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-15274c81b84ab99810d3c83e905965b62dfd4b1ad9b48054a8088fc24730b556 | Anna and Bea each choose an integer from $1$ to $120$ at random. What is the probability that neither integer is the square of the other? | [
"\\(\\frac{719}{720}\\)"
] | [
"Anna and Bea each choose an integer from $1$ to $120$ at random. What is the probability that neither integer is the square of the other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c0c2eb5500e845d4cbf95620340af0e733b12d24af4632c59ef70510c2529166 | Anna and Becky play a game where they each roll a die. If the sum of their dice equals a composite number, Anna wins that round, and if the sum equals a prime number, Becky wins that round. They play a best out of 5 games, so whoever wins 3 or more games wins. The probability that Anna wins can be expressed as $\frac{m... | [
"569"
] | [
"Anna and Becky play a game where they each roll a die. If the sum of their dice equals a composite number, Anna wins that round, and if the sum equals a prime number, Becky wins that round. They play a best out of 5 games, so whoever wins 3 or more games wins. The probability that Anna wins can be expressed as $\\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-d00349495fb21c4ecf706beb1005957037a98c971260e26b3ae38ed2eeabfa73 | Anna and Bella are playing a game where Anna starts. In each round, a player chooses \( n-1 \) of the \( n \) boxes and adds one ball to each chosen box. The player who, after her turn, has a box with \( 2n+1 \) balls wins. Determine which player has a winning strategy. | [
"Bella has the winning strategy."
] | [
"Anna and Bella are playing a game where Anna starts. In each round, a player chooses \\( n-1 \\) of the \\( n \\) boxes and adds one ball to each chosen box. The player who, after her turn, has a box with \\( 2n+1 \\) balls wins. Determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-06e865236fc9e2a0d26f79470d4fec9681f3ee7a5bdb23084e374cbee6ef799c | Anna and Berta play a game where they take turns removing marbles from a table. Anna starts first. At the beginning of each turn, there are \( n \geq 1 \) marbles on the table. The player whose turn it is can remove \( k \) marbles, where \( k \geq 1 \) is either an even number with \( k \leq \frac{n}{2} \) or an odd n... | [
"131070"
] | [
"Anna and Berta play a game where they take turns removing marbles from a table. Anna starts first. At the beginning of each turn, there are \\( n \\geq 1 \\) marbles on the table. The player whose turn it is can remove \\( k \\) marbles, where \\( k \\geq 1 \\) is either an even number with \\( k \\leq \\frac{n}{2... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 17
} | 65 | 0 | 0 |
omr-ws-f679b5dc90369623e512eef073f0ad77123ac9d32ac655e6fe07bcb25c029f6a | Anna and Bob alternately color the integers of the set \( S = \{1, 2, \ldots, 2022\} \) red or blue, with Anna starting first. The game ends when all numbers in \( S \) are colored. Let \( N \) be the number of pairs \((a, b)\) where \( a \) and \( b \) are elements of \( S \), have the same color, and \( b - a = 3 \).... | [
"\\( 3 \\times 337 - 3 = 1011 - 3 = 1008 \\)"
] | [
"Anna and Bob alternately color the integers of the set \\( S = \\{1, 2, \\ldots, 2022\\} \\) red or blue, with Anna starting first. The game ends when all numbers in \\( S \\) are colored. Let \\( N \\) be the number of pairs \\((a, b)\\) where \\( a \\) and \\( b \\) are elements of \\( S \\), have the same color... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-c8113f2b1c8e88646c874b87fedf8e6ed4634d78e45974b5ddc745fd9b1ba752 | Anna and Bob are playing a game where the number $2$ is initially written on the blackboard. Anna plays first, and they alternately double the number or square it. The player who first writes a number greater than $2023^{10}$ wins. Determine which player has a winning strategy. | [
"Bob has the winning strategy."
] | [
"Anna and Bob are playing a game where the number $2$ is initially written on the blackboard. Anna plays first, and they alternately double the number or square it. The player who first writes a number greater than $2023^{10}$ wins. Determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 19
} | 32 | 0 | 0 |
omr-ws-ee0aa5a20045b56e82df635413379164f6bbec6e66b10940da6f2e0fb24b4634 | Anna and Bob play a game involving 2008 chips. They take turns removing a number of chips equal to a power of 2 (not including 1). Assuming Anna plays perfectly, what is the largest number of moves necessary for her to win? | [
"23"
] | [
"Anna and Bob play a game involving 2008 chips. They take turns removing a number of chips equal to a power of 2 (not including 1). Assuming Anna plays perfectly, what is the largest number of moves necessary for her to win?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92b91dc4d6c28d856b32585917192b492fc2eb85c35486ea57aedfb1450c5f20 | Anna and Bob play a game on an infinite board. Anna starts by placing 7 crosses, and Bob responds by placing 3 zeros, with each move occupying different free cells. A ladder is any set of 6 consecutive crosses diagonally, and a bridge is any set of 6 consecutive crosses horizontally or vertically. Anna wins if after he... | [
"3"
] | [
"Anna and Bob play a game on an infinite board. Anna starts by placing 7 crosses, and Bob responds by placing 3 zeros, with each move occupying different free cells. A ladder is any set of 6 consecutive crosses diagonally, and a bridge is any set of 6 consecutive crosses horizontally or vertically. Anna wins if aft... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-5f854eba6c663b4eb90d3ddeb05b1deec2bebd427b599a15dd8cf6edd3b09f88 | Anna and Bob play a game on an infinite board. Anna starts by placing 7 crosses, and Bob responds by placing 3 zeros, with each move occupying different free cells. A ladder is any set of 6 consecutive crosses diagonally, and a bridge is any set of 6 consecutive crosses horizontally or vertically. Anna wins if after he... | [
"4"
] | [
"Anna and Bob play a game on an infinite board. Anna starts by placing 7 crosses, and Bob responds by placing 3 zeros, with each move occupying different free cells. A ladder is any set of 6 consecutive crosses diagonally, and a bridge is any set of 6 consecutive crosses horizontally or vertically. Anna wins if aft... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-fea2ac8279bd7f10efadad98ee4c98b7a73866d4a108d91f7acefcc6407ecbea | Anna and Boris move simultaneously towards each other from points A and B respectively, with constant but not necessarily equal speeds. If Anna had started 30 minutes earlier, they would have met 2 kilometers nearer to B. If Boris had started 30 minutes earlier instead, they would have met some distance nearer to A. Ca... | [
"2 km"
] | [
