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-d1139b14639fdff20280a6f75d6c32b88f7a5f3b1dab87fb729ccfbe16ad3583 | After traveling 50 miles by taxi, Ann is charged a fare of $\$120$. Assuming the taxi fare is directly proportional to distance traveled, how much would Ann be charged (in dollars) if she had traveled 70 miles? | [
"168"
] | [
"After traveling 50 miles by taxi, Ann is charged a fare of $\\$120$. Assuming the taxi fare is directly proportional to distance traveled, how much would Ann be charged (in dollars) if she had traveled 70 miles?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5072321af29bfc5ad47d025dc837278a0935b7cd51ec1fbc955d6bfb13f7526b | After two days of matches where each of the 32 football clubs plays one match per day, determine the maximum number of clubs that can be selected such that no two of these clubs have played against each other. | [
"16"
] | [
"After two days of matches where each of the 32 football clubs plays one match per day, determine the maximum number of clubs that can be selected such that no two of these clubs have played against each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f793f0fd88205b04a596ebc3d2ae9178086e7642502354ef41e2470394dea424 | After winning $60,000 in the lottery, a retired teacher gave $10,000 to her grandchildren. She invested part of the remainder in a growth fund that earned 4.4% annually and the rest in certificates of deposit paying 5.8% annually. The total interest received from these two investments at the end of the first year was $... | [
"$11,000 in the growth fund and $39,000 in certificates of deposit."
] | [
"After winning $60,000 in the lottery, a retired teacher gave $10,000 to her grandchildren. She invested part of the remainder in a growth fund that earned 4.4% annually and the rest in certificates of deposit paying 5.8% annually. The total interest received from these two investments at the end of the first year ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 24
} | 24 | 0 | 0 |
omr-ws-c11b44ed2eeb132e29dc49c077a6e2cd41143c763805551b8500a90cc9f66a70 | Agraphena writes down the decimal representation of $3^{20072008}$ on a board. She then divides the number into 3 parts of random lengths. If all 3 parts are powers of 3, what is the sum of the exponents of these three powers of 3? | [
"20072008"
] | [
"Agraphena writes down the decimal representation of $3^{20072008}$ on a board. She then divides the number into 3 parts of random lengths. If all 3 parts are powers of 3, what is the sum of the exponents of these three powers of 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-c28a10c1f5523e2ec8af107f71b65e09439dd05e66cff5f64bcd38d8afe1609e | Agung found that the average score of the three math tests he followed was $81$. The first test score was $85$. The third test score was $4$ lower than the second test score. What is the value of Agung's second test? | [
"81"
] | [
"Agung found that the average score of the three math tests he followed was $81$. The first test score was $85$. The third test score was $4$ lower than the second test score. What is the value of Agung's second test?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60cf010e52d946cdcd0a36a21ccbee50d6b161e7825bbc9c6fe52b45626fe35a | Agus and Jaka use an escalator to go up to the $2^{nd}$ floor. Agus took $25$ steps and Jaka took $9$ steps to get to the $2^{nd}$ floor. The time it took Jaka was twice the time it took Agus. When the escalator is stopped, how many steps indicate the distance from the $1^{st}$ floor to the $2^{nd}$ floor? | [
"41"
] | [
"Agus and Jaka use an escalator to go up to the $2^{nd}$ floor. Agus took $25$ steps and Jaka took $9$ steps to get to the $2^{nd}$ floor. The time it took Jaka was twice the time it took Agus. When the escalator is stopped, how many steps indicate the distance from the $1^{st}$ floor to the $2^{nd}$ floor?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7acac30520114bc664fb7cc74ff0d404b7a1e079d32942cf548fa0198443c529 | Agustina and Santiago play a game on a rectangular sheet. Agustina says a natural number $n$, and Santiago marks $n$ points on the sheet. Agustina then chooses some of these points. Santiago wins if he can draw a rectangle with sides parallel to the edges of the sheet that contains all the points chosen by Agustina and... | [
"5"
] | [
"Agustina and Santiago play a game on a rectangular sheet. Agustina says a natural number $n$, and Santiago marks $n$ points on the sheet. Agustina then chooses some of these points. Santiago wins if he can draw a rectangle with sides parallel to the edges of the sheet that contains all the points chosen by Agustin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19cc99cb222c16234bb01f082466ca4826cd7600a413b3e2aa0c9b71b1fa9db3 | Ahmad has 20 chocolates and will share them among his friends Budi, Cika, Dhani, and Eko such that each friend receives a different number of chocolates. In how many ways can Ahmad share the chocolates such that Cika receives more chocolates than one of her friends? | [
"414"
] | [
"Ahmad has 20 chocolates and will share them among his friends Budi, Cika, Dhani, and Eko such that each friend receives a different number of chocolates. In how many ways can Ahmad share the chocolates such that Cika receives more chocolates than one of her friends?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-e5b6bee632dbeef214f9478898128b97b2feb4cdd4de81f491c583e971ad9f5d | Ahmed and Beth have respectively $p$ and $q$ marbles, with $p > q$. Starting with Ahmed, each in turn gives to the other as many marbles as the other already possesses. After $2n$ such transfers, Ahmed has $q$ marbles and Beth has $p$ marbles. Find $\frac{p}{q}$ in terms of $n$. | [
"\\(\\frac{2 \\times 4^n + 1}{4^n + 2}\\)"
] | [
"Ahmed and Beth have respectively $p$ and $q$ marbles, with $p > q$. Starting with Ahmed, each in turn gives to the other as many marbles as the other already possesses. After $2n$ such transfers, Ahmed has $q$ marbles and Beth has $p$ marbles. Find $\\frac{p}{q}$ in terms of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 3
} | 48 | 0 | 0 |
omr-ws-538780a4cc5c8a94522ad2ebfd35fc3a62d0eefd493e3f944adf258a95cccd8a | Ahmed and Beth have respectively \( p \) and \( q \) marbles, with \( p > q \). Starting with Ahmed, each in turn gives to the other as many marbles as the other already possesses. After \( 2n \) such transfers, Ahmed has \( q \) marbles and Beth has \( p \) marbles. Find \( \frac{p}{q} \) in terms of \( n \). | [
"\\(\\frac{2 \\times 4^n + 1}{4^n + 2}\\)"
] | [
"Ahmed and Beth have respectively \\( p \\) and \\( q \\) marbles, with \\( p > q \\). Starting with Ahmed, each in turn gives to the other as many marbles as the other already possesses. After \\( 2n \\) such transfers, Ahmed has \\( q \\) marbles and Beth has \\( p \\) marbles. Find \\( \\frac{p}{q} \\) in terms ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 4,
"tir": 8
} | 58 | 0 | 0 |
omr-ws-444d8c032526aaabcc0dc1c7867619692a8e6eabbe5c4cd1c8488b4739c063d7 | Ahmed and Beth have respectively \( p \) and \( q \) marbles, with \( p > q \). Starting with Ahmed, each in turn gives to the other as many marbles as the other already possesses. After \( 2n \) such transfers, Ahmed has \( q \) marbles and Beth has \( p \) marbles. Find \(\frac{p}{q}\) in terms of \( n \). | [
"\\dfrac{2 \\cdot 4^n + 1}{4^n + 2}"
] | [
"Ahmed and Beth have respectively \\( p \\) and \\( q \\) marbles, with \\( p > q \\). Starting with Ahmed, each in turn gives to the other as many marbles as the other already possesses. After \\( 2n \\) such transfers, Ahmed has \\( q \\) marbles and Beth has \\( p \\) marbles. Find \\(\\frac{p}{q}\\) in terms of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5983acd42a9af5bf0d391b6da7130ebeb4427f681023146923491ac405a0a349 | Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get? | [
"380"
] | [
"Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-cdef46adc1f1b9c15a824c46055284a16940e2bd180c33aed38e707f6b28fff0 | Ai and Beth each have $2023$ dollars. They also each have a biased coin that flips heads with probability \( 51\% \). Every second, they flip their coins. On every flip, Ai bets a dollar that her coin comes up heads, whereas Beth bets a dollar that her coin comes up tails. Ai and Beth eventually both go broke. Who was ... | [
"Beth"
] | [
