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-bafd1f3e8519f2668d47af996ba4c9c858c11b91985e9c6460737481fc9b0047 | Alberto places \( N \) checkers in a circle, some of which are black and the others white. Betul places new checkers in between the pairs of adjacent checkers in Alberto’s ring: she places a white checker between every two of the same color, and a black checker between every pair of opposite color. She then removes Alb... | [
"N"
] | [
"Alberto places \\( N \\) checkers in a circle, some of which are black and the others white. Betul places new checkers in between the pairs of adjacent checkers in Alberto’s ring: she places a white checker between every two of the same color, and a black checker between every pair of opposite color. She then remo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33be1dba859ae49f8e62c24495dfee5ebedff467112156451b133d7b8a38b4e5 | Alberto places \( N \) checkers in a circle, where each checker is either black or white. Betul places new checkers between each pair of adjacent checkers: a white checker between two checkers of the same color and a black checker between two checkers of opposite color. She then removes Alberto's original checkers, lea... | [
"3"
] | [
"Alberto places \\( N \\) checkers in a circle, where each checker is either black or white. Betul places new checkers between each pair of adjacent checkers: a white checker between two checkers of the same color and a black checker between two checkers of opposite color. She then removes Alberto's original checke... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-d6a3616556b7aec7751d9bf185ab62a7f0eb2b59574815eb0d6b1467c1763be5 | Albrecht fills in each cell of an $8 \times 8$ table with a $0$ or a $1$. Then at the end of each row and column, he writes down the sum of the $8$ digits in that row or column, and then he erases the original digits in the table. Given only the sums, Albrecht finds that he can restore the $64$ digits in the table uniq... | [
"2"
] | [
"Albrecht fills in each cell of an $8 \\times 8$ table with a $0$ or a $1$. Then at the end of each row and column, he writes down the sum of the $8$ digits in that row or column, and then he erases the original digits in the table. Given only the sums, Albrecht finds that he can restore the $64$ digits in the tabl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-3d86918720179e16b6fe7767c01bd6224c2a579e7f6e500088e59b5ebfa732f7 | Albrecht is travelling at a constant speed on the motorway. At noon, they passed milestone $XY$ (where $X$ and $Y$ are digits), at $12:42$ they passed milestone $YX$, and at $1$pm they arrived at milestone $X0Y$. What is the speed of the car? | [
"90 km/h"
] | [
"Albrecht is travelling at a constant speed on the motorway. At noon, they passed milestone $XY$ (where $X$ and $Y$ are digits), at $12:42$ they passed milestone $YX$, and at $1$pm they arrived at milestone $X0Y$. What is the speed of the car?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 4
} | 22 | 0 | 0 |
omr-ws-a503991dd5286a0501cb58c0594664ce6b15a8917e189637ca3a873fa46ece1e | Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases, the number written on point $(a, b)$ is one greater than the average of the numbers written on points $(a+1, b-1)$ and $(a-1, b+... | [
"\\(121 \\times 212\\)"
] | [
"Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases, the number written on point $(a, b)$ is one greater than the average of the numbers written on points $(a+1, b-1)$ and $(a-... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
omr-ws-9ded1e715d068ab6fea4d9cdc10560f942829f8f2e342f0e2c8033b35e1dead5 | Alculator11 is driving at 25 meters per second, 250 meters behind a line of cars that starts accelerating at a constant rate of 2.5 meters per second squared until they reach 25 meters per second. Alculator11 can decelerate at up to -5 meters per second squared and accelerate at up to 5 meters per second squared. What ... | [
"0"
] | [
"Alculator11 is driving at 25 meters per second, 250 meters behind a line of cars that starts accelerating at a constant rate of 2.5 meters per second squared until they reach 25 meters per second. Alculator11 can decelerate at up to -5 meters per second squared and accelerate at up to 5 meters per second squared. ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-fce98aae5fc8ad7492896af8e0784208e7f5127e66b0597bac19dfd4ed1eaf74 | Alec must purchase 14 identical shirts and only has $\$130$. There is a flat $\$2$ entrance fee for shopping at the warehouse store where he plans to buy the shirts. The price of each shirt is the same whole-dollar amount. Assuming a $5\%$ sales tax is added to the price of each shirt, what is the greatest possible pri... | [
"8"
] | [
"Alec must purchase 14 identical shirts and only has $\\$130$. There is a flat $\\$2$ entrance fee for shopping at the warehouse store where he plans to buy the shirts. The price of each shirt is the same whole-dollar amount. Assuming a $5\\%$ sales tax is added to the price of each shirt, what is the greatest p... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-32df2b742b330977ece577bd81a89b1f37a4de4e645696744793c97ba52df97f | Alec, Bob, and Carlo each roll a 6-sided die. Alec gains a point if the sum of their rolls is a multiple of $4$, Bob gains a point if the sum is a multiple of $5$, and Carlo gains a point if the sum is a multiple of $6$. The first to reach $3$ points wins. What is the probability that the first winner is the only playe... | [
"\\dfrac{107838}{1295029}"
] | [
"Alec, Bob, and Carlo each roll a 6-sided die. Alec gains a point if the sum of their rolls is a multiple of $4$, Bob gains a point if the sum is a multiple of $5$, and Carlo gains a point if the sum is a multiple of $6$. The first to reach $3$ points wins. What is the probability that the first winner is the only ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-c76877d23a5bb146be5be00c2772394b3d22a889401465a8d2ed3277319eb904 | Alec, Brittany, Cindy, and Derek are planning to run a marathon as a relay team in that order. The rules mandate that each runner run at least 2 miles and that handoffs can only be made at the mile markers. In how many different ways can they run the marathon? (A marathon can be assumed to be 26 miles). | [
"1330"
] | [
"Alec, Brittany, Cindy, and Derek are planning to run a marathon as a relay team in that order. The rules mandate that each runner run at least 2 miles and that handoffs can only be made at the mile markers. In how many different ways can they run the marathon? (A marathon can be assumed to be 26 miles)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-32e2320de100f8707765d406d1332f69da74612af75a284c5e8235090d9b7b02 | Alejandra is distributing candies to several children. The first child receives 1 candy, the second child receives 2 candies, the third child receives twice as many as the second, and so on. If Alejandra has 2007 candies, what is the minimum number of additional candies needed to distribute them exactly, and how many c... | [
"40 candies are missing, and 11 children would receive candy."