"Anna and Boris move simultaneously towards each other from points A and B respectively, with constant but not necessarily equal speeds. If Anna had started 30 minutes earlier, they would have met 2 kilometers nearer to B. If Boris had started 30 minutes earlier instead, they would have met some distance nearer to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-893fce3f89c7b3d22831efc8d3bd170871c2f3b3e93dbe39f38f43766a246553 | Anna and Lisa are 30 kilometers from their final destination when Anna's bike breaks down. They decide to alternate riding Lisa's bike and walking to reach the destination at the same time. Anna walks at 4 km/h and cycles at 15 km/h, while Lisa walks at 5 km/h and cycles at 20 km/h. How long does it take them to cover ... | [
"4.75"
] | [
"Anna and Lisa are 30 kilometers from their final destination when Anna's bike breaks down. They decide to alternate riding Lisa's bike and walking to reach the destination at the same time. Anna walks at 4 km/h and cycles at 15 km/h, while Lisa walks at 5 km/h and cycles at 20 km/h. How long does it take them to c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d29c73789147ff26927a083d61dafebcffaac82ce91b3cd47726743a09089d14 | Anna considered positive integer numbers from $1$ to $10^{2016}$, multiplied the digits in each number, and then added up those multiplied numbers. What is the sum of these products? | [
"\\( 45 \\times 46^{2015} \\)"
] | [
"Anna considered positive integer numbers from $1$ to $10^{2016}$, multiplied the digits in each number, and then added up those multiplied numbers. What is the sum of these products?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e311a512ff75e020dbe851def622d9f1734ff99fb86e902dc07da8b94421080 | Anna has 365 cards with different numbers on them, all face-down. For one dollar, Anna can choose three cards and ask Jenny to arrange them in ascending order. How many dollars at least does Anna need to spend to ensure that all 365 cards are arranged in ascending order? | [
"364"
] | [
"Anna has 365 cards with different numbers on them, all face-down. For one dollar, Anna can choose three cards and ask Jenny to arrange them in ascending order. How many dollars at least does Anna need to spend to ensure that all 365 cards are arranged in ascending order?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-913226cc93684499dc06f0a4cf69ee80424e58ca601493bc17efa7655870bde7 | Anna has a \( 5 \times 5 \) grid of pennies. How many ways can she arrange them so that exactly two pennies show heads in each row and in each column? | [
"2040"
] | [
"Anna has a \\( 5 \\times 5 \\) grid of pennies. How many ways can she arrange them so that exactly two pennies show heads in each row and in each column?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f5b693d635d8982be944ac46eb616eb955b99b9bace842e22cec2eab204b4631 | Anna has x marbles. She gave 30% of them to her teacher, 50% of the remaining marbles to her sister, and 20% of the remaining marbles to her brother. She has 12 remaining marbles. How many did she have in the beginning? | [
"\\(\\frac{300}{7}\\)"
] | [
"Anna has x marbles. She gave 30% of them to her teacher, 50% of the remaining marbles to her sister, and 20% of the remaining marbles to her brother. She has 12 remaining marbles. How many did she have in the beginning?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-8e60556bf3ef185da8f6be557771161b6795a194656628ee6f79c1ec5f1c6231 | Anna is standing at \( A_0=(-1,0) \) on the coordinate plane. She is walking in a straight line toward \( A_1=(0,d) \) for some \( 0<d<1 \). After she reaches \( A_1 \), she turns 90 degrees to the right, and continues moving in that direction until she hits a coordinate axis at \( A_2 \), turns 90 degrees to the right... | [
"5"
] | [
"Anna is standing at \\( A_0=(-1,0) \\) on the coordinate plane. She is walking in a straight line toward \\( A_1=(0,d) \\) for some \\( 0<d<1 \\). After she reaches \\( A_1 \\), she turns 90 degrees to the right, and continues moving in that direction until she hits a coordinate axis at \\( A_2 \\), turns 90 degre... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dfdb8a06f608bcc3dd595dcb388dead87e5128bfd73436a958f94da400440d51 | Anna must write 7 positive integers, not necessarily distinct, around a circle such that the sum of the seven numbers equals 36. Additionally, if two numbers are neighbors, the difference between the largest and the smallest is equal to 2 or 3. Find the maximum value of the largest of the numbers that Anna can write. | [
"9"
] | [
"Anna must write 7 positive integers, not necessarily distinct, around a circle such that the sum of the seven numbers equals 36. Additionally, if two numbers are neighbors, the difference between the largest and the smallest is equal to 2 or 3. Find the maximum value of the largest of the numbers that Anna can wri... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-21b29304ce11cd7c8768fab06e312d0dd0098021f7cd63ad8cb89a09d493e115 | Anna randomly chooses a positive divisor of the number $15!$ while Bob randomly chooses another positive divisor of the same number, not necessarily distinct from the one chosen by Anna. What is the probability that the sum of Anna's and Bob's divisors will be odd? | [
"\\(\\frac{11}{72}\\)"
] | [
"Anna randomly chooses a positive divisor of the number $15!$ while Bob randomly chooses another positive divisor of the same number, not necessarily distinct from the one chosen by Anna. What is the probability that the sum of Anna's and Bob's divisors will be odd?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ff532f4cf0277ff4fdb9255c2ae862ee435d96141e950bb6cba88e30ad4a249c | Anna thinks of a natural number from the set \( S = \{1, 2, 3, \ldots, n\} \). Sasha can ask questions by choosing a nonempty subset \( X \) of \( S \) and asking if the chosen number belongs to \( X \). Anna must answer yes or no truthfully. Determine the minimum number of questions Sasha needs to ask to identify the ... | [
"\\(\\lceil \\log_2{n} \\rceil\\)"
] | [
"Anna thinks of a natural number from the set \\( S = \\{1, 2, 3, \\ldots, n\\} \\). Sasha can ask questions by choosing a nonempty subset \\( X \\) of \\( S \\) and asking if the chosen number belongs to \\( X \\). Anna must answer yes or no truthfully. Determine the minimum number of questions Sasha needs to ask ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-70c627ac46c06e5ca85375a3be256a0e48e5c58cf772c4e72b5a79231d4a533f | Anna thinks of a number from the set \( X = \{1, 2, \ldots, 144\} \). Jenny can choose a subset \( A \) of \( X \) and ask Anna if the number is in \( A \). If Anna says "YES," Jenny pays 2 dollars, and if Anna says "NO," Jenny pays 1 dollar. How many dollars at least does Jenny need to pay to determine the number Anna... | [
"16"
] | [