"Ai and Beth each have $2023$ dollars. They also each have a biased coin that flips heads with probability \\( 51\\% \\). Every second, they flip their coins. On every flip, Ai bets a dollar that her coin comes up heads, whereas Beth bets a dollar that her coin comes up tails. Ai and Beth eventually both go broke. ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-00be958718efdb6b35f6e06b53bcb39424fdedce6b949546583807d9995ceea1 | Aibek marked a point inside a regular hexagon and connected it with segments from each of the vertices. He painted the resulting six triangles alternately in red and green. If the area of the regular hexagon is $H$, calculate the absolute difference between the sum of the areas of the red triangles and the sum of the a... | [
"0"
] | [
"Aibek marked a point inside a regular hexagon and connected it with segments from each of the vertices. He painted the resulting six triangles alternately in red and green. If the area of the regular hexagon is $H$, calculate the absolute difference between the sum of the areas of the red triangles and the sum of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f1a7c26cfe969e9888a04c412895c3176f0faafa2a81c428c05210a3c8d25ac | Aida made three cubes with positive integer side lengths \(a\), \(b\), and \(c\). She divided them into unit cubes and attempted to construct a cube of side \(a+b+c\). She was 648 blocks short. How many possibilities of the ordered triple \((a, b, c)\) are there? | [
"7"
] | [
"Aida made three cubes with positive integer side lengths \\(a\\), \\(b\\), and \\(c\\). She divided them into unit cubes and attempted to construct a cube of side \\(a+b+c\\). She was 648 blocks short. How many possibilities of the ordered triple \\((a, b, c)\\) are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 5
} | 26 | 0 | 0 |
omr-ws-d92b2d6d7bf9bd06c4d501484b1f701018a61c63e2c6ead81c6b74b7a28a5e7e | Aimee flips a coin 8 times. Each time she flips heads, she writes down a 1, and each time she flips tails, she writes down a 0. What is the expected value of the binary number formed by the 8 flips? | [
"127.5"
] | [
"Aimee flips a coin 8 times. Each time she flips heads, she writes down a 1, and each time she flips tails, she writes down a 0. What is the expected value of the binary number formed by the 8 flips?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-373420a0d94378e8fb4556877d083a27beddb896aedb3488266b244a9284e767 | Aimee plays at least one game of chess a day for eight weeks, but she plays no more than 11 games in any 7-day period. Let $N$ be the number of different possible values of $k$ such that there exists a period of $k$ consecutive days in which she plays exactly 23 games. Find the value of $N$. | [
"9"
] | [
"Aimee plays at least one game of chess a day for eight weeks, but she plays no more than 11 games in any 7-day period. Let $N$ be the number of different possible values of $k$ such that there exists a period of $k$ consecutive days in which she plays exactly 23 games. Find the value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-78fb22c94f770e0d7a5b90cf0c121103c5aea1a0be9dd6cabd4f5a73cd264b69 | Aina the Algebraist needs to buy tickets for 1096 days of public transport in a European city. The city offers 7-day tickets for 7.03 euros and 30-day tickets for 30 euros. What is the cheapest way for Aina to buy tickets to cover all 1096 days? | [
"1096.84 euros"
] | [
"Aina the Algebraist needs to buy tickets for 1096 days of public transport in a European city. The city offers 7-day tickets for 7.03 euros and 30-day tickets for 30 euros. What is the cheapest way for Aina to buy tickets to cover all 1096 days?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 1,
"tir": 31
} | 78 | 0 | 0 |
omr-ws-ca23ee330327f3d34aa22477eb4c80e940b2f723f7460eda2786d9f754e31912 | Air is being pumped into a spherical balloon at a rate of 5 cm³/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 100 cm. | [
"\\(\\frac{1}{6000\\pi}\\) cm/sec"
] | [
"Air is being pumped into a spherical balloon at a rate of 5 cm³/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 100 cm."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b98a1605243383dc5e0f8aaa1b72cae4ec985e710b9af8d49a058c07a8c23a1c | Aircraft A is flying due west at 150 mph, and Aircraft B is 15 miles due north of Aircraft A, flying south at 400 mph. How close will the two aircraft come to each other? | [
"\\(\\frac{45}{8}\\) miles"
] | [
"Aircraft A is flying due west at 150 mph, and Aircraft B is 15 miles due north of Aircraft A, flying south at 400 mph. How close will the two aircraft come to each other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71418790387d322a61f2c5283c1ef805b2caa0419348017515874f87baa9d09c | Aisling and Brendan take alternate moves in a game starting with the number $x = 2023$. A move consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game. | [
"Aisling has a winning strategy."
] | [
"Aisling and Brendan take alternate moves in a game starting with the number $x = 2023$. A move consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game."... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-619112fa5c652b737a83e7f7299bb430b6781d0e562e33c169296be81c31e9ef | Aisyah received $300,000$ in pocket money in early January. She saves half of her allowance every month. Her mother increases her allowance to $\frac{6}{5}$ of the previous month's allowance each month. Determine the amount of Aisyah's savings until the 10th month. | [
"3893802.32"
] | [
"Aisyah received $300,000$ in pocket money in early January. She saves half of her allowance every month. Her mother increases her allowance to $\\frac{6}{5}$ of the previous month's allowance each month. Determine the amount of Aisyah's savings until the 10th month."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42c1f6eefe40b19cba3fe418ff1451eeb6987bd92fdea84a9d7912aa84e5e841 | Ajay and Joey are playing a card game. Ajay has cards labelled 2, 4, 6, 8, and 10, and Joey has cards labelled 1, 3, 5, 7, 9. Each of them takes a hand of 4 random cards and picks one to play. If one of the cards is at least twice as big as the other, whoever played the smaller card wins. Otherwise, the larger card win... | [
"47"
] | [
"Ajay and Joey are playing a card game. Ajay has cards labelled 2, 4, 6, 8, and 10, and Joey has cards labelled 1, 3, 5, 7, 9. Each of them takes a hand of 4 random cards and picks one to play. If one of the cards is at least twice as big as the other, whoever played the smaller card wins. Otherwise, the larger car... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-0883daa49e43ebcfb7bfde2e0d4707a4c41b8db1d6c34b6e3ef440ab720721c6 | Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degre... | [
"120^\\circ"
] | [
"Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\\circ$ latitude and $110^\\circ \\text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\\circ \\text{ N}$ latitude and $115^\\circ \\text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90739a215dadac8048b48a61ef49e728478c0c3345b7f550796ff1bed98ea32a | Ajit arrives home at a random time between 3:00 and 5:00 PM and stays awake for 50 minutes before going to bed. Akshaj arrives home at a random time between 4:00 and 6:00 PM and stays awake for 70 minutes before going to bed. What is the probability that Ajit and Akshaj are awake together for more than 10 minutes and l... | [
"\\(\\dfrac{5}{36}\\)",
"\\dfrac{5}{36}"
] | [
"Ajit arrives home at a random time between 3:00 and 5:00 PM and stays awake for 50 minutes before going to bed. Akshaj arrives home at a random time between 4:00 and 6:00 PM and stays awake for 70 minutes before going to bed. What is the probability that Ajit and Akshaj are awake together for more than 10 minutes ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 27,
"tir": 26
} | 53 | 0 | 0 |
omr-ws-84e4e8099dc6494c6ec844b0d0086b6d8d1b6613d43a2e3d4838e1c3aa91dff5 | Akari, Kris, Anna, Lahrri, Mylanda, and Elaine are standing in line at a cafeteria. Anna and Elaine are twins and are indistinguishable from the point of view of a third party. From this person's point of view, how many visually different ways can this group of people line up? | [
"360"
] | [
"Akari, Kris, Anna, Lahrri, Mylanda, and Elaine are standing in line at a cafeteria. Anna and Elaine are twins and are indistinguishable from the point of view of a third party. From this person's point of view, how many visually different ways can this group of people line up?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5eee4fad12be082d0241cbb1dc9a1a5b11f7abda588fe215ea2149ff2f1495ce | Akash has a calculator that can multiply a number by 2, multiply a number by 2 and add 1, or multiply a number by 4 and add 1. How many ways can Akash use the calculator to get from 1 to 341? | [