] | [
"Alejandra is distributing candies to several children. The first child receives 1 candy, the second child receives 2 candies, the third child receives twice as many as the second, and so on. If Alejandra has 2007 candies, what is the minimum number of additional candies needed to distribute them exactly, and how m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 22
} | 24 | 0 | 0 |
omr-ws-c188ccae7d7fb86e26ff088f9255e7054304d56b92e785c9d554cfb120cb36a9 | Alesha and Belinda each write the same sequence \( S \), a 2020-term strictly increasing geometric sequence with an integer common ratio. Every second, Alesha replaces the two smallest terms with their geometric mean, while Belinda replaces the two largest terms with their geometric mean. They continue until Alesha is ... | [
"2018"
] | [
"Alesha and Belinda each write the same sequence \\( S \\), a 2020-term strictly increasing geometric sequence with an integer common ratio. Every second, Alesha replaces the two smallest terms with their geometric mean, while Belinda replaces the two largest terms with their geometric mean. They continue until Ale... | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-ba3250d4fc85e6ce2332d3e7764c1c798c2df710165b3a7e6f6a6ec649af5a69 | Alex Lishkov is playing a game with a 2009-letter long sequence of ternary digits (0, 1, or 2). He guesses each digit, and if he guesses incorrectly, the oracle reveals the next $k$ digits, where $k$ is the correct digit. Let $P_n$ be the probability that Lishkov gets the $n$-th digit correct. Find $P_{2009}$. | [
"\\dfrac{1}{5}"
] | [
"Alex Lishkov is playing a game with a 2009-letter long sequence of ternary digits (0, 1, or 2). He guesses each digit, and if he guesses incorrectly, the oracle reveals the next $k$ digits, where $k$ is the correct digit. Let $P_n$ be the probability that Lishkov gets the $n$-th digit correct. Find $P_{2009}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-1c6f7ad9e1b54e047d04e618944f2f513ab4300ab368e402736e608e8e9aa924 | Alex and Anka go to see stars. Alex counts thirteen fewer stars than Anka, and both of the numbers of stars they counted are square numbers. It is known that exactly eight stars were counted by both of them. How many distinct stars did they count in total? | [
"77"
] | [
"Alex and Anka go to see stars. Alex counts thirteen fewer stars than Anka, and both of the numbers of stars they counted are square numbers. It is known that exactly eight stars were counted by both of them. How many distinct stars did they count in total?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-17646dc87eadcb1d3ea2104091b063d1d3ccb4158e1630dce13ce0bf2129a63a | Alex and Becky are racing on a 1 km circular track, moving in opposite directions. Alex runs at 4 m/s and Becky runs at 5 m/s. How many times would they have passed each other if the race stops after 1 hour? | [
"32"
] | [
"Alex and Becky are racing on a 1 km circular track, moving in opposite directions. Alex runs at 4 m/s and Becky runs at 5 m/s. How many times would they have passed each other if the race stops after 1 hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-c4155740bec89d779f55dc16f28b5fbd44d0207dc0d2b1edc8fe759f3d53e1e3 | Alex and Betty want to meet each other at the airport. Alex will arrive at the airport between 12:00 and 13:15, and will wait for Betty for 15 minutes before he leaves. Betty will arrive at the airport between 12:30 and 13:10, and will wait for Alex for 10 minutes before she leaves. The chance that they arrive at any t... | [
"319"
] | [
"Alex and Betty want to meet each other at the airport. Alex will arrive at the airport between 12:00 and 13:15, and will wait for Betty for 15 minutes before he leaves. Betty will arrive at the airport between 12:30 and 13:10, and will wait for Alex for 10 minutes before she leaves. The chance that they arrive at ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-6d09fabdfd047e2e4b42cb538922d07b799a3c2fc48368c156f737b56af9393d | Alex and Bob are playing a game where they start with a number \( n_0 = 2 \) on a blackboard. On each turn, a player changes the number \( n \) to \( n' = n + m \), where \( m \) is a positive divisor of \( n \) different from \( n \) (including \( m = 1 \)). The player who first makes \( n' \geq 2016 \) wins. Determin... | [
"Alex"
] | [
"Alex and Bob are playing a game where they start with a number \\( n_0 = 2 \\) on a blackboard. On each turn, a player changes the number \\( n \\) to \\( n' = n + m \\), where \\( m \\) is a positive divisor of \\( n \\) different from \\( n \\) (including \\( m = 1 \\)). The player who first makes \\( n' \\geq 2... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 22
} | 40 | 0 | 0 |
omr-ws-81e0f89bffc0f2bcf6fcfb1ca03bedcc3576a0cdfe258975784403adb0352e57 | Alex and Bob are playing a game with $n$ glasses arranged in a circle, where $k(n)$ glasses contain poison and the rest contain water. Alex knows which glasses contain poison, but Bob does not. They take turns drinking from the glasses, and they cannot communicate during the game. Determine the largest value of $k(n)$ ... | [
"\\left\\lceil \\frac{n}{2} \\right\\rceil"
] | [
"Alex and Bob are playing a game with $n$ glasses arranged in a circle, where $k(n)$ glasses contain poison and the rest contain water. Alex knows which glasses contain poison, but Bob does not. They take turns drinking from the glasses, and they cannot communicate during the game. Determine the largest value of $k... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-e8bc1a3919644c30ae34e3082b60a60428142e3a72c63e1cb63fefc8bb65f773 | Alex and Bob play a game on a 2015 x 2015 checkered board. Initially, the board is empty. Players take turns placing either a red or blue token into any unoccupied square, with Alex moving first. A player wins if, after their move, there appears a row of three consecutive tokens of the same color (the row may be vertic... | [
"Alex",
"Alex has a winning strategy."
] | [
"Alex and Bob play a game on a 2015 x 2015 checkered board. Initially, the board is empty. Players take turns placing either a red or blue token into any unoccupied square, with Alex moving first. A player wins if, after their move, there appears a row of three consecutive tokens of the same color (the row may be v... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 12
} | 43 | 0 | 0 |
omr-ws-cb2dd642042cbc87d1e3ab3fad4f8e2d5c6c543ae24e345b31a9cc15a93c100c | Alex and Bobby are walking through a train tunnel and have covered 45% of its length when they hear a train approaching the tunnel entrance at a constant speed of 100 km/h. They both run at identical constant rates: Alex towards the entrance and Bobby towards the exit. They both narrowly escape the tunnel and avoid bei... | [
"10 km/h"
] | [
"Alex and Bobby are walking through a train tunnel and have covered 45% of its length when they hear a train approaching the tunnel entrance at a constant speed of 100 km/h. They both run at identical constant rates: Alex towards the entrance and Bobby towards the exit. They both narrowly escape the tunnel and avoi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 23
} | 42 | 0 | 0 |
omr-ws-834a71d139a3aaacbe5e39d462404801730aac0abbc5ed9a0a798c17b9010df2 | Alex and Brian take turns shooting free throws until they each shoot twice. Alex has an 80% chance of making a free throw, and Brian has a 60% chance. If Brian shoots first, what is the probability that after each free throw they take, Alex has made at least as many free throws as Brian? | [
"508/625"
] | [
"Alex and Brian take turns shooting free throws until they each shoot twice. Alex has an 80% chance of making a free throw, and Brian has a 60% chance. If Brian shoots first, what is the probability that after each free throw they take, Alex has made at least as many free throws as Brian?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-411258465cad0964c03c3a82fbf1bd3351e14364980736d6dd955ebd2b37c0ac | Alex and I are playing a game where we flip a coin. If it comes up heads, I pay Alex \$1. If it comes up tails, Alex pays me \$2. I start off with \$9, and Alex starts off with \$20. What is the expected number of flips before one of us runs out of money? | [
"180"
] | [
"Alex and I are playing a game where we flip a coin. If it comes up heads, I pay Alex \\$1. If it comes up tails, Alex pays me \\$2. I start off with \\$9, and Alex starts off with \\$20. What is the expected number of flips before one of us runs out of money?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-ee2e71ed38f83f651ab1407b00885d328dd02274c2b053e4186396a3ee04b0a6 | Alex and Kate take turns writing on an $8$ by $8$ board. Each turn, Alex must write $\textsf A$ on an empty cell, and Kate must write two $\textsf K$s in two empty adjacent cells (having a common edge). When one player cannot move, the game ends and Kate's score is the number of $\textsf K$s on the board. What is the m... | [