"Anna thinks of a number from the set \\( X = \\{1, 2, \\ldots, 144\\} \\). Jenny can choose a subset \\( A \\) of \\( X \\) and ask Anna if the number is in \\( A \\). If Anna says \"YES,\" Jenny pays 2 dollars, and if Anna says \"NO,\" Jenny pays 1 dollar. How many dollars at least does Jenny need to pay to deter... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa129fe2b4d8e8d3cc5e7bdd3cea368e8327acd40d2871f9967388986d7b94e9 | Anna took a number \(N\), which is written in base 10 and has fewer than 9 digits, and duplicated it by writing another \(N\) to its left, creating a new number with twice as many digits. Bob computed the sum of all integers from 1 to \(N\). It turned out that Anna's new number is 7 times as large as the sum computed b... | [
"285"
] | [
"Anna took a number \\(N\\), which is written in base 10 and has fewer than 9 digits, and duplicated it by writing another \\(N\\) to its left, creating a new number with twice as many digits. Bob computed the sum of all integers from 1 to \\(N\\). It turned out that Anna's new number is 7 times as large as the sum... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-172a0c2e98908ac338256c59799bbe0f0de65b2f3469fd903033f7f78cacbc11 | Anna wants to buy boxes of candy from Sugar Station, where each box contains a different kind of candy. There are 44 different kinds of candies, and the total cost to buy one of each kind is \$1.51. If Anna has \$0.75, what is the maximum number of boxes she can buy, assuming each box costs a positive integer number of... | [
"43"
] | [
"Anna wants to buy boxes of candy from Sugar Station, where each box contains a different kind of candy. There are 44 different kinds of candies, and the total cost to buy one of each kind is \\$1.51. If Anna has \\$0.75, what is the maximum number of boxes she can buy, assuming each box costs a positive integer nu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-4fcf5c9627043859b38b3413fd429af71322c1dbea371941f79161725ba93097 | Anna writes a sequence of integers starting with the number 12. Each subsequent integer she writes is chosen randomly with equal chance from among the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first time... | [
"\\(\\frac{121}{30}\\)"
] | [
"Anna writes a sequence of integers starting with the number 12. Each subsequent integer she writes is chosen randomly with equal chance from among the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 8,
"tir": 12
} | 22 | 0 | 0 |
omr-ws-32c47b265aaa39658fa98d903359b48beed872af9cdc1d36e587b05f4c6cb619 | Anna writes all non-empty subsets of the set \(\{1, 2, \ldots, n\}\) and calculates the product of the elements in each subset. She then writes the reciprocals of these products and sums them. What is the sum obtained by Anna? | [
"\\( n \\)"
] | [
"Anna writes all non-empty subsets of the set \\(\\{1, 2, \\ldots, n\\}\\) and calculates the product of the elements in each subset. She then writes the reciprocals of these products and sums them. What is the sum obtained by Anna?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65dc5ed4ff4876854bfa1084351494593816731a67842aefe834548e8421febe | Anna wrote down fifty different natural numbers. Twenty-five of the numbers do not exceed $50$, and the rest are greater than $50$ but do not exceed $100$. At the same time, no two of Anna's numbers differ by exactly $50$. Find the sum of Anna's numbers. | [
"2525"
] | [
"Anna wrote down fifty different natural numbers. Twenty-five of the numbers do not exceed $50$, and the rest are greater than $50$ but do not exceed $100$. At the same time, no two of Anna's numbers differ by exactly $50$. Find the sum of Anna's numbers."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-da466ee24e9e7b51fe939436a1138d3733c0332a2cce4715693d905bbaaae277 | Anna, Berta, Charlie, David, and Elisa baked cookies over two days. The total number of cookies baked by each person over the two days were 24, 25, 26, 27, and 28 respectively. One person baked twice as many cookies on Saturday as on Friday, one person baked three times as many, one person baked four times as many, one... | [
"David"
] | [
"Anna, Berta, Charlie, David, and Elisa baked cookies over two days. The total number of cookies baked by each person over the two days were 24, 25, 26, 27, and 28 respectively. One person baked twice as many cookies on Saturday as on Friday, one person baked three times as many, one person baked four times as many... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1f5d4f96a981c2a1c4dca88f2aa7d694ad388e070f8ef17ba7b313752a32038a | Anna, Beth, and Carl have favorite 3-digit numbers. Moving the last digit of Anna's number to the beginning results in Beth's number, and moving the last digit of Beth's number to the beginning results in Carl's number. Anna's and Beth's numbers are perfect squares, but Carl's number is not a perfect square, and the su... | [
"562"
] | [
"Anna, Beth, and Carl have favorite 3-digit numbers. Moving the last digit of Anna's number to the beginning results in Beth's number, and moving the last digit of Beth's number to the beginning results in Carl's number. Anna's and Beth's numbers are perfect squares, but Carl's number is not a perfect square, and t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8530633dcab08635e9beab4bd68c0824b36c8907fe24a4bd9903bf191367d5ad | Anna, Bobby, Carol, and David are siblings. Their ages are all different positive integers and are greater than 5, while the sum of their ages is 55. Anna is the youngest, Bobby is the second youngest, Carol is the second oldest, and David is the oldest. What is the sum of all of Bobby's possible ages? | [
"99"
] | [
"Anna, Bobby, Carol, and David are siblings. Their ages are all different positive integers and are greater than 5, while the sum of their ages is 55. Anna is the youngest, Bobby is the second youngest, Carol is the second oldest, and David is the oldest. What is the sum of all of Bobby's possible ages?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c93195e239617b199855e5450af0ae8885f2d52408c8b48621b5195ac013412e | Anna, Stephanie, and James all start running around a track at 8:00. Anna completes a lap every 4 minutes, Stephanie finishes a lap every 7 minutes, and James finishes a lap every 6 minutes. What is the earliest time when all three meet back at the beginning? | [
"9:24"
] | [
"Anna, Stephanie, and James all start running around a track at 8:00. Anna completes a lap every 4 minutes, Stephanie finishes a lap every 7 minutes, and James finishes a lap every 6 minutes. What is the earliest time when all three meet back at the beginning?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8a508d45771c11a86bd6bc563374102949845f4379b2ae70246cb85d70747f6 | Annah has a serious and genuine honesty policy for the AOIME. If Annah sees a geometry problem on problem number $n$, the probability she uses geogebra for that problem is $\frac{n}{15}$. The AOIME has $14$ non-geo problems and #$15$ is a geo problem. Calculate the expected score for Annah on the AOIME, assuming that A... | [