"16"
] | [
"Akash has a calculator that can multiply a number by 2, multiply a number by 2 and add 1, or multiply a number by 4 and add 1. How many ways can Akash use the calculator to get from 1 to 341?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6bb9614af0c30f7947e57fbcb28df32f2f20a2e1e9410a9a064392f10667766 | Akash's birthday cake is in the form of a 4 × 4 × 4 inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into 64 smaller cubes, each measuring 1 × 1 × 1 inch. How many of the small pieces will have icing on exactly two sides? | [
"20"
] | [
"Akash's birthday cake is in the form of a 4 × 4 × 4 inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into 64 smaller cubes, each measuring 1 × 1 × 1 inch. How many of the small pieces will have icing on exactly two sides?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-a97b21800858bac5a2bc006d1bdcdf9b1b38a45b04de384228450943c01663a4 | Akbar and Birbal together have \( n \) marbles, where \( n > 0 \). Akbar says to Birbal that if he gives Birbal some marbles, Birbal will have twice as many marbles as Akbar. Birbal says to Akbar that if he gives Akbar some marbles, Akbar will have thrice as many marbles as Birbal. Find the minimum possible value of \(... | [
"12"
] | [
"Akbar and Birbal together have \\( n \\) marbles, where \\( n > 0 \\). Akbar says to Birbal that if he gives Birbal some marbles, Birbal will have twice as many marbles as Akbar. Birbal says to Akbar that if he gives Akbar some marbles, Akbar will have thrice as many marbles as Birbal. Find the minimum possible va... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-9caf5d4f05ac57ca31bb2381d96153cecfb6eb605b8bcb23de6e6a6583a4ff3c | Akello divides a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes the value of its width divided by its height, and within each red rectangle, she writes the value of its height divided by its width. She then calculates $x$,... | [
"2"
] | [
"Akello divides a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes the value of its width divided by its height, and within each red rectangle, she writes the value of its height divided by its width. She then calculates... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-86a89a05d97e44424817a0f1c8c4447e18024d1fa86012e5200c84b951ae6392 | Akshaj has a special coin. If $n$ is the largest non-negative integer such that his coin's last $n$ flips were all heads, then his coin will flip heads with a probability of $\dfrac{1}{2^{n+1}}$ (and tails with probability $1 - \frac{1}{2^{n+1}}$). For example, on the first flip, the coin will flip heads with a probabi... | [
"840"
] | [
"Akshaj has a special coin. If $n$ is the largest non-negative integer such that his coin's last $n$ flips were all heads, then his coin will flip heads with a probability of $\\dfrac{1}{2^{n+1}}$ (and tails with probability $1 - \\frac{1}{2^{n+1}}$). For example, on the first flip, the coin will flip heads with a ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-fc08713b3d41b74d26baddbde09f21fd3bb7d8a65aac879d31bc343e6614df1e | Akshaj has a special coin. If \( n \) is the largest non-negative integer such that his coin's last \( n \) flips were all heads, then his coin will flip heads with a probability of \( \frac{1}{2^{n+1}} \) (and tails with probability \( 1 - \frac{1}{2^{n+1}} \)). For example, on the first flip, the coin will flip heads... | [
"840"
] | [
"Akshaj has a special coin. If \\( n \\) is the largest non-negative integer such that his coin's last \\( n \\) flips were all heads, then his coin will flip heads with a probability of \\( \\frac{1}{2^{n+1}} \\) (and tails with probability \\( 1 - \\frac{1}{2^{n+1}} \\)). For example, on the first flip, the coin ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-0435deca5a064adf77ad80c3905a004e0cdd059128a62e65a054d7b3b8a81610 | Akshaj made three cubes with positive integer side lengths \(x\), \(y\), and \(z\), respectively. He divided them into unit cubes and attempted to construct a cube of side \(x + y + z\). Unfortunately, he was short by 648 blocks. How many possibilities of the ordered triple \((x, y, z)\) are there? | [
"7"
] | [
"Akshaj made three cubes with positive integer side lengths \\(x\\), \\(y\\), and \\(z\\), respectively. He divided them into unit cubes and attempted to construct a cube of side \\(x + y + z\\). Unfortunately, he was short by 648 blocks. How many possibilities of the ordered triple \\((x, y, z)\\) are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-97a4546f81611babe2106c3d500eed9d1101fe4bea874ef1d6e515080de4d498 | Al and Barb both tried to solve the same quadratic equation of the form \(A(x^2) + bx + c = 0\). Al's solutions were -2 and -1.5, but he copied \(A\) wrong. Barb's solutions were 2 and 3, but she copied \(b\) wrong. What are the solutions of the correct equation? | [
"-1 and -6"
] | [
"Al and Barb both tried to solve the same quadratic equation of the form \\(A(x^2) + bx + c = 0\\). Al's solutions were -2 and -1.5, but he copied \\(A\\) wrong. Barb's solutions were 2 and 3, but she copied \\(b\\) wrong. What are the solutions of the correct equation?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 21
} | 29 | 0 | 0 |
omr-ws-0e287945b8562ca42323d4d8696336a1e8e46a692316e0526d1694d08a6d621d | Al and Bill are playing a game where Al thinks of a number from the set $\{1, 2, \ldots, n\}$, and Bill tries to guess the number. If Bill's guess is incorrect, Al changes the number by either increasing or decreasing it by 1 (but the number must remain positive). Let $f(n)$ be the minimum number of guesses Bill needs ... | [
"50"
] | [
"Al and Bill are playing a game where Al thinks of a number from the set $\\{1, 2, \\ldots, n\\}$, and Bill tries to guess the number. If Bill's guess is incorrect, Al changes the number by either increasing or decreasing it by 1 (but the number must remain positive). Let $f(n)$ be the minimum number of guesses Bil... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-0deefada99e9b4b54c90fc83cce0c1f8a468370f3f1d76ffdd22dbbad05994b7 | Al and Bill play a game involving a fair six-sided die. The die is rolled until either there is a number less than $5$ rolled on consecutive tosses, or there is a number greater than $4$ on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be rel... | [
"29"
] | [
"Al and Bill play a game involving a fair six-sided die. The die is rolled until either there is a number less than $5$ rolled on consecutive tosses, or there is a number greater than $4$ on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 2
} | 23 | 0 | 0 |
omr-ws-e1d2ffa8ffa33e281f280234aed052cba76aadc8cc120e49b46da4f76d1e793b | Al and Bo together have $168. Each bets the same fraction of his money as the other bets. If Al wins, he'll have double what Bo then has. If Bo wins, he'll have triple what Al then has. How many dollars does Al have at the start? | [
"72"
] | [
"Al and Bo together have $168. Each bets the same fraction of his money as the other bets. If Al wins, he'll have double what Bo then has. If Bo wins, he'll have triple what Al then has. How many dollars does Al have at the start?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-15d89b402d751aed8e8f9e96669ff23838b678d90aa99a23593f27c5437d97d8 | Al and Bo together have $168. They each bet the same fraction of their money. If Al wins, he will have twice what Bo has left. If Bo wins, he will have three times what Al has left. Find the amount in dollars that Al starts with. | [
"72"
] | [
"Al and Bo together have $168. They each bet the same fraction of their money. If Al wins, he will have twice what Bo has left. If Bo wins, he will have three times what Al has left. Find the amount in dollars that Al starts with."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 7
} | 21 | 0 | 0 |
omr-ws-cc390e8ceeffabd6e35520f0d9c7fa43e8ec716714f274c545255fd5d77a3acf | Al and Bob are playing a game with a coin where they take turns flipping. Each turn, if a player flips a head, they flip again; otherwise, it is the other player’s turn. Al goes first. The probability that Al has at least as many heads as Bob after 10 total flips is $\frac{m}{n}$, where $m$ and $n$ are relatively prime... | [
"171"
] | [
"Al and Bob are playing a game with a coin where they take turns flipping. Each turn, if a player flips a head, they flip again; otherwise, it is the other player’s turn. Al goes first. The probability that Al has at least as many heads as Bob after 10 total flips is $\\frac{m}{n}$, where $m$ and $n$ are relatively... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-a92045df2a5b12edd510267a793d96fa28249732c0dcc84ebc203cb3fb049594 | Al and Bob are running around a track in the same direction. Al completes a lap in 1.5 minutes, while Bob completes a lap in 2.5 minutes. After 2 hours, how many times has Al passed Bob (excluding the starting point)? | [