"42"
] | [
"Alex and Kate take turns writing on an $8$ by $8$ board. Each turn, Alex must write $\\textsf A$ on an empty cell, and Kate must write two $\\textsf K$s in two empty adjacent cells (having a common edge). When one player cannot move, the game ends and Kate's score is the number of $\\textsf K$s on the board. What ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-fbb64690928d707b6973d2820c3304501dbd4c4795a6d79737aafbf42b760333 | Alex and Katy play a game on an $8 \times 8$ square grid made of $64$ unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cannot move. Katy's score is the number of... | [
"32"
] | [
"Alex and Katy play a game on an $8 \\times 8$ square grid made of $64$ unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cannot move. Katy's score is the num... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-74f0a18ffe8f6188f16ce011703683f97ba6981ebaeec09df21766e3cf32ae1b | Alex and Katy play a game on an 8 x 8 square grid made of 64 unit cells. Alex writes 'A' in an empty cell on his turn, and Katy writes 'K' in two empty cells that share an edge on her turn. The game ends when one player cannot move. Katy’s score is the number of 'K's on the grid at the end of the game. What is the high... | [
"42"
] | [
"Alex and Katy play a game on an 8 x 8 square grid made of 64 unit cells. Alex writes 'A' in an empty cell on his turn, and Katy writes 'K' in two empty cells that share an edge on her turn. The game ends when one player cannot move. Katy’s score is the number of 'K's on the grid at the end of the game. What is the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-32a667ec36b482a37bd488f330ffb0e5a3e6f9d1fdbc46b0bc14e7a2f12b6e8d | Alex and Sam are playing a game with an unfair coin where the probability of tossing a head is $h>0$. The first person to toss a head wins. Alex starts and they alternate turns. Alex flips the coin once on his turn and Sam flips the coin twice on his turn. Given that the game is fair, find $h$. | [
"\\(\\frac{3-\\sqrt{5}}{2}\\)"
] | [
"Alex and Sam are playing a game with an unfair coin where the probability of tossing a head is $h>0$. The first person to toss a head wins. Alex starts and they alternate turns. Alex flips the coin once on his turn and Sam flips the coin twice on his turn. Given that the game is fair, find $h$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4bcf54d23423ee4087a9eb8a2a48d0a9102a7cfd1152b2d16547760572154d11 | Alex and Sun are playing a game where Alex picks a positive integer not greater than $1000$, and then Sun picks a positive integer strictly smaller than that. They continue taking turns to pick progressively smaller positive integers until someone picks $1$. The sum of all picked numbers is calculated. The player who p... | [
"9"
] | [
"Alex and Sun are playing a game where Alex picks a positive integer not greater than $1000$, and then Sun picks a positive integer strictly smaller than that. They continue taking turns to pick progressively smaller positive integers until someone picks $1$. The sum of all picked numbers is calculated. The player ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-868d2eb14cc4aba27067b4fae29228aab0bb1ef823889eca61ba33fc778038f6 | Alex and his brother Bob drove from Summerville, GA to Whinterville at a constant speed of 55 mph for \( n \) hours, where \( n \) is a positive integer. The odometer read \( abc \) when the trip began and \( cba \) when the trip ended. What is \( n \)? | [
"9"
] | [
"Alex and his brother Bob drove from Summerville, GA to Whinterville at a constant speed of 55 mph for \\( n \\) hours, where \\( n \\) is a positive integer. The odometer read \\( abc \\) when the trip began and \\( cba \\) when the trip ended. What is \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5aa175b00b8b6f5afd62c2dabdb926d1be7d2c4f5bb32fa6d033c681a3cc787 | Alex and his two brothers, Rob and Paul, are painting the 4 walls of Alex's cubical house. Alex and Rob will paint the first side, Rob and Paul will paint the second, Paul and Alex will paint the third, and all three together will paint the 4th. If Alex paints 1 wall in 6 hours, Rob paints one wall in 2 hours, and Paul... | [
"Alex = \\( \\frac{3}{16} \\), Rob = \\( \\frac{37}{80} \\), Paul = \\( \\frac{7}{20} \\)"
] | [
"Alex and his two brothers, Rob and Paul, are painting the 4 walls of Alex's cubical house. Alex and Rob will paint the first side, Rob and Paul will paint the second, Paul and Alex will paint the third, and all three together will paint the 4th. If Alex paints 1 wall in 6 hours, Rob paints one wall in 2 hours, and... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 29
} | 32 | 0 | 0 |
omr-ws-0344a2968d1ede29e362650075313c7785a4d684ef8569838996f42487c2215e | Alex bakes a total of $24$ pies, and each pie is apple, blueberry, or cherry. The ratio of apple to blueberry to cherry pies is $1:4:3$. How many cherry pies did Alex bake? | [
"9"
] | [
"Alex bakes a total of $24$ pies, and each pie is apple, blueberry, or cherry. The ratio of apple to blueberry to cherry pies is $1:4:3$. How many cherry pies did Alex bake?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d22697e0e003ec7f5e58c8ba0f1595cd85439d24f904ed700f5db8850e955851 | Alex calculated the value of the function \( f(n) = n^2 + n + 1 \) for each integer from 1 to 100. Marina calculated the value of the function \( g(n) = n^2 - n + 1 \) for the same numbers. Who has the greater product of values, and what is their ratio? | [
"10101"
] | [
"Alex calculated the value of the function \\( f(n) = n^2 + n + 1 \\) for each integer from 1 to 100. Marina calculated the value of the function \\( g(n) = n^2 - n + 1 \\) for the same numbers. Who has the greater product of values, and what is their ratio?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-6a91cc0929ee991419dbd618e203d9b0a9adc46502990c359bbb948b1b914517 | Alex copied two two-digit numbers from the board and multiplied them. He copied one of the numbers incorrectly: he wrote the tens digit in the units place and the units digit in the tens place. Because of this mistake, he obtained a result that was 1449 units more than he should have. Determine the possible numbers on ... | [
"\\(23 \\times 81\\) and \\(23 \\times 92\\)"
] | [
"Alex copied two two-digit numbers from the board and multiplied them. He copied one of the numbers incorrectly: he wrote the tens digit in the units place and the units digit in the tens place. Because of this mistake, he obtained a result that was 1449 units more than he should have. Determine the possible number... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 11
} | 14 | 0 | 0 |
omr-ws-55529deea1d116f0549d7c1b26203a45061531a499c9485b5c4572c8e0dc28fd | Alex dissects a paper triangle into two triangles. Each minute after this, he dissects one of the obtained triangles into two triangles. After some time (at least one hour), it appeared that all obtained triangles were congruent. Find all initial triangles for which this is possible. | [
"\\text{Right-angled isosceles triangle}"
] | [
"Alex dissects a paper triangle into two triangles. Each minute after this, he dissects one of the obtained triangles into two triangles. After some time (at least one hour), it appeared that all obtained triangles were congruent. Find all initial triangles for which this is possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-708fbadbbabad445892a2de3dd9f0b7db7af960d730d574c5185817880fb678b | Alex distributes some cookies into several boxes and records the number of cookies in each box, with each number recorded only once if it appears more than once. Serge takes one cookie from each box and places them on the first plate, then repeats this process with the remaining cookies until all boxes are empty. Serge... | [
"5"
] | [
"Alex distributes some cookies into several boxes and records the number of cookies in each box, with each number recorded only once if it appears more than once. Serge takes one cookie from each box and places them on the first plate, then repeats this process with the remaining cookies until all boxes are empty. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-49ce0e7d8b4a596875d13f800b2fca797bbb2f48853690473e694bd651034bb0 | Alex distributes some cookies into several boxes and records the number of cookies in each box. If the same number appears more than once, it is recorded only once. Serge takes one cookie from each box and puts them on the first plate. Then he takes one cookie from each box that is still non-empty and puts the cookies ... | [
"28"
] | [