"1"
] | [
"Annah has a serious and genuine honesty policy for the AOIME. If Annah sees a geometry problem on problem number $n$, the probability she uses geogebra for that problem is $\\frac{n}{15}$. The AOIME has $14$ non-geo problems and #$15$ is a geo problem. Calculate the expected score for Annah on the AOIME, assuming ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cc9fd843f6f315de427abda14bdc4f32bf1547507aa6d7cde7b147d32a113f50 | Anne and Bill decide to play a game together. At the beginning, they choose a positive integer $n$. Starting from a positive integer $\mathcal{N}_0$, Anne subtracts an integer $k$-th power (possibly $0$) of $n$ less than or equal to $\mathcal{N}_0$. The resulting number $\mathcal{N}_1 = \mathcal{N}_0 - n^k$ is then pas... | [
"500"
] | [
"Anne and Bill decide to play a game together. At the beginning, they choose a positive integer $n$. Starting from a positive integer $\\mathcal{N}_0$, Anne subtracts an integer $k$-th power (possibly $0$) of $n$ less than or equal to $\\mathcal{N}_0$. The resulting number $\\mathcal{N}_1 = \\mathcal{N}_0 - n^k$ is... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-b3646e9aef914bca02846f563743983e18fcf44a9d43961b04997ba47a2006ad | Anne went into the fortune-teller's tent and asked whether she would be chosen as the Harvest Queen. The fortune-teller told her that she would be chosen. Given that nine out of ten fortune-tellers are always right and one is always wrong, what are Anne's chances of being chosen as the Harvest Queen? | [
"\\(\\frac{9}{370}\\)"
] | [
"Anne went into the fortune-teller's tent and asked whether she would be chosen as the Harvest Queen. The fortune-teller told her that she would be chosen. Given that nine out of ten fortune-tellers are always right and one is always wrong, what are Anne's chances of being chosen as the Harvest Queen?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9e641118e0cd4d937fd18bb99b642d0328615ecc66f439fca67edb43389b2ea | Anne writes down the nine consecutive integers from $-4$ to $4$. She then performs a series of operations where she identifies two numbers that differ by two, decreases the largest by one, and increases the smaller by one so that the two numbers are now equal. After a while she has nine zeros left and stops. How many o... | [
"30"
] | [
"Anne writes down the nine consecutive integers from $-4$ to $4$. She then performs a series of operations where she identifies two numbers that differ by two, decreases the largest by one, and increases the smaller by one so that the two numbers are now equal. After a while she has nine zeros left and stops. How m... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 9
} | 30 | 0 | 0 |
omr-ws-8cc18adcb83ff7b28714c8c1f1828a2cd453ec10f9cc711d586e3a5d4992e5da | Anne, Bernard, and Charlotte each had some pennies. Bernard gave one-half of his pennies to Charlotte and kept the rest. Anne then gave one-third of her pennies to Bernard and one-sixth of them to Charlotte. At the end, each had 27 pennies. How many pennies did Charlotte have originally? | [
"9"
] | [
"Anne, Bernard, and Charlotte each had some pennies. Bernard gave one-half of his pennies to Charlotte and kept the rest. Anne then gave one-third of her pennies to Bernard and one-sixth of them to Charlotte. At the end, each had 27 pennies. How many pennies did Charlotte have originally?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-38688631906a78f5d2e6e4173c9c64502f7d130aecf651f492758812ef5b5dcf | Anne, Beth, and Carmen are running on a circular track of length 150 meters with speeds \( s_1 \), \( s_2 \), and \( s_3 \) respectively, where \( s_1 \neq s_2 \) and \( s_1 \neq s_3 \). Calculate the minimum time \( t \) such that Anne is at least 100 meters away from both Beth and Carmen, considering all possible dir... | [
"\\dfrac{100}{\\min(s_1 + s_2, s_1 + s_3)}"
] | [
"Anne, Beth, and Carmen are running on a circular track of length 150 meters with speeds \\( s_1 \\), \\( s_2 \\), and \\( s_3 \\) respectively, where \\( s_1 \\neq s_2 \\) and \\( s_1 \\neq s_3 \\). Calculate the minimum time \\( t \\) such that Anne is at least 100 meters away from both Beth and Carmen, consideri... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-8c9fb539a772886946a5333fb798c16b0a34fcc2727102604a2eb8f63565fccc | Anne, Bob, Carla, Dario, Erika, and Fernando go to the movies and find an empty bench with 9 seats. How many ways can they sit if the women must occupy the odd-numbered seats and the men must occupy the even-numbered seats, and Anne and Bob must sit together? | [
"576"
] | [
"Anne, Bob, Carla, Dario, Erika, and Fernando go to the movies and find an empty bench with 9 seats. How many ways can they sit if the women must occupy the odd-numbered seats and the men must occupy the even-numbered seats, and Anne and Bob must sit together?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-9e22dec07f4221c509df0a0b98a3ab21a92f9a4351d696596d6ac36f915cdd81 | Annie and Forest wanted to add a question about a cubical structure in front of the nanotech center to the Penn Math Contest. Annie made a suggestion: draw a vanishing point. Forest draws parallel lines to indicate three dimensions, making a regular hexagon with side length one. What is the maximum distance between two... | [
"2"
] | [
"Annie and Forest wanted to add a question about a cubical structure in front of the nanotech center to the Penn Math Contest. Annie made a suggestion: draw a vanishing point. Forest draws parallel lines to indicate three dimensions, making a regular hexagon with side length one. What is the maximum distance betwee... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8e16987ad764a20672682cb6d1d3b686539203fbfd6d68b940f3354071e63637 | Annie assembles a box using 60 unit cubes with dimensions 3 by 4 by 5. She randomly selects a face and paints it, repeating this process four times (with the possibility of painting the same face multiple times). After disassembling the box, what is the expected number of unit cubes that have no paint on them, rounded ... | [
"22"
] | [
"Annie assembles a box using 60 unit cubes with dimensions 3 by 4 by 5. She randomly selects a face and paints it, repeating this process four times (with the possibility of painting the same face multiple times). After disassembling the box, what is the expected number of unit cubes that have no paint on them, rou... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a00c7deb452ab47f2dfbbe10ae2f49280b50a4b5d5892aa5bb208d24ccf9724e | Annie chooses a permutation of the numbers $1$ through $100$. Yannick can make guesses in the form of a permutation, and Annie will respond with the number of positions in which Yannick's guess matches Annie's permutation. What is the minimum number of guesses Yannick needs to guarantee identifying Annie's permutation? | [
