates a new pile with one piece if all piles are empty. W... | [
"199"
] | [
"Angel has a warehouse with 100 piles of 100 pieces of rubbish each. Each morning, Angel can either clear every piece of rubbish from a single pile or one piece of rubbish from each pile. Each evening, a demon adds one piece of rubbish to each non-empty pile or creates a new pile with one piece if all piles are emp... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-020dafa8ff1da8289f01cb12bc1125f6c19336ca1729304db3db17ebec476d34 | Angel has a warehouse with 100 piles of 100 pieces of rubbish each. Each morning, Angel performs one of the following moves: (a) clears every piece of rubbish from a single pile, or (b) clears one piece of rubbish from each pile. Each evening, a demon performs one of the following moves: (a) adds one piece of rubbish t... | [
"100",
"199"
] | [
"Angel has a warehouse with 100 piles of 100 pieces of rubbish each. Each morning, Angel performs one of the following moves: (a) clears every piece of rubbish from a single pile, or (b) clears one piece of rubbish from each pile. Each evening, a demon performs one of the following moves: (a) adds one piece of rubb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-6c603bba873a61e316522a9f16cb84502af9e131abafd1d85cad7e13e95a70d5 | Angela has $a$ marbles, Brian has twice as many marbles as Angela, Caden has three times as many marbles as Brian, and Daryl has five times the number of marbles Caden has. If in total Angela, Brian, Caden and Daryl have 78 marbles, what is the value of $a?$ | [
"2"
] | [
"Angela has $a$ marbles, Brian has twice as many marbles as Angela, Caden has three times as many marbles as Brian, and Daryl has five times the number of marbles Caden has. If in total Angela, Brian, Caden and Daryl have 78 marbles, what is the value of $a?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b90cfdd2b3fd082aec29dd26a3cc07dde1764b11c29be8663268f1b8a75b16c | Angela has deposited $\$8,\!000$ into an account that pays $6\%$ interest compounded annually. Bob has deposited $\$10,\!000$ into an account that pays $7\%$ simple annual interest. In $20$ years Angela and Bob compare their respective balances. To the nearest dollar, what is the positive difference between their balan... | [
"\\$1,\\!657"
] | [
"Angela has deposited $\\$8,\\!000$ into an account that pays $6\\%$ interest compounded annually.\n\nBob has deposited $\\$10,\\!000$ into an account that pays $7\\%$ simple annual interest.\n\nIn $20$ years Angela and Bob compare their respective balances. To the nearest dollar, what is the positive difference be... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-207e3589be539e8ec640df5a23ceb0a11b8a362932146f503f0052b9ddb8f650 | Angela's wardrobe contains 12 shirts, 8 skirts, and 11 shoes. What fraction of her wardrobe must she draw out at random to guarantee at least two of each item? Items drawn are not put back. | [
"\\dfrac{25}{31}"
] | [
"Angela's wardrobe contains 12 shirts, 8 skirts, and 11 shoes. What fraction of her wardrobe must she draw out at random to guarantee at least two of each item? Items drawn are not put back."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-66007c47341b6b413b79a06cf2561fa62aca1cca1dcf548ab34389f6fb18f190 | Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1,2,3,4,5,6,7,8 \}$ that consists of consecutive integers. The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"421"
] | [
"Angela, Bill, and Charles each independently and randomly choose a subset of $\\{ 1,2,3,4,5,6,7,8 \\}$ that consists of consecutive integers. The expected number of elements in the intersection of the three chosen sets is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-9d76a6795f57fb2ebc90696b5e9d12238d98062f0defbe99513cfe124678f79f | Angelina drove at an average rate of 80 kph and then stopped for 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time \( t \) in hours that she drove before her stop... | [