"Alex distributes some cookies into several boxes and records the number of cookies in each box. If the same number appears more than once, it is recorded only once. Serge takes one cookie from each box and puts them on the first plate. Then he takes one cookie from each box that is still non-empty and puts the coo... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-560d40fca7880af6796e17c519e976cf656dfee49cbffc77d629cf999d7a8295 | Alex flips $3$ fair coins in a row. The probability that the first and last flips are the same can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. | [
"3"
] | [
"Alex flips $3$ fair coins in a row. The probability that the first and last flips are the same can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-dd807ca455f1274caef524da9329aabe212a0a9b9fb45ce3e452f7258e17c710 | Alex has 10 different kinds of lunch meat and 9 different kinds of cheese. If he wants to make a sandwich with one kind of meat and two kinds of cheese, how many different sandwiches could he make? (It does not matter in which order he chooses the two types of cheese.) | [
"360"
] | [
"Alex has 10 different kinds of lunch meat and 9 different kinds of cheese. If he wants to make a sandwich with one kind of meat and two kinds of cheese, how many different sandwiches could he make? (It does not matter in which order he chooses the two types of cheese.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2230d4b7d686134e7cdcd88e5663cf63b2933cbf14d98df708e8b9ba2c98ed07 | Alex has 12 friends and 63 coins. What is the minimum number of additional coins he needs so that he can give each friend at least one coin and no two friends receive the same number of coins? | [
"15"
] | [
"Alex has 12 friends and 63 coins. What is the minimum number of additional coins he needs so that he can give each friend at least one coin and no two friends receive the same number of coins?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0028731a6694497938f58d632b4c5de8f5e8e547224b35dadbf9018d2f53126 | Alex has 5 pieces of paper, each with a different math problem. In how many ways can he give these problems to his 10 friends (he can give each friend more than one problem)? | [
"100,\\!000"
] | [
"Alex has 5 pieces of paper, each with a different math problem. In how many ways can he give these problems to his 10 friends (he can give each friend more than one problem)?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97bc306d343a4f3f1ef634325214a767be0eb0b50259597d3b23eb399fd33fad | Alex has a $4 \times 4$ grid of squares. Let $N$ be the number of ways that Alex can fill out each square with one of the letters $A$, $B$, $C$, or $D$ such that in every row and column, the number of $A$'s and $B$'s are the same, and the number of $C$'s and $D$'s are the same. Find the remainder when $N$ is divided by... | [
"1"
] | [
"Alex has a $4 \\times 4$ grid of squares. Let $N$ be the number of ways that Alex can fill out each square with one of the letters $A$, $B$, $C$, or $D$ such that in every row and column, the number of $A$'s and $B$'s are the same, and the number of $C$'s and $D$'s are the same. Find the remainder when $N$ is divi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-1ed3294fd1f60d24e81b85ba722059e4ef08ea8574d852ca0af48c2bbcf63b73 | Alex has a cube with an arrow drawn on one face, parallel to one set of edges. The cube is initially placed on the ground with the arrow on the top face pointing north. Alex rolls the cube from face to face across the ground, eventually returning the cube to its original location with the arrow face on top again. Let $... | [
"18"
] | [
"Alex has a cube with an arrow drawn on one face, parallel to one set of edges. The cube is initially placed on the ground with the arrow on the top face pointing north. Alex rolls the cube from face to face across the ground, eventually returning the cube to its original location with the arrow face on top again. ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-83c560f12dd648b36be54bac74b49e5fddbc509a6461e0f68eafd452a54f6c9a | Alex has a piece of cheese. He chooses a positive number $a \neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a$. His goal is to divide the cheese into two piles such that the mass of one pile is exactly $k$ times the mass of the other pile. If $a ... | [
"\\dfrac{9}{7}"
] | [
"Alex has a piece of cheese. He chooses a positive number $a \\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a$. His goal is to divide the cheese into two piles such that the mass of one pile is exactly $k$ times the mass of the other pile. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-9e9739fb40f0669b0d0e1b2ba6bf745d0bad9a71517dee83e907962bd92dff55 | Alex has a piece of cheese. He chooses a positive number \(a \neq 1\) and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio \(1:a\). After making a series of cuts, he divides the pieces into two piles such that the total mass of each pile is equal. If the origina... | [
"2\\sqrt{2}"
] | [
"Alex has a piece of cheese. He chooses a positive number \\(a \\neq 1\\) and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio \\(1:a\\). After making a series of cuts, he divides the pieces into two piles such that the total mass of each pile is equal. If t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-b222846610fafb5cf4ff5fa0757a7dabca619c6fa2f91bfd089263473eea6e0c | Alex has a sum of money and places bets on the color of the cube in each of 100 boxes, each containing either a red or a blue cube. The bet can be any amount from $0$ up to his current total. If he wins, he doubles his bet; if he loses, he loses the bet. If Alex knows there are exactly $k$ blue cubes, where $1 < k \leq... | [
"\\(\\frac{2^{100}}{2^{100} - 101}\\)"
] | [
"Alex has a sum of money and places bets on the color of the cube in each of 100 boxes, each containing either a red or a blue cube. The bet can be any amount from $0$ up to his current total. If he wins, he doubles his bet; if he loses, he loses the bet. If Alex knows there are exactly $k$ blue cubes, where $1 < k... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2094840973bd2f461c3b446740d05562689ffb857070d409980bbdd4f110ef72 | Alex has a sum of money and places bets on the color of the cube in each of 100 boxes, each containing either a red or a blue cube. The bet can be any amount from $0$ up to his current total. If he wins, he doubles his bet; if he loses, he loses the bet. If Alex knows there is exactly 1 blue cube, what is the maximum f... | [
"\\(\\frac{2^{98}}{25}\\)"
] | [
"Alex has a sum of money and places bets on the color of the cube in each of 100 boxes, each containing either a red or a blue cube. The bet can be any amount from $0$ up to his current total. If he wins, he doubles his bet; if he loses, he loses the bet. If Alex knows there is exactly 1 blue cube, what is the maxi... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-75e1cfe3d934a2f04b8f8a10c752acd720603eca8805f16a4d3ff24df8b4057a | Alex has four bags of candy weighing 1, 2, 3, and 4 pounds. He will pick up two adjacent bags, compare their weights on a balance scale, and put them back with the heavier one to the left. What is the maximum number of times he could possibly have to swap two bags before they are in order from heaviest to lightest? | [
"6"
] | [
"Alex has four bags of candy weighing 1, 2, 3, and 4 pounds. He will pick up two adjacent bags, compare their weights on a balance scale, and put them back with the heavier one to the left. What is the maximum number of times he could possibly have to swap two bags before they are in order from heaviest to lightest... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7852bf1fcfb8189cb025f50326e25f7cbd27d1ff650b581635719a6b90fd5464 | Alex is a lazy student. His teacher, Stephen, is going to give him a test with 500 questions, and Alex must answer 10 of them. Stephen gives him a list of 2020 questions beforehand, guaranteeing that the questions on the test will be picked from that list. Alex wants to ensure that he can answer 10 questions correctly,... | [
"1530"
] | [
"Alex is a lazy student. His teacher, Stephen, is going to give him a test with 500 questions, and Alex must answer 10 of them. Stephen gives him a list of 2020 questions beforehand, guaranteeing that the questions on the test will be picked from that list. Alex wants to ensure that he can answer 10 questions corre... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f706f439c308876702e1871c4ee566718739510b6d377a49a2c38d195237897a | Alex is at the candy store buying jellybeans. He wants to buy at least 100 jellybeans. He wants to buy the least amount such that he would have exactly $11$ leftover after dividing the jellybeans evenly among $13$ people. How many jellybeans should Alex buy? | [