"100"
] | [
"Annie chooses a permutation of the numbers $1$ through $100$. Yannick can make guesses in the form of a permutation, and Annie will respond with the number of positions in which Yannick's guess matches Annie's permutation. What is the minimum number of guesses Yannick needs to guarantee identifying Annie's permuta... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-1a9276a3470016d4a5824142553b900b6c03e6efe0c1f33a53fcce2d258ce5ee | Annie covers an $m \times n$ board completely using two types of shapes: shape 1 and shape 2. Bob then attempts to cover the same $m \times n$ board completely using only shape 1. If it is possible for Bob to do so, let $f(m, n)$ be the minimum number of shape 1 needed to cover the board. If it is impossible, let $f(m,... | [
"40"
] | [
"Annie covers an $m \\times n$ board completely using two types of shapes: shape 1 and shape 2. Bob then attempts to cover the same $m \\times n$ board completely using only shape 1. If it is possible for Bob to do so, let $f(m, n)$ be the minimum number of shape 1 needed to cover the board. If it is impossible, le... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5cf31302a130df77d6316502144364ce1388dc88318a94a9d07e70a327a66046 | Annie drives 25 miles east and then 50 miles north. From that point, she then drives 15 miles east and 20 miles south. What is Annie's distance now from her starting point? | [
"50"
] | [
"Annie drives 25 miles east and then 50 miles north. From that point, she then drives 15 miles east and 20 miles south. What is Annie's distance now from her starting point?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc50f021a65fdc51eef50a418d3f7981ee4d5d1af8fa41db5d70760dcaf2a6ad | Annie has 24 letter tiles in a bag; 8 C’s, 8 M’s, and 8 W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw? | [
"18"
] | [
"Annie has 24 letter tiles in a bag; 8 C’s, 8 M’s, and 8 W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1407fe3d4f925f278a16b2911c36251d2221c3e37363311b3643cabce28fb1a7 | Annie is currently in room B205 in a building with 6 rooms on the second floor (B201 to B206) and 8 rooms on the first floor (B101 to B108). Each minute, she either stays or changes floors with equal probability and then chooses a room on that floor at random. Rooms B104, B108, and B203 have teachers who will scold her... | [
"5"
] | [
"Annie is currently in room B205 in a building with 6 rooms on the second floor (B201 to B206) and 8 rooms on the first floor (B101 to B108). Each minute, she either stays or changes floors with equal probability and then chooses a room on that floor at random. Rooms B104, B108, and B203 have teachers who will scol... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e52ba7ce4b0be77f88992bdca5fa4c96388fba52f370d5a2bb45244288d3dc2e | Annie is given 8 documents to type, each consisting of 10 pages. Find the probability that at most 2 of the documents are error-free. | [
"\\( P(X \\leq 2) = (1 - e^{-1})^8 + 8 e^{-1} (1 - e^{-1})^7 + 28 e^{-2} (1 - e^{-1})^6 \\)"
] | [
"Annie is given 8 documents to type, each consisting of 10 pages. Find the probability that at most 2 of the documents are error-free."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ea0f5ca7c78e7ee4a5cf6e473b8ad0317d74e44259be2324925781e61a8590e | Annie is given 8 documents to type, each consisting of 10 pages. The number of errors Annie makes can be modeled by a Poisson distribution with a mean of 0.1 errors per page. Find the probability that at most 2 of the documents are error-free. | [
"0.4717"
] | [
"Annie is given 8 documents to type, each consisting of 10 pages. The number of errors Annie makes can be modeled by a Poisson distribution with a mean of 0.1 errors per page. Find the probability that at most 2 of the documents are error-free."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0db3498035dfe70f31f154765f7fbe441411f727ccc7bbe3c254fafca27031f6 | Annie is given 8 documents to type, each consisting of 10 pages. The number of errors Annie makes can be modeled by a Poisson distribution with a mean of 0.1 per page. Find the probability that at most 2 of the documents are error-free. | [
"\\(\\left(1 - \\frac{1}{e}\\right)^8 + 8 \\left(\\frac{1}{e}\\right) \\left(1 - \\frac{1}{e}\\right)^7 + 28 \\left(\\frac{1}{e}\\right)^2 \\left(1 - \\frac{1}{e}\\right)^6\\)"
] | [
"Annie is given 8 documents to type, each consisting of 10 pages. The number of errors Annie makes can be modeled by a Poisson distribution with a mean of 0.1 per page. Find the probability that at most 2 of the documents are error-free."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 27
} | 37 | 0 | 0 |
omr-ws-8d03e8ef35a7b398442c20be0136ad90b6459e48bd16406da00d8943c0916396 | Annie is given a 10-page document to type, and Betty is given a 15-page document to type. The number of errors Annie makes can be modeled by a Poisson distribution with a mean of 0.1 errors per page, and the number of errors Betty makes can be modeled by a Poisson distribution with a mean of 0.07 errors per page. Find ... | [
"0.848"
] | [
"Annie is given a 10-page document to type, and Betty is given a 15-page document to type. The number of errors Annie makes can be modeled by a Poisson distribution with a mean of 0.1 errors per page, and the number of errors Betty makes can be modeled by a Poisson distribution with a mean of 0.07 errors per page. ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-cd47c1262aa3858f37cffdf48bdc7be4cc9be3f9467ab1c6f5ab1ebf7639a5e8 | Annie is waiting in a long line to get food at Hillel on Dairy Night when she notices that the food is starting to run out. There are $8$ portions of mac and cheese, $16$ portions of mashed potatoes, $3$ portions of broccoli cheddar soup, and $2$ portions of penne pasta still left. Annie quickly starts counting the num... | [
"11"
] | [
"Annie is waiting in a long line to get food at Hillel on Dairy Night when she notices that the food is starting to run out. There are $8$ portions of mac and cheese, $16$ portions of mashed potatoes, $3$ portions of broccoli cheddar soup, and $2$ portions of penne pasta still left. Annie quickly starts counting th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3957cacdc9b2605e98f4063496ccc9c73ed8b61ace8fa0e56748950908a3e0c4 | Annie picks a positive integer \( n \leq 100! \) at random. Then, Bonnie will randomly choose one of the first \( n \) positive integers. Before Bonnie chooses, Connie computes the probability \( p \) that Bonnie will pick a multiple of 100. What is the probability that \( p < \frac{1}{100} \)? | [
"\\(\\frac{99}{100}\\)"
] | [