"16"
] | [
"Al and Bob are running around a track in the same direction. Al completes a lap in 1.5 minutes, while Bob completes a lap in 2.5 minutes. After 2 hours, how many times has Al passed Bob (excluding the starting point)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 6
} | 11 | 0 | 0 |
omr-ws-b63e7df0ca868c1aab2f1a6260c26ec4df387c5d704e9043ea2f1a9f2a1ff71e | Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game? | [
"\\(\\frac{2}{729}\\)"
] | [
"Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f370b594a1f9666a699d5cee53f527b7f01b016a0f946c4b0d05eb30b31f26ad | Al and Bob play a game where Al rolls a six-sided die until he gets a $1$, and then Bob does the same. They alternate turns. After $2005$ total rolls, let the expected number of turns that Al has completed be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is div... | [
"17"
] | [
"Al and Bob play a game where Al rolls a six-sided die until he gets a $1$, and then Bob does the same. They alternate turns. After $2005$ total rolls, let the expected number of turns that Al has completed be $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2c7211939d79362d3e299684b1ef106228dc9967d145c86908f34682dcae1495 | Al has 2019 red jelly beans, Bob has 2018 green jelly beans, and Carl has \( x \) blue jelly beans. The minimum number of jelly beans that must be drawn to guarantee at least 2 jelly beans of each color is 4041. Compute \( x \). | [
"2020"
] | [
"Al has 2019 red jelly beans, Bob has 2018 green jelly beans, and Carl has \\( x \\) blue jelly beans. The minimum number of jelly beans that must be drawn to guarantee at least 2 jelly beans of each color is 4041. Compute \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d71fe06b8a496f3734c5d2a2ce6b078c6eac37be5ea3c75d68f76cea0b109ed2 | Al has 2019 red jelly beans. Bob has 2018 green jelly beans. Carl has \( x \) blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee 2 jelly beans of each color is 4041. Compute \( x \). | [
"2020"
] | [
"Al has 2019 red jelly beans. Bob has 2018 green jelly beans. Carl has \\( x \\) blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee 2 jelly beans of each color is 4041. Compute \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-122edf2eb0153d8c9f77964f8e977784a184f8548cecb7d2539d008e240f3d6f | Al invents a new game called $Z-Y-X-W-V$. Two players each choose to play either $Z, Y, X, W$, or $V$. If they play the same thing, the result is a tie. Al randomly decides a winner for each of the 10 pairs of distinct plays. What is the probability that after Al makes all of these 10 choices, the game is balanced, tha... | [
"\\dfrac{3}{128}"
] | [
"Al invents a new game called $Z-Y-X-W-V$. Two players each choose to play either $Z, Y, X, W$, or $V$. If they play the same thing, the result is a tie. Al randomly decides a winner for each of the 10 pairs of distinct plays. What is the probability that after Al makes all of these 10 choices, the game is balanced... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-8b3fe660ab9a67d3f56c8167b41d024c07f6b7ef1a6ebe33dec7704edef80222 | Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination? | [
"\\(\\sqrt{5} + \\sqrt{13}\\)"
] | [
"Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination?"
] | [
"aops_c4_high_school_math"
] | [
"genselect",
"tir"
] | {
"genselect": 7,
"tir": 13
} | 20 | 0 | 0 |
omr-ws-da18393805efd75757f65234d5d1bab7fc03031a932601838735e3381c026aa8 | Al usually arrives at the train station on the commuter train at 6:00, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at 5:00. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Jane ar... | [
"45 mph"
] | [
"Al usually arrives at the train station on the commuter train at 6:00, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at 5:00. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Ja... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e9242307093ce3ffb964a1461ca2908d6d24af463e0cfaa4edb173da823f2bba | Al usually arrives at the train station on the commuter train at 6:00, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at 5:00. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Jane ar... | [
"45 mph"
] | [
"Al usually arrives at the train station on the commuter train at 6:00, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at 5:00. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Ja... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-e66b9767d027e0c3587bbf1ca81d9fce4894a8ac2e0edb51d322acb098465159 | Al walks down to the bottom of an escalator that is moving up and counts 150 steps. Bob walks up to the top of the escalator and counts 75 steps. If Al's speed is 3 times Bob's speed, how many steps are visible on the escalator at any given time? | [
"120"
] | [
"Al walks down to the bottom of an escalator that is moving up and counts 150 steps. Bob walks up to the top of the escalator and counts 75 steps. If Al's speed is 3 times Bob's speed, how many steps are visible on the escalator at any given time?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 3
} | 25 | 0 | 0 |
omr-ws-222228cfbb37eca8ecf06b25327b11f1e01e268f3fbbae1190976de9d515e443 | Al walks down to the bottom of an escalator that is moving up and counts 150 steps. Bob walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is ... | [
"120"
] | [
"Al walks down to the bottom of an escalator that is moving up and counts 150 steps. Bob walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this valu... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-90ceac2d09dd350de40863c4633264e642f6b6fefbbefc45eb901a1e6c75ec17 | Al walks down to the bottom of an escalator that is moving up and counts 150 steps. Bob walks up to the top of the escalator and counts 75 steps. If Al's walking speed (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is con... | [
"120"
] | [
"Al walks down to the bottom of an escalator that is moving up and counts 150 steps. Bob walks up to the top of the escalator and counts 75 steps. If Al's walking speed (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value i... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-0cd9b73c9281057045a52fce8ed84438f93d9dfbe03df164ad02524e3db3ef8a | Al walks down to the bottom of an escalator that is moving up and counts 150 steps. Bob walks up to the top of the same escalator and counts 75 steps. If Al's walking speed (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value i... | [
"120"
] | [
"Al walks down to the bottom of an escalator that is moving up and counts 150 steps. Bob walks up to the top of the same escalator and counts 75 steps. If Al's walking speed (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this va... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-128960e8fc306dee003f55c0b324258428d2427b5fe88a3693b4d72f469cff2e | Al walks down to the bottom of an escalator that is moving up and counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al’s walking speed (in steps per unit time) is three times Bob’s walking speed, how many steps are visible on the escalator at a given time? (Assume that this... | [
"120"
] | [
"Al walks down to the bottom of an escalator that is moving up and counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al’s walking speed (in steps per unit time) is three times Bob’s walking speed, how many steps are visible on the escalator at a given time? (Assume that... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-5bc154a3dd054a64fc86d2f08942ac1a455b1c31f15f2759eca4ee0f74b871b4 | Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume tha... | [
"120"
] | [
"Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assum... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-b10f8e2708fd6d1fb0a62755115b81c2b4d9e9e502574425915d7216fcdf4e46 | Al, Bert, and Carl are to divide a pile of Halloween candy in a ratio of $3:2:1$, respectively. Each believes they are the first to arrive and takes what they think is their correct share. What fraction of the candy goes unclaimed? | [
"\\(\\frac{5}{18}\\)"
] | [