"102"
] | [
"Alex is at the candy store buying jellybeans. He wants to buy at least 100 jellybeans. He wants to buy the least amount such that he would have exactly $11$ leftover after dividing the jellybeans evenly among $13$ people. How many jellybeans should Alex buy?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7387ea762b4a8010bccb8fea163d0fe953d88e8db883b39469bfa78cbb0a5ae | Alex is happy if there is at least one subsequence of "abc" in any given word. How many distinct strings of length 8 consisting of characters 'a', 'b', and 'c' will make Alex happy? | [
"3489"
] | [
"Alex is happy if there is at least one subsequence of \"abc\" in any given word. How many distinct strings of length 8 consisting of characters 'a', 'b', and 'c' will make Alex happy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-9a23f08d6227370ea0e158b93311cb355a15b2982296589dacb491fc058ebfa1 | Alex is participating in a tournament where each series is best-of-7, i.e., each game has a winner and the first person to win 4 games wins the series. After winning $w$ series and losing $\ell$ series in the tournament, Alex has won 20 games and lost 12 games. Given this information, how many possible ordered pairs of... | [
"7"
] | [
"Alex is participating in a tournament where each series is best-of-7, i.e., each game has a winner and the first person to win 4 games wins the series. After winning $w$ series and losing $\\ell$ series in the tournament, Alex has won 20 games and lost 12 games. Given this information, how many possible ordered pa... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-62c3be0fd90e13aebf92e37d3342c769b57a56cb64a1ff7af206983c8f3abe82 | Alex is selling cookies to receive her girl-scouts badge. She sells two types of cookies, bookies and nookies. Bookies cost $\$5$ each, while nookies cost $\$7$ each. How many ways are there for Alex to meet her goal of earning exactly $\$234$ by selling cookies? (Assume she cannot sell fractions of a box) | [
"7"
] | [
"Alex is selling cookies to receive her girl-scouts badge. She sells two types of cookies, bookies and nookies. Bookies cost $\\$5$ each, while nookies cost $\\$7$ each. How many ways are there for Alex to meet her goal of earning exactly $\\$234$ by selling cookies? (Assume she cannot sell fractions of a box)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2683783cfb8461beb7b1812a43eeb0204959952113ecd42ec70f5300ae8b228e | Alex leaves his house on his dirt bike. He rides north for one mile, then he turns and rides west for half a mile, then he turns south and rides for a quarter mile, then east for an eighth of a mile, then north again for a sixteenth of a mile, west for $1/32$ of a mile, and so on. Eventually he reaches a point where he... | [
"\\dfrac{2\\sqrt{5}}{5}"
] | [
"Alex leaves his house on his dirt bike. He rides north for one mile, then he turns and rides west for half a mile, then he turns south and rides for a quarter mile, then east for an eighth of a mile, then north again for a sixteenth of a mile, west for $1/32$ of a mile, and so on. Eventually he reaches a point whe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2dc3efe702fdca548a560a20284b6628dbbbfcb8a4948a0f3861ee89e1dd3100 | Alex moves from \((0, 0)\) to \((4, 4)\) in the coordinate plane, moving one unit to the right or one unit up each step. There are chutes-ladders between \((1, 2)\) and \((2, 0)\) and between \((1, 3)\) and \((4, 0)\). How many ways are there for Alex to reach \((4, 4)\)? | [
"37"
] | [
"Alex moves from \\((0, 0)\\) to \\((4, 4)\\) in the coordinate plane, moving one unit to the right or one unit up each step. There are chutes-ladders between \\((1, 2)\\) and \\((2, 0)\\) and between \\((1, 3)\\) and \\((4, 0)\\). How many ways are there for Alex to reach \\((4, 4)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-172ebc1b2fc4a8b99af32bf4d3d5319061a4beebcfb2ff264fb78e388c953f3e | Alex needs to borrow $\$10,\!000$ from the bank. The bank gives him two options. 1. A ten-year loan with an annual interest rate of $10\%$ compounded quarterly, with the condition that at the end of 5 years, Alex must make a payment equal to half of what he owes. The other half continues to accrue interest, and at the ... | [
"382 \\text{ dollars}"
] | [
"Alex needs to borrow $\\$10,\\!000$ from the bank. The bank gives him two options.\n\n1. A ten-year loan with an annual interest rate of $10\\%$ compounded quarterly, with the condition that at the end of 5 years, Alex must make a payment equal to half of what he owes. The other half continues to accrue interest,... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be688f6e7094dfabde49b6d171e7e572cb062eae319a7c3443d186c83e5575d2 | Alex needs to catch a train. The train arrives randomly some time between 1:00 and 2:00, waits for 10 minutes, and then leaves. If Alex also arrives randomly between 1:00 and 2:00, what is the probability that the train will be there when Alex arrives? | [
"\\frac{11}{72}"
] | [
"Alex needs to catch a train. The train arrives randomly some time between 1:00 and 2:00, waits for 10 minutes, and then leaves. If Alex also arrives randomly between 1:00 and 2:00, what is the probability that the train will be there when Alex arrives?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50fc7b48793b3955cff2ab5bcf344eb0646b0f6bcf6942f45f7e15cf1bbfd4b0 | Alex picks a number $n$. He tells his friend that $6n$ has 96 positive integer factors, $10n$ has 120 positive integer factors, and $15n$ has 126 positive factors. How many factors does $n$ have? | [
"70"
] | [
"Alex picks a number $n$. He tells his friend that $6n$ has 96 positive integer factors, $10n$ has 120 positive integer factors, and $15n$ has 126 positive factors. How many factors does $n$ have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-227c8a0a12366c39db38be9eb37995e84d83c71452250dbd23c70c6145814b13 | Alex picks any 5 cards from a deck (without Jokers). After inspecting and shuffling, he hands the 5 cards to Peter (who cannot see them). Peter selects 1 card and returns it to Alex, then arranges the remaining 4 cards in a specific order and places them face down in a neat pile for Alex. Using only the order and the f... | [
"2598960"
] | [
"Alex picks any 5 cards from a deck (without Jokers). After inspecting and shuffling, he hands the 5 cards to Peter (who cannot see them). Peter selects 1 card and returns it to Alex, then arranges the remaining 4 cards in a specific order and places them face down in a neat pile for Alex. Using only the order and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-49c6c4e6ee7485ed8ffcc8bf3a492554d8c0c1c854cb41ae171ebe2b705c9162 | Alex rolls a dice twice. If he rolls a sum of 7, 8, 10, 11, or 12, he rerolls twice again until he obtains a sum of 2, 3, 4, 5, 6, or 9. What is the probability that Alex rolls a sum of 3, 5, or 6 on his last pair of rolls? | [
"\\(\\frac{11}{19}\\)"
] | [
"Alex rolls a dice twice. If he rolls a sum of 7, 8, 10, 11, or 12, he rerolls twice again until he obtains a sum of 2, 3, 4, 5, 6, or 9. What is the probability that Alex rolls a sum of 3, 5, or 6 on his last pair of rolls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9cef7d967f7dbf32927ce9b8e02637a0a634e879b9a22b65482021a52d49526c | Alex rolls three fair 6-sided dice and adds up the values to get \( A \). Bob rolls one fair 16-sided die numbered 1, 2, ..., 16 to get \( B \). What is the probability that \( A > B + 2 \)? | [
"\\(\\frac{15}{32}\\)"
] | [
"Alex rolls three fair 6-sided dice and adds up the values to get \\( A \\). Bob rolls one fair 16-sided die numbered 1, 2, ..., 16 to get \\( B \\). What is the probability that \\( A > B + 2 \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-e2f9a365087950346197d787a58aa075c280cd549e949094828c3eb99d60de79 | Alex sees the TV tower by azimuth $60^o$, the water-tower by azimuth $90^o$, and the belfry by azimuth $120^o$. For Boris, these azimuths are $270^o$, $240^o$, and $X$ respectively. Find all the possible values for $X$. | [
"\\( X = 330^\\circ \\text{ or } 270^\\circ \\)"
] | [
"Alex sees the TV tower by azimuth $60^o$, the water-tower by azimuth $90^o$, and the belfry by azimuth $120^o$. For Boris, these azimuths are $270^o$, $240^o$, and $X$ respectively. Find all the possible values for $X$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3943c93b7d4f8457ccb91d79ebf3c0d3b54e607ee7d53b6d257eb6d6f2f87a93 | Alex the Kat writes the number 27000 on a blackboard. Each minute, he replaces the number with a randomly chosen positive divisor of the current number, including itself. Find the probability that, after 2024 minutes, the number on the blackboard is 1. | [
"\\(\\left( \\frac{1}{5^{2024}} - 4\\left(\\frac{1}{4^{2024}}\\right) + 6\\left(\\frac{1}{3^{2024}}\\right) - 4\\left(\\frac{1}{2^{2024}}\\right) + 1 \\right)^3\\)"