"Annie picks a positive integer \\( n \\leq 100! \\) at random. Then, Bonnie will randomly choose one of the first \\( n \\) positive integers. Before Bonnie chooses, Connie computes the probability \\( p \\) that Bonnie will pick a multiple of 100. What is the probability that \\( p < \\frac{1}{100} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-964927e496ffd618027f5894801c7287bdc63e22941c4eed70b44006e77971eb | Annie picks a triple of single-digit positive integers and gives Britta the following hints: 1) Her numbers are all distinct; 2) The sum of her numbers is greater than or equal to 20; 3) The product of her numbers is divisible by 32. Britta computes the sum of all single-digit positive integers that Annie didn’t choose... | [
"24"
] | [
"Annie picks a triple of single-digit positive integers and gives Britta the following hints: 1) Her numbers are all distinct; 2) The sum of her numbers is greater than or equal to 20; 3) The product of her numbers is divisible by 32. Britta computes the sum of all single-digit positive integers that Annie didn’t c... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ae3d841cd750dbd778b3e814f085bb2fe2548179f60af6f170f63e42b6f1579 | Annie selects a permutation of the numbers $1$ through $10$. Yannick is allowed to make guesses in the form of permutations. After each guess, Annie tells Yannick how many numbers are in their correct positions. What is the minimum number of guesses Yannick needs to guarantee he can determine Annie's permutation? | [
"10"
] | [
"Annie selects a permutation of the numbers $1$ through $10$. Yannick is allowed to make guesses in the form of permutations. After each guess, Annie tells Yannick how many numbers are in their correct positions. What is the minimum number of guesses Yannick needs to guarantee he can determine Annie's permutation?"... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-396853f91dcacf7afbea444eb6b9842e59cb06a936f6aec27c50db48ad368677 | Annie starts at (0,0) and moves 1 space up or right with equal probability every minute. Benny starts at (5,7) and moves 1 space down or left every minute with equal probability. What is the probability they meet? Note: The problem statement is extracted directly from the forum post, and the context of the user's metho... | [
"\\(\\frac{792}{64^2}\\)"
] | [
"Annie starts at (0,0) and moves 1 space up or right with equal probability every minute. Benny starts at (5,7) and moves 1 space down or left every minute with equal probability. What is the probability they meet?\n\nNote: The problem statement is extracted directly from the forum post, and the context of the user... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-04162fb1352ba3ef238f60926e05328df2b1a2a38f68d43b87f1ac886c7930c3 | Annie takes a 6-question test, with each question having two parts, each worth 1 point. She receives one of nine letter grades $\{\text{A,B,C,D,E,F,G,H,I}\}$ for each part, corresponding to a unique numerical score. She knows that $\text{A}$ corresponds to $1$, $\text{E}$ corresponds to $0.5$, and $\text{I}$ correspond... | [
"11"
] | [
"Annie takes a 6-question test, with each question having two parts, each worth 1 point. She receives one of nine letter grades $\\{\\text{A,B,C,D,E,F,G,H,I}\\}$ for each part, corresponding to a unique numerical score. She knows that $\\text{A}$ corresponds to $1$, $\\text{E}$ corresponds to $0.5$, and $\\text{I}$... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-7cec392da093f2879e24365dcd28cef59e595e3ad3e2a0d1ddc7d13b702a9e0c | Annie the Ant starts at the lattice point (0,0) and each minute moves 1 space up or 1 space to the right (with equal probability). Benny the Beetle starts at (5,7) and each minute moves 1 space down or 1 space to the left (with equal probability). What is the probability that they meet? | [
"\\(\\frac{99}{512}\\)"
] | [
"Annie the Ant starts at the lattice point (0,0) and each minute moves 1 space up or 1 space to the right (with equal probability). Benny the Beetle starts at (5,7) and each minute moves 1 space down or 1 space to the left (with equal probability). What is the probability that they meet?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-a844ac55fcab0cdcf8cd12ccde068e16a6ff5c91f938e6fb1ad8c459e5573c5c | Annie the pogo woman is standing next to a ladder with a rung at every meter. Her jumps are $5$ meters high, and after her jump, she can fall down either $0$ meters, $1$ meter, or $2$ meters. If Annie takes at most two jumps, find the number of distinct heights she can end up at. | [
"8"
] | [
"Annie the pogo woman is standing next to a ladder with a rung at every meter. Her jumps are $5$ meters high, and after her jump, she can fall down either $0$ meters, $1$ meter, or $2$ meters. If Annie takes at most two jumps, find the number of distinct heights she can end up at."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-cb092e166620a0a04f93aa9f0e0c80ede52c9419a8fc083a6a4d6f4f7eb48214 | Another game is played with ten prisoners, each wearing a red or blue hat chosen uniformly at random. Each prisoner can see the hats of those in front of them but not their own or those behind them. Each prisoner can only bet 0 or 1 points and guess the color of their hat. The prisoners collectively win if their total ... | [
"\\(\\frac{1023}{1024}\\)"
] | [
"Another game is played with ten prisoners, each wearing a red or blue hat chosen uniformly at random. Each prisoner can see the hats of those in front of them but not their own or those behind them. Each prisoner can only bet 0 or 1 points and guess the color of their hat. The prisoners collectively win if their t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-77798fce9ad60ef97980c15b6ffe515441d3489c1a94e5ea53e9ce14223edb31 | Anselmo and Bonifacio play a game where they alternately replace a number on a board with either the number of its divisors or the difference between the number and its number of divisors. Anselmo starts with the number $1036$. The player who first writes the number $0$ wins. Determine which player has a winning strate... | [
"Bonifacio has a winning strategy."
] | [
"Anselmo and Bonifacio play a game where they alternately replace a number on a board with either the number of its divisors or the difference between the number and its number of divisors. Anselmo starts with the number $1036$. The player who first writes the number $0$ wins. Determine which player has a winning s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 19
} | 36 | 0 | 0 |
omr-ws-bbf7b0da03cf89cc352884bf11bc99b550f8a1184fd59315106a7fd4aaeb479d | Anselmo and Claudio are playing a game with a box initially containing 32 fruits. Anselmo plays first, and each turn consists of taking away 1, 2, or 3 fruits from the box or taking away \(\frac{2}{3}\) of the fruits from the box (only possible when the number of fruits left is a multiple of 3). The player who takes aw... | [
"Anselmo has the winning strategy."