"Al, Bert, and Carl are to divide a pile of Halloween candy in a ratio of $3:2:1$, respectively. Each believes they are the first to arrive and takes what they think is their correct share. What fraction of the candy goes unclaimed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-ea3069ef0171c18f3924cb18f037892a1cdfa885bb2169c8e9da7af396d722e7 | Al, Betty, and Clare split $\$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $\$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $\$100$. What was Al's original portion? | [
"400"
] | [
"Al, Betty, and Clare split $\\$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $\\$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $\\$100$. What was Al's original portion?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b4ce39980e4b65b1ddd2b935df14b04fedeafc6cf9f71ec18ba906bda3d0340 | Al, Bob, Carl, and D’Angelo play a team game of Rock Paper Scissors. A game is called perfect if some two of the four play the same thing, and the other two also play the same thing, but something different. What is the probability of a perfect game? | [
"\\(\\frac{2}{9}\\)"
] | [
"Al, Bob, Carl, and D’Angelo play a team game of Rock Paper Scissors. A game is called perfect if some two of the four play the same thing, and the other two also play the same thing, but something different. What is the probability of a perfect game?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-f833e67b1d38d4564dfd283772cd258afc067535f6411af8a2d7c94690c55d7c | Al, Bob, and Carl each have different amounts of money. They share it as follows: Al gives Bob and Carl each 1/3 of his money. Bob then gives Al and Carl each 1/3 of his new total. Carl then gives Al and Bob each 1/3 of his new total. The result is that Al ends up with $45 more than Bob and Bob ends with $45 more than ... | [
"131"
] | [
"Al, Bob, and Carl each have different amounts of money. They share it as follows: Al gives Bob and Carl each 1/3 of his money. Bob then gives Al and Carl each 1/3 of his new total. Carl then gives Al and Bob each 1/3 of his new total. The result is that Al ends up with $45 more than Bob and Bob ends with $45 more ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6075c89116c50bf3a65cf2bcd6e489f61f824fbc6d5d5e99e1eaedcabaf4402 | Aladdin has a set of coins with weights $1, 2, \ldots, 20$ grams. He can ask the Genie to compare any two coins from the set, but he must pay the Genie one coin from the set for each comparison. Can Aladdin find two coins from the set with a total weight of at least $28$ grams? | [
"Yes, Aladdin can find two coins from the set with a total weight of at least 28 grams. \n\nSince the question about 30 grams is not answered in the provided discussions, we focus on the confirmed solution."
] | [
"Aladdin has a set of coins with weights $1, 2, \\ldots, 20$ grams. He can ask the Genie to compare any two coins from the set, but he must pay the Genie one coin from the set for each comparison. Can Aladdin find two coins from the set with a total weight of at least $28$ grams?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-531bdfd449520c3da90889f5bc65063121f9e8aa176c862a04bc9cf89eb9464f | Aladdin has a set of coins with weights \( 1, 2, \ldots, 20 \) grams. He can ask the Genie about any two coins from the set which one is heavier, but he must pay the Genie some other coin from the set before each question. Can Aladdin find two coins from the set with a total weight of at least 28 grams? | [
"Yes"
] | [
"Aladdin has a set of coins with weights \\( 1, 2, \\ldots, 20 \\) grams. He can ask the Genie about any two coins from the set which one is heavier, but he must pay the Genie some other coin from the set before each question. Can Aladdin find two coins from the set with a total weight of at least 28 grams?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5c6e5a2f834242c0d5c90a4883ad1351b33f56bacec52df0ee61aade1b07ee18 | Aladdin visited every point of the equator, moving sometimes to the west, sometimes to the east, and sometimes being instantaneously transported by genies to the diametrically opposite point on the Earth. Let $d$ be the difference between the distances moved by Aladdin to the west and to the east during the period of t... | [
"\\dfrac{L}{2}"
] | [
"Aladdin visited every point of the equator, moving sometimes to the west, sometimes to the east, and sometimes being instantaneously transported by genies to the diametrically opposite point on the Earth. Let $d$ be the difference between the distances moved by Aladdin to the west and to the east during the period... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-734512b170a80c03cd98884e154fa44638f9d3235695b3b0f7a62c951ac1e20c | Alan and Barbara play a game in which they take turns filling entries of an initially empty $4 \times 4$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Let $D$ be the determinant of the resulting matrix. If Barbara has a ... | [
"0"
] | [
"Alan and Barbara play a game in which they take turns filling entries of an initially empty $4 \\times 4$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Let $D$ be the determinant of the resulting matrix. If Barbara ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c2032e664e9f018a962d2351532ada92e3e0f61bd323361a2c5d1d1b220a4a1 | Alan and Brian play a badminton match where the first to win two sets in a row or two out of three sets wins the match. Alan wins the first set with probability $\frac{2}{3}$, and his probability of winning any subsequent set is half of that of winning the previous set. If Alan wins the match, Brian pays him $\$b$ per ... | [
"\\(\\frac{59}{79}\\)"
] | [
"Alan and Brian play a badminton match where the first to win two sets in a row or two out of three sets wins the match. Alan wins the first set with probability $\\frac{2}{3}$, and his probability of winning any subsequent set is half of that of winning the previous set. If Alan wins the match, Brian pays him $\\$... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0aaccb94c54f758a596b5241dbababe9aa79235e0eeef5f3b07ae659a298946d | Alan and Charlie are located at vertices A and C of a rectangular track ABCD with dimensions AB=100 m and BC=50 m. Alan starts biking towards vertex B at a speed of 10 m/s, and Charlie starts biking towards vertex D at a speed of 15 m/s. How long before they meet? | [
"30 seconds"
] | [
"Alan and Charlie are located at vertices A and C of a rectangular track ABCD with dimensions AB=100 m and BC=50 m. Alan starts biking towards vertex B at a speed of 10 m/s, and Charlie starts biking towards vertex D at a speed of 15 m/s. How long before they meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-c6286e271450ede9536addde05514353e7e3e1317d8e58cb13759baef4f1eb45 | Alan and Charlie are located at vertices A and C of a rectangular track ABCD with dimensions AB=100 m and BC=50 m. Alan starts biking towards vertex B at a speed of 10 m/s, and Charlie starts biking towards vertex D at a speed of 15 m/s. Where on the track is the meeting point? | [
"A"
] | [
"Alan and Charlie are located at vertices A and C of a rectangular track ABCD with dimensions AB=100 m and BC=50 m. Alan starts biking towards vertex B at a speed of 10 m/s, and Charlie starts biking towards vertex D at a speed of 15 m/s. Where on the track is the meeting point?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-dd41bf05346206e5b058c0b16c98130b098e5636f8e29a24e98926a9f893f9e3 | Alan earns 4 dollars more in 4 hours than Joy earns in 3 hours. Joy earns 0.50 dollars more in 4 hours than Alan earns in 5 hours. How much does Alan earn per hour? | [
"17.50"
] | [
"Alan earns 4 dollars more in 4 hours than Joy earns in 3 hours. Joy earns 0.50 dollars more in 4 hours than Alan earns in 5 hours. How much does Alan earn per hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-040c83a4db651557ff55b01ee14d1cb9b1a5936229c5c07ec201c3c619b3f12f | Alan has 3 more dollars than Bill. Bill has 5 more dollars than Charles and Deb combined. Deb has 7 dollars more than Charlie, who has 5 dollars. How much money does Alan have? | [
"25"
] | [
"Alan has 3 more dollars than Bill. Bill has 5 more dollars than Charles and Deb combined. Deb has 7 dollars more than Charlie, who has 5 dollars. How much money does Alan have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d12b47311664cd7c23f6555f66765e74a60638f85ad056cdb6c7324897eb8d6 | Alan leaves home when the clock in his cardboard box says 7:35 AM and his watch says 7:41 AM. When he arrives at school, his watch says 7:47 AM and the 7:45 AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? | [
"4 minutes"
] | [
"Alan leaves home when the clock in his cardboard box says 7:35 AM and his watch says 7:41 AM. When he arrives at school, his watch says 7:47 AM and the 7:45 AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock?"