] | [
"Alex the Kat writes the number 27000 on a blackboard. Each minute, he replaces the number with a randomly chosen positive divisor of the current number, including itself. Find the probability that, after 2024 minutes, the number on the blackboard is 1."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8098337aec8af5f463c1d8d43e623c43f9f85e7f4426bd9bccbcb4edbea6dbb | Alex the frog is travelling on the $xy$ plane in a series of $2^{2017} - 1$ steps, starting at the origin. At the $n^{th}$ step, if $n$ is odd, then Alex jumps one unit to the right. If $n$ is even, Alex jumps $m$ units up, where $m$ is the greatest integer such that $2^m|n$. If Alex begins at the origin, what is the a... | [
"\\( 2^{2016}(2^{2016} - 1) - 2016 \\cdot 2^{2015} \\)"
] | [
"Alex the frog is travelling on the $xy$ plane in a series of $2^{2017} - 1$ steps, starting at the origin. At the $n^{th}$ step, if $n$ is odd, then Alex jumps one unit to the right. If $n$ is even, Alex jumps $m$ units up, where $m$ is the greatest integer such that $2^m|n$. If Alex begins at the origin, what is ... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 1
} | 1 | 0 | 0 |
omr-ws-90fdaac41c63798627127f2d77e076c8d9cd5c6ee682a5c6a449bd95de547916 | Alex thinks of a number less than 1000. Bob can ask questions by providing a number between 10 and 20, and Alex responds with the smallest common multiple of the number in his mind and Bob's number. How many questions does Bob need to ask to determine Alex's number with certainty? | [
"2"
] | [
"Alex thinks of a number less than 1000. Bob can ask questions by providing a number between 10 and 20, and Alex responds with the smallest common multiple of the number in his mind and Bob's number. How many questions does Bob need to ask to determine Alex's number with certainty?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-d85cfccef50ddcb7a3e23c775e63dafd18c1896ba8e36eb971548c50241a1ef3 | Alex was evaluating a polynomial \( f(x) \) with integer coefficients at different points and obtained the following values: \( f(-2) = -56 \), \( f(1) = -2 \), \( f(3) = 53 \), \( f(6) = 528 \). One of these values was wrong. Which one was it? | [
"\\( f(3) = 53 \\)"
] | [
"Alex was evaluating a polynomial \\( f(x) \\) with integer coefficients at different points and obtained the following values: \\( f(-2) = -56 \\), \\( f(1) = -2 \\), \\( f(3) = 53 \\), \\( f(6) = 528 \\). One of these values was wrong. Which one was it?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 13
} | 25 | 0 | 0 |
omr-ws-5441c89485bd370b521fef5da56f0c8ca391cd5ea47065eb076819469cadaeb6 | Alex's cylindrical cans are 6 cm in diameter and 12 cm high, while Felicia's are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans? | [
"\\dfrac{1}{2}"
] | [
"Alex's cylindrical cans are 6 cm in diameter and 12 cm high, while Felicia's are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b4d7fabd20207b5f1a41bb4fb51aa5d4e7bd5a82285826b4d5560455d4ffc399 | Alex, Alec, Aleck, and a few other friends are standing around a circle. Alex always stands next to Aleck, but Alec never stands next to Aleck. Define an $ordering$ to be an arrangement of Alex, Alec, and Aleck around the circle so that the conditions are satisfied, and that two $orderings$ are the same if one can be r... | [
"2"
] | [
"Alex, Alec, Aleck, and a few other friends are standing around a circle. Alex always stands next to Aleck, but Alec never stands next to Aleck. Define an $ordering$ to be an arrangement of Alex, Alec, and Aleck around the circle so that the conditions are satisfied, and that two $orderings$ are the same if one can... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-20d3936fe747bfe031ee27ff19febf33f33d22b016fc839351d27365e8c1373f | Alex, Ben, and Charles each have a label with one of the numbers $2, 3, 4, 5, 6, 7, 8$ on their foreheads. They can see the numbers on the others' foreheads but not their own. Mary tells them that the numbers on their foreheads are not all distinct and their product is a perfect square. Can any of the friends determine... | [
"Yes"
] | [
"Alex, Ben, and Charles each have a label with one of the numbers $2, 3, 4, 5, 6, 7, 8$ on their foreheads. They can see the numbers on the others' foreheads but not their own. Mary tells them that the numbers on their foreheads are not all distinct and their product is a perfect square. Can any of the friends dete... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-a363610bd8e7a4a772bda4bb6ac637f48a1ae2405786f7d0da3b0ff4b3cedbad | Alex, Betty, Cathy, and Daniel are participating in a footrace. In how many different orders can the four of them finish if ties are allowed? | [
"75"
] | [
"Alex, Betty, Cathy, and Daniel are participating in a footrace. In how many different orders can the four of them finish if ties are allowed?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b0b579c0706d6424a8abda56eb571996ec455921e94114fd22a4d670c32697b | Alex, Blaire, and Clara have decided to settle their differences with a three-way duel. In some randomly-chosen firing order, they will take turns shooting at another duelist until there is one person left standing. Alex and Blaire are expert markswomen, hitting all their shots with 100% accuracy, whereas Clara can onl... | [
"Clara"
] | [
"Alex, Blaire, and Clara have decided to settle their differences with a three-way duel. In some randomly-chosen firing order, they will take turns shooting at another duelist until there is one person left standing. Alex and Blaire are expert markswomen, hitting all their shots with 100% accuracy, whereas Clara ca... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 1
} | 24 | 0 | 0 |
omr-ws-1ee8cbec819a650a7da84e27b4c10325e66841c64e5fdec1275c8be5718b9413 | Alex, Bob, and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game, the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of the tournament, Alex played a total of $10$ games, Bob played $15$ games, a... | [
"Alex"
] | [
"Alex, Bob, and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game, the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of the tournament, Alex played a total of $10$ games, Bob played $15$ gam... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-3e17fe29883d780239b6e38d1bca19d391e26052e13d8de69b9f17c785b28576 | Alex, Bob, and Chad are playing a tennis tournament. During a game, two boys play while the other rests. In the next game, the loser rests and the rester plays the winner. By the end of the tournament, Alex played 10 games, Bob played 15 games, and Chad played 17 games. Who lost the second game? | [
"Alex lost the second game"
] | [
"Alex, Bob, and Chad are playing a tennis tournament. During a game, two boys play while the other rests. In the next game, the loser rests and the rester plays the winner. By the end of the tournament, Alex played 10 games, Bob played 15 games, and Chad played 17 games. Who lost the second game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-ba1d44f09ce17bc90e41e93c36d80a17f296d0e9b3f7eb93b1849998eebe10d5 | Alex, Bruce, and Ryan pass around a hot potato in a circle, starting with Alex. Each person holds the potato for a random duration between 2 and 5 seconds before passing it to the next person. If the potato explodes after 10 seconds, the probability that Alex loses the game can be expressed as the common fraction $\tfr... | [
"665"
] | [
"Alex, Bruce, and Ryan pass around a hot potato in a circle, starting with Alex. Each person holds the potato for a random duration between 2 and 5 seconds before passing it to the next person. If the potato explodes after 10 seconds, the probability that Alex loses the game can be expressed as the common fraction ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-d42892720fad1557d3a6daf1aba5c7de241b43d9a685567d2a282e34901e0d66 | Alex, Irvine, Max, and Edwin arrive at a party at a random time between 9:00PM and 11:00PM. Alex stays for 10 minutes to see if the 3 other people are there and if they aren't, he leaves. Irvine waits for 35 minutes to see if 2 other people are already there and if not, leaves as well. Max waits for 5 minutes and stays... | [
"\\dfrac{2213}{331776}"
] | [
"Alex, Irvine, Max, and Edwin arrive at a party at a random time between 9:00PM and 11:00PM. Alex stays for 10 minutes to see if the 3 other people are there and if they aren't, he leaves. Irvine waits for 35 minutes to see if 2 other people are already there and if not, leaves as well. Max waits for 5 minutes and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-3714bd19e165c10e09a7f79b0011c024274021fda05bdacca255c12cf548fa4b | Alex, Michael, and Will write 2-digit perfect squares $A, M, W$ on the board. They notice that the 6-digit number $10000A + 100M + W$ is also a perfect square. Given that $A < W$, find the square root of the 6-digit number. | [