"\\( 80t + 100\\left(\\frac{8}{3} - t\\right) = 250 \\)"
] | [
"Angelina drove at an average rate of 80 kph and then stopped for 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time \\( t \\) in hours that she drove before h... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 19
} | 42 | 0 | 0 |
omr-ws-4c18ebba21c72f43d76dfdba74ed662ba3e31e50b749cb8cf907aa0a3b60d010 | Angie can paddle her canoe at 6 miles per hour in still water. When she tries to paddle directly across an 80-foot wide river, she ends up on the opposite bank 60 feet downstream from the starting point. How fast is the river flowing? | [
"4.5 miles/hour"
] | [
"Angie can paddle her canoe at 6 miles per hour in still water. When she tries to paddle directly across an 80-foot wide river, she ends up on the opposite bank 60 feet downstream from the starting point. How fast is the river flowing?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-81537f13e5498297823aa18f250343e887cd6c5130cfde9905f307160b173f99 | Angie starts with 8 pieces of paper. At least one piece is cut into 8 smaller pieces, then at least one of these smaller pieces is cut into 8 smaller pieces, and then at least one of these smallest pieces is cut into 8 smaller pieces. If $x$ represents the final number of pieces of paper that Angie has, find the sum of... | [
"2255"
] | [
"Angie starts with 8 pieces of paper. At least one piece is cut into 8 smaller pieces, then at least one of these smaller pieces is cut into 8 smaller pieces, and then at least one of these smallest pieces is cut into 8 smaller pieces. If $x$ represents the final number of pieces of paper that Angie has, find the s... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16408853d86cf78897bcbbdd3c33eccaa563eb735b83dd7a9587c2dccca5a74e | Angie's class has 2 girls for every 3 boys. If there are 20 students in the class, how many girls are in Angie's class? | [
"8"
] | [
"Angie's class has 2 girls for every 3 boys. If there are 20 students in the class, how many girls are in Angie's class?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9bbe0130e8bbcba6b11ac9ac3e03675b47d3900ecca414453e293ccdb7c5d8b2 | Angie, Beth, and Chester each start with 100 tokens. Each turn, one of the players is randomly selected and gives 5 tokens to each of the other two players. After six turns, what is the probability that each player has exactly 100 tokens? Express the probability in the form $\dfrac{a}{b}$, where $a$ and $b$ are coprime... | [
"91"
] | [
"Angie, Beth, and Chester each start with 100 tokens. Each turn, one of the players is randomly selected and gives 5 tokens to each of the other two players. After six turns, what is the probability that each player has exactly 100 tokens? Express the probability in the form $\\dfrac{a}{b}$, where $a$ and $b$ are c... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-7b2276c2bf3bfec5364c2a5aa67859fb47d9664a191e9e3ef392a71a520e6d36 | Angle \(\angle AOB = 60^\circ\). On the side \(OA\) at the point \(C\) is the first pedestrian moving in the direction of the point \(A\) with a speed of \(v\). On the side \(OB\) at the point \(D\) there is a second pedestrian, moving in the direction of the point \(O\) also with the speed \(v\), with \(OC = 1\), \(OD... | [
"3"
] | [
"Angle \\(\\angle AOB = 60^\\circ\\). On the side \\(OA\\) at the point \\(C\\) is the first pedestrian moving in the direction of the point \\(A\\) with a speed of \\(v\\). On the side \\(OB\\) at the point \\(D\\) there is a second pedestrian, moving in the direction of the point \\(O\\) also with the speed \\(v\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d2b7bce0b110271094f80c52b0596efc9b73b5f67c54e26063dfbbb473772d9d | Angle bisector \(AD\) is drawn in an isosceles triangle \(ABC\) (\(AB = BC\)). For what values of angle \(\angle A\) on line \(AC\) is there a point \(E\) such that \(BD = DE = EC\)? | [