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb6da2678492485921a2ea8c94026661b258b111d9dd1336c640f1014f964ac2 | Alan rolls one six-sided die and Brandon rolls two six-sided dice. Alan wins if his number is greater than at least one of Brandon's numbers. Brandon wins if both of his numbers are greater than or equal to Alan's number. What is the probability that Alan wins? | [
"\\(\\frac{125}{216}\\)"
] | [
"Alan rolls one six-sided die and Brandon rolls two six-sided dice. Alan wins if his number is greater than at least one of Brandon's numbers. Brandon wins if both of his numbers are greater than or equal to Alan's number. What is the probability that Alan wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-2f13bf85723a04c6230d64b118bf4b9fbf8a07a5dea74f6bd029ec118b7029f4 | Alan starts with the numbers $1, 2, ..., 10$ written on a blackboard. He chooses three of the numbers, and erases the second largest one of them. He repeats this process until there are only two numbers left on the board. What are all possible values for the sum of these two numbers? | [
"11"
] | [
"Alan starts with the numbers $1, 2, ..., 10$ written on a blackboard. He chooses three of the numbers, and erases the second largest one of them. He repeats this process until there are only two numbers left on the board. What are all possible values for the sum of these two numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7f460c71aaded80197354a5adfa09bd00cb37db3939fca31482eb93deacef71 | Alan writes down a 23-element subset \( T \) of \( S = \{1, 2, \dots, 2000\} \). Bernard tries to guess \( T \) by asking questions where he specifies a subset \( R \) of \( S \), and Alan responds whether \( T \) is a subset of \( R \). What is the minimum number of questions Bernard must ask to determine \( T \) with... | [
"253"
] | [
"Alan writes down a 23-element subset \\( T \\) of \\( S = \\{1, 2, \\dots, 2000\\} \\). Bernard tries to guess \\( T \\) by asking questions where he specifies a subset \\( R \\) of \\( S \\), and Alan responds whether \\( T \\) is a subset of \\( R \\). What is the minimum number of questions Bernard must ask to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-e1867efa98ed5b8c2ab890e22e45166e20d6e8ad689750fcf34b182be2910c87 | Alan writes the numbers 1 to 64 in binary on a piece of paper without leading zeroes. How many more times will he have written the digit 1 than the digit 0? | [
"4"
] | [
"Alan writes the numbers 1 to 64 in binary on a piece of paper without leading zeroes. How many more times will he have written the digit 1 than the digit 0?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-64f372ceeb25c208011985ea91d24db8bbd51e2087a2c3f84f619c0e6e866dee | Alan, Ben, and Catherine start working on January 1st, 2009. Alan works for 3 days and takes 1 day off, Ben works for 5 days and takes 2 days off, and Catherine works for 7 days and takes 3 days off. On how many days in 2009 will none of them be working? | [
"11"
] | [
"Alan, Ben, and Catherine start working on January 1st, 2009. Alan works for 3 days and takes 1 day off, Ben works for 5 days and takes 2 days off, and Catherine works for 7 days and takes 3 days off. On how many days in 2009 will none of them be working?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2485c8ade253b7ea8a1b58bf9e12fffbce26bd0444d4363fbb5858d60d74b6d3 | Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with 2 tokens. In each round, the first person to solve the MafsCounts question wins the round and steals one token from each of the other players. If a player runs out of tokens, they are removed from the game. The last player remai... | [
"\\(\\frac{1}{2}\\)"
] | [
"Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with 2 tokens. In each round, the first person to solve the MafsCounts question wins the round and steals one token from each of the other players. If a player runs out of tokens, they are removed from the game. The last player ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-ae78b9942bf394f6b3d45cc92d76c4010c8ae1e51c715e594548673980c4d639 | Alan, Steve, and Tunan play the following game: (a) First, Alan flips a coin and wins if he flips heads. (b) If Alan doesn’t win, Steve flips a coin and wins if he flips heads. (c) If neither Alan nor Steve win, Tunan flips a coin and wins if he flips heads. (d) If none of the three win, the process starts over. Let th... | [
"4"
] | [
"Alan, Steve, and Tunan play the following game:\n(a) First, Alan flips a coin and wins if he flips heads.\n(b) If Alan doesn’t win, Steve flips a coin and wins if he flips heads.\n(c) If neither Alan nor Steve win, Tunan flips a coin and wins if he flips heads.\n(d) If none of the three win, the process starts ove... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8bf105f5dd2e12179df996d0e2fec6583ede52ff060f2e4192774e6e45b1cda9 | Alba, Bety, Cory, Dana, and Elsa have a store that opens from Monday to Friday. Two people are needed every day to attend it. They decide to make a work plan for a week specifying who will work each day, fulfilling the following conditions: - Each person works exactly two days a week. - The 5 couples assigned for the w... | [
"1440"
] | [
"Alba, Bety, Cory, Dana, and Elsa have a store that opens from Monday to Friday. Two people are needed every day to attend it. They decide to make a work plan for a week specifying who will work each day, fulfilling the following conditions:\n- Each person works exactly two days a week.\n- The 5 couples assigned fo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-f3f8a01ed281c0c703043cc738f561214ccfcb2ae0ad05a4a74c77f355c3a3c7 | Albert Einstein, Paul Erdős, and Nikola Tesla each have a positive integer in their heads. They can see the numbers in the heads of the others but not their own. They know one of these numbers and the sum of the other two. They are repeatedly asked whether they know the number in their head. If the numbers in their hea... | [
"1"
] | [
"Albert Einstein, Paul Erdős, and Nikola Tesla each have a positive integer in their heads. They can see the numbers in the heads of the others but not their own. They know one of these numbers and the sum of the other two. They are repeatedly asked whether they know the number in their head. If the numbers in thei... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-b27b4eb1a356e4c93372de6e70537d649bd4a5e3d8a5616aa6d82ef90cb0fe66 | Albert and Barbara play a game with 1999 sticks. Each player, starting with Barbara, removes at least one stick but no more than half of the remaining sticks. The player who leaves exactly one stick loses. Determine which player has a winning strategy and describe the strategy. | [
"Barbara has a winning strategy by removing 464 sticks initially."