"408"
] | [
"Alex, Michael, and Will write 2-digit perfect squares $A, M, W$ on the board. They notice that the 6-digit number $10000A + 100M + W$ is also a perfect square. Given that $A < W$, find the square root of the 6-digit number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a2d60633e565903d2d26aa77c667b853afebb0181fe1300edf84b2c06de5102 | Alexa wrote the first 16 numbers of a sequence: \[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, \ldots\] She continued following the same pattern until she had 2015 numbers in total. What was the last number she wrote? | [
"1344"
] | [
"Alexa wrote the first 16 numbers of a sequence:\n\\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, \\ldots\\]\nShe continued following the same pattern until she had 2015 numbers in total. What was the last number she wrote?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 7
} | 54 | 0 | 0 |
omr-ws-1814df03f8091fd966c44b5d21818204e6b727d2144dd7035715e88f0db63f4b | Alexander and Louise are a couple of burglars. Everyday morning, Louise steals one third of Alexander’s money, but later in the afternoon, she gives him half of her money. If Louise starts with no money and begins stealing from the first day, what is the least integer amount of money Alexander must carry initially so t... | [
"\\( 4 \\cdot 3^{2011} \\)"
] | [
"Alexander and Louise are a couple of burglars. Everyday morning, Louise steals one third of Alexander’s money, but later in the afternoon, she gives him half of her money. If Louise starts with no money and begins stealing from the first day, what is the least integer amount of money Alexander must carry initially... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 7,
"tir": 8
} | 36 | 0 | 0 |
omr-ws-bc31a92d73f86305f7b4c59685c9af3e02a04de172c17cff19bc8a8c77aec9ef | Alexander and Louise are a pair of burglars. Every morning, Louise steals one third of Alexander's money, but later in the afternoon, she gives him half of all the money she has. If Louise starts with no money and begins stealing on the first day, what is the least positive integer amount of money Alexander must have s... | [
"\\( 3^{2012} \\)"
] | [
"Alexander and Louise are a pair of burglars. Every morning, Louise steals one third of Alexander's money, but later in the afternoon, she gives him half of all the money she has. If Louise starts with no money and begins stealing on the first day, what is the least positive integer amount of money Alexander must h... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 5
} | 7 | 0 | 0 |
omr-ws-9981afaca73e500c1759420ed58cc8a379b0720a217b3c8ffab62560f3b31d31 | Alexander has chosen a natural number \( N > 1 \) and has written down all its positive divisors \( d_1 < d_2 < \ldots < d_s \) (where \( d_1 = 1 \) and \( d_s = N \)). For each pair of neighboring numbers, he has found their greatest common divisor. The sum of all these \( s-1 \) greatest common divisors is equal to \... | [
"\\( N = 3 \\)"
] | [
"Alexander has chosen a natural number \\( N > 1 \\) and has written down all its positive divisors \\( d_1 < d_2 < \\ldots < d_s \\) (where \\( d_1 = 1 \\) and \\( d_s = N \\)). For each pair of neighboring numbers, he has found their greatest common divisor. The sum of all these \\( s-1 \\) greatest common diviso... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36f030ce81623e2e0e37e8ab93660bd986de563b946d807cff1d30fe2dcb03e3 | Alexander is eating sliced olives for dinner. Olives are sliced into thirds so that each olive is composed of two indistinguishable “end“ pieces and one “middle“ piece. For each olive slice that Alexander puts on his plate, there is a $\frac{2}{3}$ chance that it is an end piece and a $\frac{1}{3}$ chance that it is a ... | [
"5"
] | [
"Alexander is eating sliced olives for dinner. Olives are sliced into thirds so that each olive is composed of two indistinguishable “end“ pieces and one “middle“ piece. For each olive slice that Alexander puts on his plate, there is a $\\frac{2}{3}$ chance that it is an end piece and a $\\frac{1}{3}$ chance that i... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-68573004e9963bfa2b93a60f083c2cc2da60f4243b583f5eb7447795a01c411b | Alexander is juggling four balls: a red ball, a blue ball, a yellow ball, and a green ball. Each second, he tosses a ball from one hand to the other. If Alexander juggles for 20 seconds and the number of ways for Alexander to toss the balls such that he has all four balls in his right hand at the end is \( N \), find t... | [
"640"
] | [
"Alexander is juggling four balls: a red ball, a blue ball, a yellow ball, and a green ball. Each second, he tosses a ball from one hand to the other. If Alexander juggles for 20 seconds and the number of ways for Alexander to toss the balls such that he has all four balls in his right hand at the end is \\( N \\),... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-7eb0d318261f4c3e8b6b5e014a0dc39efb01387865dfecfcf70acde697f9d900 | Alexandra completes the letter A by adding to Figure 1. She draws the horizontal part of the letter A along the line $y=c$. The area of the shaded region inside the letter A and above the line with equation $y=c$ is $\frac{4}{9}$ of the total area of the region above the $x$-axis and between the left and right sides. D... | [
"2"
] | [
"Alexandra completes the letter A by adding to Figure 1. She draws the horizontal part of the letter A along the line $y=c$. The area of the shaded region inside the letter A and above the line with equation $y=c$ is $\\frac{4}{9}$ of the total area of the region above the $x$-axis and between the left and right si... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b1ed313e52996249837fea4c1fcf9d20769c0bcff3c14dfeb59623f0aef3afd | Alexandra lives near mall A, so the probability that she goes to mall A is $\frac{5}{9}$. The probability that Alexandra goes to mall B is the same as the probability that she goes to mall C. Barbara lives far from mall B, so the probability that she goes to mall B is $\frac{1}{7}$. The probability that Barbara goes to... | [
"\\(\\frac{166}{189}\\)"
] | [
"Alexandra lives near mall A, so the probability that she goes to mall A is $\\frac{5}{9}$. The probability that Alexandra goes to mall B is the same as the probability that she goes to mall C. Barbara lives far from mall B, so the probability that she goes to mall B is $\\frac{1}{7}$. The probability that Barbara ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-d57bd2d03d5eb336c09b4d1dba9541c332a6b924fb6dbdb222041df46385206e | Alexandre and Bernado are playing a game with $n$ balls in a bag. On the $k$-th turn, the player can take between $1$ and $k$ balls. Alexandre starts by taking one ball, and they alternate turns. The player who takes the last ball and makes the bag empty wins. Determine for each value of $n$ which player has the winnin... | [
"Alexandre wins if \\( n = k^2 + t \\) where \\( 0 \\leq t \\leq k-1 \\), and Bernado wins if \\( n = k^2 + t \\) where \\( k \\leq t \\leq 2k \\)."
] | [
"Alexandre and Bernado are playing a game with $n$ balls in a bag. On the $k$-th turn, the player can take between $1$ and $k$ balls. Alexandre starts by taking one ball, and they alternate turns. The player who takes the last ball and makes the bag empty wins. Determine for each value of $n$ which player has the w... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 18
} | 23 | 0 | 0 |
omr-ws-8dde49aa4b9aea2d2da711a7d3115395fc8b2b796abd08976899720b1445c87b | Alexandre and Herculano are at Campanha station and decide to calculate the length of a freight train that passes through the station without changing its speed. When the front of the train passes them, Alexandre starts walking in the direction of the train's movement and Herculano starts walking in the opposite direct... | [
"75"
] | [
"Alexandre and Herculano are at Campanha station and decide to calculate the length of a freight train that passes through the station without changing its speed. When the front of the train passes them, Alexandre starts walking in the direction of the train's movement and Herculano starts walking in the opposite d... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c94c4f35eee5576db5bb4e5e3a7a17ac0a2cd2e695c7a1b1305f51c5ab6b3a40 | Alexey and Bogdan play a game with two piles of stones. Initially, one pile contains 2021 stones, and the other is empty. In one move, a player must pick up an even number of stones from one pile, transfer half of these stones to the other pile, and remove the other half from the game. The player who cannot make a move... | [
"Bogdan wins."