"\\(60^\\circ\\)"
] | [
"Angle bisector \\(AD\\) is drawn in an isosceles triangle \\(ABC\\) (\\(AB = BC\\)). For what values of angle \\(\\angle A\\) on line \\(AC\\) is there a point \\(E\\) such that \\(BD = DE = EC\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-fb5e68b795e5f00fce485f5447b2b4108e43890847a77c4d9602ba00d513d708 | Angles $A$ and $B$ are supplementary. If the measure of angle $A$ is $8$ times angle $B$, what is the measure of angle A? | [
"160"
] | [
"Angles $A$ and $B$ are supplementary. If the measure of angle $A$ is $8$ times angle $B$, what is the measure of angle A?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-679b57a0bdbe46a88d2ecd1c4a402df6db5f3c3f46439fef6dfc57f830d84413 | Angles $B$ and $C$ are given in $\triangle ABC$. A circle is drawn through the midpoint $O$ of the side $AB$ and the vertex $A$ which touches the side $BC$. Find the ratio of the radius of the circle to the length of the side $BC$. | [
"$R_1=\\frac{\\sin C}{\\sin (B+C)}\\cdot \\frac{3-2\\sqrt 2\\cos B}{4\\sin B}$, $R_2=\\frac{\\sin C}{\\sin (B+C)}\\cdot \\frac{3+2\\sqrt 2\\cos B}{4\\sin B}$"
] | [
"Angles $B$ and $C$ are given in $\\triangle ABC$. A circle is drawn through the midpoint $O$ of the side $AB$ and the vertex $A$ which touches the side $BC$. Find the ratio of the radius of the circle to the length of the side $BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b40c40ed2ed652ad4a82c91f55edc06d4463fcf817c773596f962a4e97ff138 | Angles $B$ and $C$ are given in $\triangle ABC$. The bisector of $\angle BAC$ cuts the side $BC$ at a point $D$ and the circle circumscribed about $\triangle ABC$ at a point $E$. Find the ratio $| AE | : | DE |$. | [
"$\\frac{|AE|}{|DE|}=\\frac{\\cos ^2 ((\\beta -\\gamma )/2)} {\\cos ^2 ((\\beta +\\gamma )/2)}$, where $\\gamma =\\angle ACB$, $\\beta =\\angle ABC$"
] | [
"Angles $B$ and $C$ are given in $\\triangle ABC$. The bisector of $\\angle BAC$ cuts the side $BC$ at a point $D$ and the circle circumscribed about $\\triangle ABC$ at a point $E$. Find the ratio $| AE | : | DE |$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98b7ab0ef6f4ab156c0416a1519cba318777b5b5052989af299e1b8621febb88 | Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she comp... | [
"053"
] | [
"Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-ea383d354f2537ba1c5776490581190c2943407c956cd71d770603ac970be1fb | Anh read a book. On the first day, she read $n$ pages in $t$ minutes. On the second day, she read $n + 1$ pages in $t + 1$ minutes. Each subsequent day, she read one more page and took one more minute than the previous day until she completely read the 374-page book in a total of 319 minutes. Find $n + t$. | [
"053"
] | [
"Anh read a book. On the first day, she read $n$ pages in $t$ minutes. On the second day, she read $n + 1$ pages in $t + 1$ minutes. Each subsequent day, she read one more page and took one more minute than the previous day until she completely read the 374-page book in a total of 319 minutes. Find $n + t$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-38543c2a2fc2e4e274b91aece2f408c74d95efd9129be0b22651cfb61410fe84 | Anja and Bernd take turns removing stones from a heap initially consisting of $n$ stones ($n \ge 2$). Anja begins by removing at least one but not all the stones. In each subsequent turn, a player must remove at least one stone and at most as many stones as were removed in the preceding move. The player who removes the... | [
"Anja wins if $n$ is odd or $n$ is not a power of $2$, and Bernd wins if $n$ is a power of $2$."