] | [
"Anselmo and Claudio are playing a game with a box initially containing 32 fruits. Anselmo plays first, and each turn consists of taking away 1, 2, or 3 fruits from the box or taking away \\(\\frac{2}{3}\\) of the fruits from the box (only possible when the number of fruits left is a multiple of 3). The player who ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-83db7d234cce64c9de56394c97bd03b33a8ad54f601764ecd4d242dc647496e2 | Answer the same question for a whole number in whose decimal representation the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 are present, each of the digits is used only once, and the digit 5 stands at the end of the number. | [
"It cannot"
] | [
"Answer the same question for a whole number in whose decimal representation the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 are present, each of the digits is used only once, and the digit 5 stands at the end of the number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd75bbfc5d1ab90ef6dec20df10861430c89b06d31ac6dae2f0bb33c994d3c7f | Ant #1 is climbing up a $60^\circ$ incline. Ant #2 starts $10$ ft from the vertex and walks at the same speed as ant #1. How high will ant #1 be when it is directly above ant #2? | [
"\\(\\frac{20\\sqrt{3}}{3}\\)"
] | [
"Ant #1 is climbing up a $60^\\circ$ incline. Ant #2 starts $10$ ft from the vertex and walks at the same speed as ant #1. How high will ant #1 be when it is directly above ant #2?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-028851fe0d9506bb33e4ab2a3789a84a6de7ca17c50b3c5c6ecbc991238bd49d | Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n = 1, 2, 3$, Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0, 1)$. During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using... | [
"$\\frac{2}{3}$"
] | [
"Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n = 1, 2, 3$, Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0, 1)$. During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-b6bcaaa3c28152d28a0798bb2e68d6ebe9a91bc0ba7d6d4ff92a8590f9d7c87d | Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3$, Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1)$. During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $... | [
"\\(\\frac{2}{3}\\)"
] | [
"Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3$, Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1)$. During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 32
} | 80 | 0 | 0 |
omr-ws-85b8dc980a1c475b52793a5b69ee34bb02d70621a6bc723c3aac12b0f6ef57f0 | Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $... | [
"\\(\\frac{2}{3}\\)"
] | [
"Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 33
} | 81 | 0 | 0 |
omr-ws-8db4303ef7b06d448f7c7b127091f8f40d9d5b808c5f1b7f6dadecf4ef4e3258 | Ant Amelia starts on the number line at $0$ and crawls in the following manner. For all $n$, Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1)$. During the $n$th step, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If ... | [
"0.689"
] | [
"Ant Amelia starts on the number line at $0$ and crawls in the following manner. For all $n$, Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1)$. During the $n$th step, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-318b774effb663f9d8f3fefe89519363411b7c97e8b804ee090dc4d666ef3279 | Anthoney has an unfair $6$-sided die that lands on $6$ half of the time. The other sides have an equal chance of being landed on. If Anthoney rolls two of these dice simultaneously, let $\frac{m}{n}$ be the probability that they sum to $11$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | [
"11"
] | [
"Anthoney has an unfair $6$-sided die that lands on $6$ half of the time. The other sides have an equal chance of being landed on. If Anthoney rolls two of these dice simultaneously, let $\\frac{m}{n}$ be the probability that they sum to $11$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$."... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecb3a05e7d6d5042e0d04c12d7fc0421f4b29381b80f9fa1b45be4dbf0c33809 | Anthony Garcia rented a car for 37 dollars a day for 7 days. There is also a charge of 45 cents for every mile driven over an average of 100 miles per day of rental, and a collision waiver of 9 dollars per day. Anthony drove 825 miles in 7 days. Gasoline costs were 49 dollars; parking and tolls were 32 dollars. What wa... | [
"\\(\\frac{459.25}{825} \\approx 0.5566666666666666\\) or approximately 55.67 cents per mile."
] | [
"Anthony Garcia rented a car for 37 dollars a day for 7 days. There is also a charge of 45 cents for every mile driven over an average of 100 miles per day of rental, and a collision waiver of 9 dollars per day. Anthony drove 825 miles in 7 days. Gasoline costs were 49 dollars; parking and tolls were 32 dollars. Wh... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-e430352d18347142ba83c29231ff341ea1f9fed4ecd114b116d1d748edc84282 | Anthony denotes in sequence all positive integers which are divisible by $2$. Bertha denotes in sequence all positive integers which are divisible by $3$. Claire denotes in sequence all positive integers which are divisible by $4$. Orderly Dora denotes all numbers written by the other three, puts them in order by size,... | [
"3026"
] | [
"Anthony denotes in sequence all positive integers which are divisible by $2$. Bertha denotes in sequence all positive integers which are divisible by $3$. Claire denotes in sequence all positive integers which are divisible by $4$. Orderly Dora denotes all numbers written by the other three, puts them in order by ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-7ce9ce7e0bd4ad7e07d9706af157c944f620cf9584cb40081ba1a33b0f2620c6 | Anthony made $5$ of his first $12$ free throw attempts. If he makes $2/3$ of his next $24$ attempts, by how many percentage points will he increase his overall success rate percentage? Express your answer to the nearest whole number. | [
"17\\%"
] | [
"Anthony made $5$ of his first $12$ free throw attempts. If he makes $2/3$ of his next $24$ attempts, by how many percentage points will he increase his overall success rate percentage? Express your answer to the nearest whole number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d2743e628f324d8844ba5688ac1848b4c9fecdeb746841b07a7c339987f1d379 | Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$ on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible n... | [
"\\( n^2 + n \\)"
] | [
"Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$ on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum poss... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 6
} | 25 | 0 | 0 |
omr-ws-25f9f9b0f9caf27bed571889cfbdd41e9b59a7d42ef19b7792ac426997799c31 | Anton and Britta play a game with the set \( M = \{1, 2, \dots, n-1\} \) where \( n \geq 5 \) is an odd integer. In each step, Anton removes a number from \( M \) and puts it in his set \( A \), and Britta removes a number from \( M \) and puts it in her set \( B \). When \( M \) is empty, Anton picks two distinct numb... | [
"\\( n \\) is a prime number."
] | [
"Anton and Britta play a game with the set \\( M = \\{1, 2, \\dots, n-1\\} \\) where \\( n \\geq 5 \\) is an odd integer. In each step, Anton removes a number from \\( M \\) and puts it in his set \\( A \\), and Britta removes a number from \\( M \\) and puts it in her set \\( B \\). When \\( M \\) is empty, Anton ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-130e4a7f309986c03a5c4eb6ea85c49e48c42084e4ceb5f4d87bb2996b6a275a | Anton has two species of ants, Species A and Species B, in his ant farm. Every day, the number of ants of Species A doubles, and the number of ants of Species B triples. On Day 0, there are 30 ants in total. On Day 5, there are 3281 ants in total. How many of these ants are of Species A? | [
"608\n\nExplanation:\nFrom the forum discussion, we have the following equations based on the problem statement:\n\n1. \\( a + b = 30 \\)\n2. \\( 32a + 243b = 3281 \\)\n\nBy solving these equations, we find:\n- \\( b = 11 \\)\n- \\( a = 19 \\)\n\nOn Day 5, the number of Species A ants is \\( 2^5 \\times a = 32 \\ti... | [
"Anton has two species of ants, Species A and Species B, in his ant farm. Every day, the number of ants of Species A doubles, and the number of ants of Species B triples. On Day 0, there are 30 ants in total. On Day 5, there are 3281 ants in total. How many of these ants are of Species A?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-8d1101dea86c40eb074cfb03123643b51bb63b2808a24a84d8165edde88052c5 | Anton has two species of ants, Species A and Species B, in his ant farm. The two species are identical in appearance, but Anton knows that every day, there are twice as many ants of Species A than before, while there are three times as many ants of Species B. On Day 0, Anton counts that there are 30 ants in his ant far... | [
"608"
] | [
"Anton has two species of ants, Species A and Species B, in his ant farm. The two species are identical in appearance, but Anton knows that every day, there are twice as many ants of Species A than before, while there are three times as many ants of Species B. On Day 0, Anton counts that there are 30 ants in his an... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d04a196cf3f3160f1a6533239b5ed894a52d7017a1da93f2f1cb48394adcc4a0 | Anton the ant walks along the vertices of a cube, starting and ending at the same vertex while visiting each other vertex exactly once and not intersecting its path. Let $E$ be the minimum number of edges Anton must walk along to satisfy these conditions. Find $E$. | [
"8"
] | [
"Anton the ant walks along the vertices of a cube, starting and ending at the same vertex while visiting each other vertex exactly once and not intersecting its path. Let $E$ be the minimum number of edges Anton must walk along to satisfy these conditions. Find $E$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f7844d1458fa482013bf4565f87a9e6873ab5d1cf5b4b504438cbf365fb9614 | Anton the ant walks along the vertices of a cube, starting and ending at the same vertex while visiting each other vertex exactly once without intersecting its path. Let $E$ be the number of edges Anton walks along. Find the minimum possible value of $E$. | [
"8"
] | [
"Anton the ant walks along the vertices of a cube, starting and ending at the same vertex while visiting each other vertex exactly once without intersecting its path. Let $E$ be the number of edges Anton walks along. Find the minimum possible value of $E$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68663ef61a38b8817352595802159d925d4c3b28525f49e5cd3fb5bb64ac1d17 | Antonette gets $70\%$ on a 10-problem test, $80\%$ on a 20-problem test and $90\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is her overall score, rounded to the nearest percent? | [
"83\\%"
] | [
"Antonette gets $70\\%$ on a 10-problem test, $80\\%$ on a 20-problem test and $90\\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is her overall score, rounded to the nearest percent?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-339d59a75a71ed21ad0c725b475f00bafab6d55f7e99ee5bbcc1079f092c095c | Antonio and Beltran each have a hat with an integer between 0 and 19 (inclusive). They can see the number on the other's hat but not their own. A third person tells them that the sum of their numbers is either 6, 11, or 19. After the bell rings for the tenth time, Antonio says he knows his number. Determine the number ... | [
"Antonio has 9 and Beltran has 10."