] | [
"Alexey and Bogdan play a game with two piles of stones. Initially, one pile contains 2021 stones, and the other is empty. In one move, a player must pick up an even number of stones from one pile, transfer half of these stones to the other pile, and remove the other half from the game. The player who cannot make a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 15
} | 35 | 0 | 0 |
omr-ws-0f9cc4a607923025752ca4fd665c5ada42701d14cb49f8e9d5beacab61b48434 | Alexey has a device that measures the distance from all the vertices of a cube and the intersection points of the diagonals of the faces to a selected plane. He observed that there are only two different numbers among all the measurements, with the smallest being $2$. What is the edge length of the cube? | [
"\\( 4(\\sqrt{3} + 1) \\)"
] | [
"Alexey has a device that measures the distance from all the vertices of a cube and the intersection points of the diagonals of the faces to a selected plane. He observed that there are only two different numbers among all the measurements, with the smallest being $2$. What is the edge length of the cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8b502d652fc1d5f402e24da6a76390dd98e8bc33c88e513a1c84fda1d0b8809 | Alexia designed a logo 2 inches wide and 1.5 inches tall to be used on her school's notebooks. The school wants the logo on the notebook to be 8 inches wide. How tall, in inches, will it be if it is enlarged proportionately? | [
"6"
] | [
"Alexia designed a logo 2 inches wide and 1.5 inches tall to be used on her school's notebooks. The school wants the logo on the notebook to be 8 inches wide. How tall, in inches, will it be if it is enlarged proportionately?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e0ae919a7da6168d2773fcc6ff941e75ae45a1102ccdc93750c056ecd932920 | Alexio has 100 cards numbered 1-100, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3 or 5? Express your answer as a common fraction. | [
"\\dfrac{37}{50}"
] | [
"Alexio has 100 cards numbered 1-100, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3 or 5? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22fa1b451b420be7da7ac5f4bb1cf9de510ec7883e88d5ade57abfefdc6dec27 | Alexis and Joshua are drawing symbols in the sand. Alexis draws stars in pairs, and Joshua draws squares in trios. They want to create rows of 15 adjacent symbols. How many distinct rows of 15 symbols can they create, given that each row must consist of pairs of stars and trios of squares? | [
"28"
] | [
"Alexis and Joshua are drawing symbols in the sand. Alexis draws stars in pairs, and Joshua draws squares in trios. They want to create rows of 15 adjacent symbols. How many distinct rows of 15 symbols can they create, given that each row must consist of pairs of stars and trios of squares?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-3cf057b639b9f6449cf8d0236cd21a8deeab6d1c2888ebfb6c79776e27784917 | Alexis is at a spot on a 2-D coordinate grid that is 2400 units from the origin and 3840 units from another point. She realizes the wormhole is at a specific location based on these distances. How many units did Alexis run from her third position to the wormhole? To solve this, we need to determine the coordinates of t... | [
"4608"
] | [
"Alexis is at a spot on a 2-D coordinate grid that is 2400 units from the origin and 3840 units from another point. She realizes the wormhole is at a specific location based on these distances. How many units did Alexis run from her third position to the wormhole?\n\nTo solve this, we need to determine the coordina... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-1a07aca4dd666192c9f167cf7432461a7f9359b93cb4aa6c1b75f00ed6b1a2bd | Alexor draws an equilateral triangle of side length $1$, along with its circumcircle and incircle. Of the $7$ finite regions that result, he colors the $4$ regions with the largest area. What is the area of the colored region? | [
"\\(\\frac{5\\pi}{12} - \\frac{\\sqrt{3}}{4}\\)"
] | [
"Alexor draws an equilateral triangle of side length $1$, along with its circumcircle and incircle. Of the $7$ finite regions that result, he colors the $4$ regions with the largest area. What is the area of the colored region?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9aec974be7c03d735591d76c165368bb38b70cb6bd1ec6f283200fe8543459f6 | Alf attended an eight-year elementary school on Melmac. At the end of each school year, if he was promoted, his father gave him a number of cats equal to Alf’s age times the number of the grade he passed. Alf failed one grade and had to repeat it. After finishing elementary education, Alf found out that the total numbe... | [
"4"
] | [
"Alf attended an eight-year elementary school on Melmac. At the end of each school year, if he was promoted, his father gave him a number of cats equal to Alf’s age times the number of the grade he passed. Alf failed one grade and had to repeat it. After finishing elementary education, Alf found out that the total ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b2bad50af27cd78ffe15d886a36f51164836b138692c6de2a280c1deee876db0 | Alfonso and Benita each have 29 cards, with nine $0$s, ten $1$s, and ten $2$s. They alternate turns placing one card face up on a table. When three face-up cards sum to a multiple of $3$, they are removed. The game ends when all cards have been played. Alfonso wins if there are any non-$0$ cards left on the table at th... | [
"Benita"
] | [
"Alfonso and Benita each have 29 cards, with nine $0$s, ten $1$s, and ten $2$s. They alternate turns placing one card face up on a table. When three face-up cards sum to a multiple of $3$, they are removed. The game ends when all cards have been played. Alfonso wins if there are any non-$0$ cards left on the table ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-f0f9144f25302a988c195ffa4b24790feda36e921c7c537bff1849b1fed8fdf5 | Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the ... | [
"093"
] | [
"Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-7877f5f5773d13761ba4c4edd48170a4b85d9e898df0d67095456ac75907e4c4 | Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the ... | [
"093"
] | [
"Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-988bdc94930f140a3155fcadb0bbaadb65ba59e8d1b296f8dd6446aec61cb300 | Alfred owns a six-sided die where the probability of a number \( x \) appearing on top is directly proportional to \( x^2 \). If he rolls the die twice, what is the probability that the sum of the two numbers he rolls is equal to 7? | [
"\\dfrac{80}{1183}"
] | [
"Alfred owns a six-sided die where the probability of a number \\( x \\) appearing on top is directly proportional to \\( x^2 \\). If he rolls the die twice, what is the probability that the sum of the two numbers he rolls is equal to 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e45cb56ae0d22064e7df90d12d38e1cfbe12dee1990306b6c365b24ca717a23f | Alfred removes ten numbers from the set \(\{1, 2, \ldots, 100\}\). Phil then searches for arithmetic progressions among the remaining 90 members. What is the maximum length of an arithmetic progression that Phil is guaranteed to find? | [
"10"
] | [
"Alfred removes ten numbers from the set \\(\\{1, 2, \\ldots, 100\\}\\). Phil then searches for arithmetic progressions among the remaining 90 members. What is the maximum length of an arithmetic progression that Phil is guaranteed to find?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 60,
"genselect": 7
} | 67 | 0 | 0 |
omr-ws-84d6164652da6af2c150cf9db44ac3945ce67db8e0b01af4b30d5b25a74e4861 | Alfred the Ant lives at \( (3, 4) \) and wants to visit Gregory the Grasshopper at \( (10, 7) \). He must avoid potholes at \( (4, 7) \), \( (8, 6) \), and \( (9, 5) \). How many distinct paths can Alfred take to Gregory's house, moving only up or right? | [
"32"
] | [
"Alfred the Ant lives at \\( (3, 4) \\) and wants to visit Gregory the Grasshopper at \\( (10, 7) \\). He must avoid potholes at \\( (4, 7) \\), \\( (8, 6) \\), and \\( (9, 5) \\). How many distinct paths can Alfred take to Gregory's house, moving only up or right?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-4035f24c18e51a7ca8b2c0b4d633ab8faf9356638cc68cc7d4d788ec88a9fb88 | Alfred, Bob, and Carl plan to meet at a café between noon and 2 pm. Alfred and Bob will arrive at a random time between noon and 2 pm and will wait for 20 minutes or until 2 pm for all 3 people to show up after which they will leave. Carl will arrive at the café at noon and leave at 1:30 pm. What is the probability tha... | [
"\\dfrac{2}{9}"
] | [
"Alfred, Bob, and Carl plan to meet at a café between noon and 2 pm. Alfred and Bob will arrive at a random time between noon and 2 pm and will wait for 20 minutes or until 2 pm for all 3 people to show up after which they will leave. Carl will arrive at the café at noon and leave at 1:30 pm. What is the probabilit... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-72248fb2d1b3f1cbe350df818412f82eaa891b40c36c069b6584f4e8eac3c8dd | Algebraically determine the value of \( t \) for which \( V(t) \) is $100 or less. | [
"10"
] | [
"Algebraically determine the value of \\( t \\) for which \\( V(t) \\) is $100 or less."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-2dea3d04e8cc8397c421275a0941137d1bdd2a8da4d59e73ed37caffdbfe4657 | Ali Baba has a rectangular piece of carpet whose dimensions are unknown. He finds that if he lays it flat on the floor of either of his storerooms, each corner touches a different wall of that room. The storerooms have the same length but widths of 38 feet and 55 feet, respectively. What are the dimensions of the carpe... | [
"\\( l = 33 \\) feet, \\( w = 44 \\) feet"
] | [
"Ali Baba has a rectangular piece of carpet whose dimensions are unknown. He finds that if he lays it flat on the floor of either of his storerooms, each corner touches a different wall of that room. The storerooms have the same length but widths of 38 feet and 55 feet, respectively. What are the dimensions of the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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