] | [
"Anja and Bernd take turns removing stones from a heap initially consisting of $n$ stones ($n \\ge 2$). Anja begins by removing at least one but not all the stones. In each subsequent turn, a player must remove at least one stone and at most as many stones as were removed in the preceding move. The player who remov... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 24
} | 29 | 0 | 0 |
omr-ws-8f5b3cc33a1b4840e1099cda0cd97ad6950c869df81e45ff1872a656e3979c10 | Anja writes $2014$ integers on the board. If the arithmetic mean of any three of these numbers is also among those $2014$ numbers, what is the minimum number of distinct integers Anja could have written on the board? | [
"1"
] | [
"Anja writes $2014$ integers on the board. If the arithmetic mean of any three of these numbers is also among those $2014$ numbers, what is the minimum number of distinct integers Anja could have written on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27c41f519b0917f88e1ed4f657147ae24648c353dad9f87f23ebcf32049abbfc | Anka jumps off an 80-meter high building and bounces up to 3/4 of the previous height each time. How many times can she bounce while still moving a total distance of less than 300 meters? | [
"2"
] | [
"Anka jumps off an 80-meter high building and bounces up to 3/4 of the previous height each time. How many times can she bounce while still moving a total distance of less than 300 meters?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c56badf5c7b6bc981ed98e091a79187c0825e37abbca297ef07a16101ea6d41 | Ankit, Box, and Clark are playing a game. Clark chooses a prime number less than 100 and writes each digit of the prime number on a piece of paper, giving one digit to Ankit and one to Box without specifying which is the tens digit and which is the ones digit. The following exchange occurs: 1. Clark: There is only one ... | [
"11"
] | [
"Ankit, Box, and Clark are playing a game. Clark chooses a prime number less than 100 and writes each digit of the prime number on a piece of paper, giving one digit to Ankit and one to Box without specifying which is the tens digit and which is the ones digit. The following exchange occurs:\n\n1. Clark: There is o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-3ecb94d98057dd35350e0915c0af692e25a74cd6673e754fcdeee2fa2287d3d1 | Ann and Barbara divided 81 cards of the game SET between themselves, with Ann receiving 41 cards and Barbara receiving 40 cards. Each girl counted the number of ways they can form a SET of three cards from their own cards. What is the sum of the numbers they obtained? | [
"1080"
] | [
"Ann and Barbara divided 81 cards of the game SET between themselves, with Ann receiving 41 cards and Barbara receiving 40 cards. Each girl counted the number of ways they can form a SET of three cards from their own cards. What is the sum of the numbers they obtained?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cd33128e7fd0ac98ac75846a9cb8eeb124bad39da064fae2ca090e4cd231f99 | Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is 44 years, then what is Ann's age? | [
"24"
] | [
"Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is 44 years, then what is Ann's age?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-ed5fd56de3e8246b5bead42abf8d3ddae6652aa070573343149eea810bb6ecc7 | Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is 44, what is Ann’s age? | [
"24"
] | [
"Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is 44, what is Ann’s age?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-6a8bb9b5fc159eb01bc1ed6ad95ee05718f19965eb18dfd7833513a48b56597a | Ann and Ben play a game where Ann starts by constructing an acute-angled triangle $\Delta A_0B_0C_0$ which is not equilateral and in which every interior angle (when measured in degrees) is an integer. For each $\Delta A_nB_nC_n$ constructed by one player, the other player subsequently constructs $\Delta A_{n+1}B_{n+1}... | [
"88"
] | [
"Ann and Ben play a game where Ann starts by constructing an acute-angled triangle $\\Delta A_0B_0C_0$ which is not equilateral and in which every interior angle (when measured in degrees) is an integer. For each $\\Delta A_nB_nC_n$ constructed by one player, the other player subsequently constructs $\\Delta A_{n+1... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-ffa7b0c396073742fe0553fa082694b85e0d23edc3ac69860cb38f1ddd07eecb | Ann and Bill take turns writing digits on a sheet of paper, left to right, to form a 100-digit number. Ann starts with a nonzero digit. Bill wins if the resulting number divided by 11 leaves a remainder of 5; otherwise, Ann wins. Both players are good at mathematics. Who will win the game? | [
"Ann"
] | [