] | [
"Albert and Barbara play a game with 1999 sticks. Each player, starting with Barbara, removes at least one stick but no more than half of the remaining sticks. The player who leaves exactly one stick loses. Determine which player has a winning strategy and describe the strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 13
} | 60 | 0 | 0 |
omr-ws-4a9298f7ea1db19710774c7efaaabf2a9f0b0327cd4ba20cdb6d1a3fe0b04ff8 | Albert and Beatrice play a game with 2021 stones on a table. Starting with Albert, they take turns removing stones, where on the $n$-th turn, the active player can remove from $1$ to $n$ stones. The player who removes the last stone loses. Determine which player has a winning strategy regardless of the other player's m... | [
"Beatrice has a winning strategy."
] | [
"Albert and Beatrice play a game with 2021 stones on a table. Starting with Albert, they take turns removing stones, where on the $n$-th turn, the active player can remove from $1$ to $n$ stones. The player who removes the last stone loses. Determine which player has a winning strategy regardless of the other playe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 5
} | 22 | 0 | 0 |
omr-ws-8c55b9150425fd702e2f89cf615b258018cf4b7a8bd4528f397bb031033ceeb5 | Albert and Bernie are dividing a cake according to the following procedure. First, Albert proposes a division. If Bernie agrees, the division is implemented. Otherwise, they wait for an hour and Bernie proposes a division. If Albert agrees, the division is implemented; otherwise, they wait for another hour and Albert p... | [
"They will wait for 0 hours."
] | [
"Albert and Bernie are dividing a cake according to the following procedure. First, Albert proposes a division. If Bernie agrees, the division is implemented. Otherwise, they wait for an hour and Bernie proposes a division. If Albert agrees, the division is implemented; otherwise, they wait for another hour and Alb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-3654f5cb906c4643def36c07f3a79626b312cef3b0fcef34a37ac7ee49f76147 | Albert and Betty are playing a game with two bowls. The red bowl contains 100 blue balls, and the blue bowl contains 100 red balls. On each turn, a player must perform one of the following moves: a) Take two red balls from the blue bowl and put them in the red bowl. b) Take two blue balls from the red bowl and put them... | [
"Betty"
] | [
"Albert and Betty are playing a game with two bowls. The red bowl contains 100 blue balls, and the blue bowl contains 100 red balls. On each turn, a player must perform one of the following moves:\na) Take two red balls from the blue bowl and put them in the red bowl.\nb) Take two blue balls from the red bowl and p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 9
} | 15 | 0 | 0 |
omr-ws-364d5b00b39e46be8812a907a7a40a52264abf5a6c7669dc2729bbec522c1195 | Albert and Bob start running at the same time and same place in a circular running track, with Albert running clockwise and Bob running counterclockwise. Albert runs at 3 meters per second, and Bob takes 3 minutes and 20 seconds to finish one lap around the track. The first time Albert and Bob meet is after 2 minutes a... | [
"3000"
] | [
"Albert and Bob start running at the same time and same place in a circular running track, with Albert running clockwise and Bob running counterclockwise. Albert runs at 3 meters per second, and Bob takes 3 minutes and 20 seconds to finish one lap around the track. The first time Albert and Bob meet is after 2 minu... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-df2b6400bf361efabde5d58da26b0029dfe61d744215af3afdfbea34a8fece89 | Albert had a list of ports numbered from 1 to 1000. He visited the port with the lowest number $n$ left on the list that had not yet been visited and then removed the number $2n + 1$ (if it was less than 1000). This process continued until he reached the end of the list. How many numbers were left on the list when he f... | [
"500"
] | [
"Albert had a list of ports numbered from 1 to 1000. He visited the port with the lowest number $n$ left on the list that had not yet been visited and then removed the number $2n + 1$ (if it was less than 1000). This process continued until he reached the end of the list. How many numbers were left on the list when... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab042c8373cf0e8b52456615a3e5eb7211b6a0eae85e10ca4fe0a460e8907ac0 | Albert has 12 cylindrical weights, one for each positive divisor of 96. He places these weights on a balanced scale such that both sides have at least one weight. Find the number of ways to place the weights such that the right side weighs more than the left. | [
"2036"
] | [
"Albert has 12 cylindrical weights, one for each positive divisor of 96. He places these weights on a balanced scale such that both sides have at least one weight. Find the number of ways to place the weights such that the right side weighs more than the left."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 21
} | 31 | 0 | 0 |
omr-ws-4014e81e4c419cec702a4a1a9c868eb0c2f51b64054fbf3f1169143d6f4a84b8 | Albert has a very large bag of candies. When he splits the candies evenly amongst his 20 friends and himself, there are five left over. When Ante arrives and they redistribute the candies evenly amongst 21 people, there are three left over. If the bag contains over 500 candies, what is the fewest number of candies the ... | [
"509"
] | [
"Albert has a very large bag of candies. When he splits the candies evenly amongst his 20 friends and himself, there are five left over. When Ante arrives and they redistribute the candies evenly amongst 21 people, there are three left over. If the bag contains over 500 candies, what is the fewest number of candies... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-283c95c1290c8fa739edf30adaced8837a4c422818c38885340346ca5a6363f8 | Albert has glued together $2015^3$ unitary cubes to form a large cube with side length 2015. Boris needs to detach all the unit cubes by giving head shots, each of which removes the glue along the outer surface of a cube of integer side length. What is the minimum number of head shots Boris must give to completely deta... | [
"8056"
] | [
"Albert has glued together $2015^3$ unitary cubes to form a large cube with side length 2015. Boris needs to detach all the unit cubes by giving head shots, each of which removes the glue along the outer surface of a cube of integer side length. What is the minimum number of head shots Boris must give to completely... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-6cdf829e3aa5230942e28e1d648a35ea78c8be090b373409ac3db7203f8511f6 | Albert is borrowing 1 million dollars from a bank. He promised to pay the bank 10,000 dollars every month at the beginning of each month, starting next month, at a compounded monthly interest rate of 0.6%. How many years would it take for him to pay back his entire principal plus the interest, given that $\log(2) = 0.3... | [
"13"
] | [
"Albert is borrowing 1 million dollars from a bank. He promised to pay the bank 10,000 dollars every month at the beginning of each month, starting next month, at a compounded monthly interest rate of 0.6%. How many years would it take for him to pay back his entire principal plus the interest, given that $\\log(2)... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-da67c95c8a5679d10c7c5c23f4fbfbeed44f65502fb0166f00f6806b5a07f0a1 | Albert is shooting half-court shots with two different balls, one Harden ball and one Westbrook ball. Every shot attempt, he picks a ball at random (with equal probability) and shoots it. He has a $\frac{3}{5}$ chance of making it if he shoots the Harden ball, and a $\frac{2}{3}$ chance of making it if he shoots the We... | [
"\\(\\frac{112}{19}\\)"
] | [
"Albert is shooting half-court shots with two different balls, one Harden ball and one Westbrook ball. Every shot attempt, he picks a ball at random (with equal probability) and shoots it. He has a $\\frac{3}{5}$ chance of making it if he shoots the Harden ball, and a $\\frac{2}{3}$ chance of making it if he shoots... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c1c508ba049c8fdba727244b146b720bdc230fa8048d9909e6e592aab91eaff | Albert knows there are 6 types of leptons and 6 types of quarks but does not know their names. On a physics test, he is given a list of the names of all 12 particles and must label 6 as leptons and 6 as quarks. What is the probability that he gets at least 10 of the 12 particles correctly? | [