"Ann and Bill take turns writing digits on a sheet of paper, left to right, to form a 100-digit number. Ann starts with a nonzero digit. Bill wins if the resulting number divided by 11 leaves a remainder of 5; otherwise, Ann wins. Both players are good at mathematics. Who will win the game?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 6,
"tir": 3
} | 10 | 0 | 0 |
omr-ws-bbdacb34d02e5ba6811e6ebbca66f58c60fe7cec8e900e262faeeaff16bb7bba | Ann and Bob are part of a cross country team race where they alternate riding a horse and walking. Ann walks at a rate of \( v \) and rides at a rate of \( 17.9v \). Bob walks at a rate of \( 3.7v \) and rides at a rate of \( 4.4v \). Determine the fraction of the total race distance that Ann should ride in order to mi... | [
"\\(\\frac{112591}{143831}\\)"
] | [
"Ann and Bob are part of a cross country team race where they alternate riding a horse and walking. Ann walks at a rate of \\( v \\) and rides at a rate of \\( 17.9v \\). Bob walks at a rate of \\( 3.7v \\) and rides at a rate of \\( 4.4v \\). Determine the fraction of the total race distance that Ann should ride i... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 16
} | 36 | 0 | 0 |
omr-ws-8c57e4d767fc05213864c11a887e4803b7a4b0133f95587d41f7de99a73443b4 | Ann and Bob are playing a game where a piece of paper is divided into two regions, Region I and Region II. Ann starts by choosing a random point on the paper. If she chooses a point in Region I, she wins. If she chooses a point in Region II, Bob gets a chance to choose a point. If Bob chooses a point in Region II, he w... | [
"10"
] | [
"Ann and Bob are playing a game where a piece of paper is divided into two regions, Region I and Region II. Ann starts by choosing a random point on the paper. If she chooses a point in Region I, she wins. If she chooses a point in Region II, Bob gets a chance to choose a point. If Bob chooses a point in Region II,... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a6f1d14d776f398cd050b0868c2b2e709e1a623bb6dea65238d36a89f14e9172 | Ann and Bob are playing a game with squares labeled $1$ to $5$. On each turn, a player flips a fair coin: heads means moving to the next square, tails means staying at the current square. The first player to reach square $5$ wins. Ann starts at square $1$ and goes first, while Bob starts at square $2$. What is the prob... | [
"\\(\\frac{242}{729}\\)"
] | [
"Ann and Bob are playing a game with squares labeled $1$ to $5$. On each turn, a player flips a fair coin: heads means moving to the next square, tails means staying at the current square. The first player to reach square $5$ wins. Ann starts at square $1$ and goes first, while Bob starts at square $2$. What is the... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-ed3bf836d42860a4f0b076b1421a09b849bcf4ebbef23e178aa12ad492038dc3 | Ann and Bob play a game on the Cartesian plane starting at the origin. At each step, Ann chooses a number \( 0 \leq a \leq 1 \) and Bob moves the point in one of the four cardinal directions by a distance of \( a \). This process continues for 100 steps. Among every 100 consecutive moves, each of the four possible move... | [
"97"
] | [
"Ann and Bob play a game on the Cartesian plane starting at the origin. At each step, Ann chooses a number \\( 0 \\leq a \\leq 1 \\) and Bob moves the point in one of the four cardinal directions by a distance of \\( a \\). This process continues for 100 steps. Among every 100 consecutive moves, each of the four po... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-e279f7a5c751d432d01e3539f84c2e8a8f423f547ec950d4d3646218c8046516 | Ann and Bob play a game on the Cartesian plane starting at the origin. At each step, Ann chooses a number \( 0 \leq a \leq 1 \) and Bob moves the point in one of the four cardinal directions by a distance of \( a \). This process continues for 100 steps. Among every 100 consecutive moves, each of the four possible move... | [
"97"
] | [
"Ann and Bob play a game on the Cartesian plane starting at the origin. At each step, Ann chooses a number \\( 0 \\leq a \\leq 1 \\) and Bob moves the point in one of the four cardinal directions by a distance of \\( a \\). This process continues for 100 steps. Among every 100 consecutive moves, each of the four po... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-67e2a236a5ff85e63cf793af13953dd420771270bdcc6854b7863f666b757067 | Ann and Bruno play a game. Ann chooses 300 points in the plane and paints them red. Bruno wins if he can find 10 points that are collinear or 10 points such that no three of them are collinear. Determine which player can guarantee a victory. | [
"Bruno can guarantee a victory."
] | [
"Ann and Bruno play a game. Ann chooses 300 points in the plane and paints them red. Bruno wins if he can find 10 points that are collinear or 10 points such that no three of them are collinear. Determine which player can guarantee a victory."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-c4d5d263daa4e82b79af9f498dcafc7f85c691511faecfe2dfded394f5163063 | Ann and Max play a game on a $100 \times 100$ board. Ann writes an integer from 1 to 10,000 in each square of the board so that each number is used exactly once. Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves to reach the rightmost column, moving the token to a... | [
"500,000"
] | [
"Ann and Max play a game on a $100 \\times 100$ board. Ann writes an integer from 1 to 10,000 in each square of the board so that each number is used exactly once. Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves to reach the rightmost column, moving the toke... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdad98b97402f94726b96780a83d232791a0bfc15ec0b7830797a7da523ae88d | Ann and Max play a game on a $100 \times 100$ board. Ann writes an integer from 1 to 10000 in each square of the board so that each number is used exactly once. Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves to reach the rightmost column, moving the token to a ... | [
"495100"
] | [
"Ann and Max play a game on a $100 \\times 100$ board. Ann writes an integer from 1 to 10000 in each square of the board so that each number is used exactly once. Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves to reach the rightmost column, moving the token... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-6890bc2be48a0e4cdf4263b8f688936fe112bb3c1dfe4acb0cda589bdc7b26b4 | Ann flips 2011 fair coins and Bob flips 2010 fair coins. What is the probability that Bob flips more heads than Ann? | [
"\\dfrac{1}{2} - \\dfrac{\\dbinom{4021}{2010}}{2^{4021}}"
] | [
"Ann flips 2011 fair coins and Bob flips 2010 fair coins. What is the probability that Bob flips more heads than Ann?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-2d59972a4af03a0078fdb97a6813abeb841e9ba92ec24c8ed4c55f72d48faaa1 | Ann has 6 dice and rolls to see if at least one of them comes up six. Bob has 12 dice and rolls hoping for two or more to come up six. Who has the better chance of succeeding? What is Ann's probability of success? What is Bob's probability of success? | [
"Ann has the better chance of succeeding. Ann's probability of success is approximately \\( 0.6651 \\). Bob's probability of success is approximately \\( 0.6186 \\)."
] | [
"Ann has 6 dice and rolls to see if at least one of them comes up six. Bob has 12 dice and rolls hoping for two or more to come up six. Who has the better chance of succeeding? What is Ann's probability of success? What is Bob's probability of success?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 32
} | 32 | 0 | 0 |
omr-ws-95700db14e030ee3eeb78feff8239ee0f53c24ad4fe2c2fa0151aa6e53c3cb68 | Ann has a \(70 \times 70\) chocolate bar. Every day, Ann breaks the largest part of the chocolate into two rectangles. Find the smallest number of days after which there will not be any part of the chocolate with more than 1615 blocks. | [
"2"
] | [
"Ann has a \\(70 \\times 70\\) chocolate bar. Every day, Ann breaks the largest part of the chocolate into two rectangles. Find the smallest number of days after which there will not be any part of the chocolate with more than 1615 blocks."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-505fab35e2ae6752689eb12142b869a6162faf23cbd52248a160119eec217933 | Ann has walked 8 km at a speed of 4 km/h. She will now run at a speed of 8 km/h. How long does she have to run in order to have an overall average speed of 5 km/h? | [
"40 min"
] | [
"Ann has walked 8 km at a speed of 4 km/h. She will now run at a speed of 8 km/h. How long does she have to run in order to have an overall average speed of 5 km/h?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-a2f41d47b2c810e4fd92487bca41ee24452cf6a59ad65193d5d8adb27dea582b | Ann made a 3-step staircase using 18 toothpicks. How many additional toothpicks does she need to add to complete a 5-step staircase? | [
"22"
] | [
"Ann made a 3-step staircase using 18 toothpicks. How many additional toothpicks does she need to add to complete a 5-step staircase?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-5ddb81b28300fae26fd10d46fa9686420c5afae2321b804fbb5b2bc549bbc18d | Ann participates in a quiz where each question is worth its position (first question is 1 point, second is 2 points, etc.). She gets all questions right and receives a score of 149, but one question was counted twice. Determine which question was counted twice and what her actual score should be. | [
"Question 13 was counted twice, and her actual score is 136."