"\\dfrac{37}{924}"
] | [
"Albert knows there are 6 types of leptons and 6 types of quarks but does not know their names. On a physics test, he is given a list of the names of all 12 particles and must label 6 as leptons and 6 as quarks. What is the probability that he gets at least 10 of the 12 particles correctly?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00edf07986a932eee2802a22ee3d3223973d404f054087d8cfd588f01127d3fc | Albert plays four equal dice. He discards the die with the lowest value and sums the other three. What is the probability that the sum of the remaining three dice is greater than or equal to 16? | [
"\\(\\frac{169}{1296}\\)"
] | [
"Albert plays four equal dice. He discards the die with the lowest value and sums the other three. What is the probability that the sum of the remaining three dice is greater than or equal to 16?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 25
} | 33 | 0 | 0 |
omr-ws-5cd89eb130e5ceb04360588fab9d72edd3d268693fb304c7b2d6b0aa5b2a8ca2 | Albert rolls a fair six-sided die thirteen times. For each roll that is strictly greater than the previous roll (excluding the first roll), he gains a point. Find the expected number of points that Albert receives. | [
"5"
] | [
"Albert rolls a fair six-sided die thirteen times. For each roll that is strictly greater than the previous roll (excluding the first roll), he gains a point. Find the expected number of points that Albert receives."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-7f7886780a106d209439af8c609e6b5aedf1ac1675b65bce2804cd0dceeb1828 | Albert secretly and completely orders a set L of 20 objects. Bart can investigate by picking a triple of objects T ⊆ L and asking Albert to reveal either the maximum or the minimum element of that triplet. What is the maximum cardinality of a subset L' ⊆ L for which Bart can completely know the order? | [
"10"
] | [
"Albert secretly and completely orders a set L of 20 objects. Bart can investigate by picking a triple of objects T ⊆ L and asking Albert to reveal either the maximum or the minimum element of that triplet. What is the maximum cardinality of a subset L' ⊆ L for which Bart can completely know the order?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-b65573b3ea82dd8d4358d902cd0b08e495f3f515c5f24414be33bdf8ee1da600 | Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits he wrote (the... | [
"116"
] | [
"Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \\ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits he wrot... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e98c14579ecada5f29e647795269aee0206ad20defc0aaac19d7f7950692b8c5 | Albert, Ben, and Charles set off from Jerney towards Gark at 5 a.m., while Don and Eric set off from Gark towards Jerney at the same time. Fred set off from Gark four hours later and passed Albert at 1:10 p.m., with both reaching the opposite shore at the same time. Ben and Don waved to each other as their paths crosse... | [
"90 minutes"
] | [
"Albert, Ben, and Charles set off from Jerney towards Gark at 5 a.m., while Don and Eric set off from Gark towards Jerney at the same time. Fred set off from Gark four hours later and passed Albert at 1:10 p.m., with both reaching the opposite shore at the same time. Ben and Don waved to each other as their paths c... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 14
} | 28 | 0 | 0 |
omr-ws-3ba82cea81d46f91f7a2b5baa2cb12ea16aec1c483bc08deb8d33d0993657f7d | Albert, Bernard, and Cheryl each start with 5 red balls, 7 green balls, and 13 blue balls. In each turn, they must throw two balls of different colors and receive two balls of the remaining color from an infinite supply. After several moves, Albert claims he only has red balls, Bernard claims he only has blue balls, an... | [
"Bernard and Cheryl"
] | [
"Albert, Bernard, and Cheryl each start with 5 red balls, 7 green balls, and 13 blue balls. In each turn, they must throw two balls of different colors and receive two balls of the remaining color from an infinite supply. After several moves, Albert claims he only has red balls, Bernard claims he only has blue ball... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-a40bafbd72762130aa88647050e0727cfe545e34db106d17ca0847c5416bad7f | Albert, Bernard, and Cheryl each start with 5 red marbles, 7 green marbles, and 13 blue marbles. In each turn, each player can choose 2 marbles of different colors and replace them with 2 marbles of the third color. After a finite number of steps, Albert claims he has only red marbles, Bernard claims he has only blue m... | [
"Bernard and Cheryl are lying."
] | [
"Albert, Bernard, and Cheryl each start with 5 red marbles, 7 green marbles, and 13 blue marbles. In each turn, each player can choose 2 marbles of different colors and replace them with 2 marbles of the third color. After a finite number of steps, Albert claims he has only red marbles, Bernard claims he has only b... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 41,
"tir": 22
} | 63 | 0 | 0 |
omr-ws-fbac64b32f61069d601fced83335349d00443e37b0112b716a6a484cdc7a04a8 | Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length \(x\), Brian a branch of length \(y\), and Christine a branch of length \(z\), and the lengths follow the condition that \(x + y + ... | [
"\\dfrac{28}{27}"
] | [
"Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length \\(x\\), Brian a branch of length \\(y\\), and Christine a branch of length \\(z\\), and the lengths follow the condition that... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-08c44c14ab4df2c8b74597ff43dd7f5fb5533aad8e86b73384d21b0a86debb59 | Alberto and Barbara each choose a positive integer. Claudio reveals that the sum or the product of their numbers is 2014. After hearing this, Alberto says he cannot determine Barbara's number, and Barbara, hearing Alberto's statement, also says she cannot determine Alberto's number. What are the numbers chosen by Alber... | [
"Alberto's number is \\(2\\) or \\(1007\\) and Barbara's number is \\(1007\\). \n\nSince the problem asks for the numbers chosen by Alberto and Barbara, and the discussion indicates that Barbara's number is definitively \\(1007\\) while Alberto's number could be either \\(2\\) or \\(1007\\), the answer is:"
] | [
"Alberto and Barbara each choose a positive integer. Claudio reveals that the sum or the product of their numbers is 2014. After hearing this, Alberto says he cannot determine Barbara's number, and Barbara, hearing Alberto's statement, also says she cannot determine Alberto's number. What are the numbers chosen by ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-2c2a162302b2827f7921398f4d64e39f3d438c95e1e14cfc28a234213b9d7746 | Alberto and Barbara play a game with piles of coins. Each player, starting with Alberto, can either take a coin from an arbitrary pile or divide a pile into two non-empty piles. The player who removes the last coin wins. Determine which player has a winning strategy with respect to the initial state. | [
"Alberto wins if at least one of $n_1$ (number of piles with 1 coin) or $n_2+n_4+n_6+...$ (sum of the number of piles with even numbers of coins) is odd; otherwise, Barbara wins."
] | [
"Alberto and Barbara play a game with piles of coins. Each player, starting with Alberto, can either take a coin from an arbitrary pile or divide a pile into two non-empty piles. The player who removes the last coin wins. Determine which player has a winning strategy with respect to the initial state."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-8bad39c41cd750a17282f3d3de65df9995ad03bf87571e8412645dc7cfdc27be | Alberto chooses 2022 integers \(a_1, a_2, \dots, a_{2022}\) and places them on a \(2022 \times 2022\) table such that the cell \((i, j)\) contains the number \(a_k\) where \(k = \max\{i, j\}\). Barbara wants to determine the value of \(a_k\) for a given \(k\) (with \(1 \leq k \leq 2022\)) by asking Alberto the sum of t... | [
"2"
] | [
"Alberto chooses 2022 integers \\(a_1, a_2, \\dots, a_{2022}\\) and places them on a \\(2022 \\times 2022\\) table such that the cell \\((i, j)\\) contains the number \\(a_k\\) where \\(k = \\max\\{i, j\\}\\). Barbara wants to determine the value of \\(a_k\\) for a given \\(k\\) (with \\(1 \\leq k \\leq 2022\\)) by... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.