] | [
"Ann participates in a quiz where each question is worth its position (first question is 1 point, second is 2 points, etc.). She gets all questions right and receives a score of 149, but one question was counted twice. Determine which question was counted twice and what her actual score should be."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-5e1a6081e13e85773034fd7a0e77ab2bfbe98c4eed2f521c99b1bf3462f3c83e | Ann tosses a fair coin twice while Bob tosses the same coin three times. What is the sum of the numerator and denominator of the irreducible fraction representing the probability that they obtain the same number of heads? | [
"21"
] | [
"Ann tosses a fair coin twice while Bob tosses the same coin three times. What is the sum of the numerator and denominator of the irreducible fraction representing the probability that they obtain the same number of heads?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-6b0c457073ecf632dd8432e37b0726e7e38778c100af93c6b6ade9906be81f45 | Ann, Ben, and Cat each independently answer 2021 yes-no questions. The score of the team is the largest number of questions in which any two of them chose the same answer. Find the minimum possible score of the team. | [
"674"
] | [
"Ann, Ben, and Cat each independently answer 2021 yes-no questions. The score of the team is the largest number of questions in which any two of them chose the same answer. Find the minimum possible score of the team."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 7,
"tir": 3
} | 55 | 0 | 0 |
omr-ws-e813a79a478fcd3d2ba27bc8603a4e7f2237e9f760688b5d411ac210c12c8324 | Ann, Bon, and Carol sat for examinations in English, Maths, and Science, scoring a result out of 10. For each subject, the exam marks were one mark of \(a\), one mark of \(b\), and one mark of \(c\), where \(a\), \(b\), and \(c\) are different positive integers. The total marks obtained by each student for the 3 subjec... | [
"Ann: \\( (9, 5, 10) \\), Bon: \\( (5, 9, 1) \\), Carol: \\( (1, 5, 9) \\)"
] | [
"Ann, Bon, and Carol sat for examinations in English, Maths, and Science, scoring a result out of 10. For each subject, the exam marks were one mark of \\(a\\), one mark of \\(b\\), and one mark of \\(c\\), where \\(a\\), \\(b\\), and \\(c\\) are different positive integers. The total marks obtained by each student... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60df6f9c159cc1a7ac24ad145f4743ec0143c3b50f9e6a71a284b573845824c6 | Anna Lisa bought two dozen apples for $\$$15.60. At that same rate, how much would three dozen of these apples cost? | [
"\\$23.40"
] | [
"Anna Lisa bought two dozen apples for $\\$$15.60. At that same rate, how much would three dozen of these apples cost?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ddc20f58ab40727f8e03241e7f14dc3a4535a0938fbd3ad6bc6ace1ed16c1462 | Anna Taylor draws a rectangle \(ABCD\) with a perimeter of 50, where \(AD < AB\). She chooses a point \(E\) on side \(AB\) such that \(AE = AD\) and draws a circular arc passing through points \(B\), \(E\), and \(D\). She shades two equal-area sections of the diagram with different colors. Let \(A\) be the area of rect... | [
"113"
] | [
"Anna Taylor draws a rectangle \\(ABCD\\) with a perimeter of 50, where \\(AD < AB\\). She chooses a point \\(E\\) on side \\(AB\\) such that \\(AE = AD\\) and draws a circular arc passing through points \\(B\\), \\(E\\), and \\(D\\). She shades two equal-area sections of the diagram with different colors. Let \\(A... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 6,
"tir": 4
} | 11 | 0 | 0 |
omr-ws-ec18139cee8fc45c833831bb7623f1d84933a34266fcdf24e2136e45b941a66c | Anna and Basilis play a game where they write numbers on a board. Starting with the number 2, each player, on their turn, chooses a prime divisor \( p \) of the current number \( n \) and writes \( n + p \). The player who first writes a number greater than or equal to 31 wins. Anna plays first. Determine which player ... | [
"Anna has a winning strategy."
] | [
"Anna and Basilis play a game where they write numbers on a board. Starting with the number 2, each player, on their turn, chooses a prime divisor \\( p \\) of the current number \\( n \\) and writes \\( n + p \\). The player who first writes a number greater than or equal to 31 wins. Anna plays first. Determine wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 14
} | 37 | 0 | 0 |
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