] | [
"Antonio and Beltran each have a hat with an integer between 0 and 19 (inclusive). They can see the number on the other's hat but not their own. A third person tells them that the sum of their numbers is either 6, 11, or 19. After the bell rings for the tenth time, Antonio says he knows his number. Determine the nu... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-63e94b1997b73382d3522949221eab769bfca6eb522a5c4983ed9987989c265c | Antonio and Bernardo play a game with two piles of chips, one with $m$ chips and the other with $n$ chips. Antonio starts, and on each turn, a player can either (i) take a chip from one pile, (ii) take a chip from each of the piles, or (iii) remove a chip from one pile and put it onto the other. The player who cannot m... | [
"The first player (Antonio) has a winning strategy if at least one of \\(m\\) or \\(n\\) is odd. If both \\(m\\) and \\(n\\) are even, the first player loses."
] | [
"Antonio and Bernardo play a game with two piles of chips, one with $m$ chips and the other with $n$ chips. Antonio starts, and on each turn, a player can either (i) take a chip from one pile, (ii) take a chip from each of the piles, or (iii) remove a chip from one pile and put it onto the other. The player who can... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 6,
"tir": 11
} | 57 | 0 | 0 |
omr-ws-0efa6d95837acf744acb54efe32efca3432baa39cca4cf33ffc598dd52c31e66 | Antonio folded a square sheet of paper with a side length of 40 cm, placing one of the vertices over the midpoint of one of the opposite sides. Determine the area of the resulting colored region. | [
"\\(\\frac{128}{3}\\)"
] | [
"Antonio folded a square sheet of paper with a side length of 40 cm, placing one of the vertices over the midpoint of one of the opposite sides. Determine the area of the resulting colored region."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19896336cdfa441bc5c7dadc77c645a829dd751fce81df185763f28df3328838 | Antonio's Jeep depreciates in value at a rate of 20% per year. If his Jeep is currently valued at $9200, what will its approximate value be in three years? (Round to the nearest dollar.) | [
"4710"
] | [
"Antonio's Jeep depreciates in value at a rate of 20% per year. If his Jeep is currently valued at $9200, what will its approximate value be in three years? (Round to the nearest dollar.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-863c583ba6e56de0f01055b30cc602dc2dd154dba65d69485e4f9f0f0bd12260 | Ants named Anna and Bob are located at vertices \(A\) and \(B\) respectively of a cube \(ABCD A_1 B_1 C_1 D_1\), with a sugar cube placed at vertex \(C_1\). Bob can move at a speed of 20 meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to reach the su... | [
"32"
] | [
"Ants named Anna and Bob are located at vertices \\(A\\) and \\(B\\) respectively of a cube \\(ABCD A_1 B_1 C_1 D_1\\), with a sugar cube placed at vertex \\(C_1\\). Bob can move at a speed of 20 meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-c725bd6eec6a0832a99666f99efe5839a9ac1bab42201a9eee25a6b8ac6f62b3 | Anu has a playlist with 8 songs, $s_1$, $s_2$, ..., $s_8$. He wants to listen to $s_3$ after $s_2$, $s_2$ after $s_1$, and $s_4$ after $s_1$. How many arrangements of the playlist are there such that Anu can listen to the songs in the order he wants? | [
"5040"
] | [
"Anu has a playlist with 8 songs, $s_1$, $s_2$, ..., $s_8$. He wants to listen to $s_3$ after $s_2$, $s_2$ after $s_1$, and $s_4$ after $s_1$. How many arrangements of the playlist are there such that Anu can listen to the songs in the order he wants?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-644a824931b949618071125c088cfcd86cbcec8b3924e12a7924241bcdc85041 | Anuma and Gopal start at the same corner of a square lawn and run clockwise at constant speeds of 12 km/h and 10 km/h, respectively. Anuma completes one lap in 60 seconds. For how many seconds were Anuma and Gopal together on the same path during one lap? | [
"42 seconds"
] | [
"Anuma and Gopal start at the same corner of a square lawn and run clockwise at constant speeds of 12 km/h and 10 km/h, respectively. Anuma completes one lap in 60 seconds. For how many seconds were Anuma and Gopal together on the same path during one lap?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 9
} | 56 | 0 | 0 |
omr-ws-bb31ecaaedd49049dd31dbbc168d07ededdfa6e203a3d6460b8c7c6c561f957c | Any pair of 2 people can be classified as friends or not friends. In any group of 4 students from a class of 8 students, one can always find 2 people who are friends. If the professor wanted to invite a group of students to dinner, with the restriction that any pair of students from the group he invites must be friends... | [
"3"
] | [
"Any pair of 2 people can be classified as friends or not friends. In any group of 4 students from a class of 8 students, one can always find 2 people who are friends. If the professor wanted to invite a group of students to dinner, with the restriction that any pair of students from the group he invites must be fr... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-f18b44f9aa5127a6e94e81be1699df5d1463ebca76d2987637c418c5b0324466 | Anya and Borya are playing a game with a "pyramid" consisting of 10 rows, where row $i$ has exactly $i$ coins arranged in a line. On each turn, a player chooses any number of contiguous coins within a single row and removes them. The player who removes the last coin wins. Anya goes first and then the players alternate.... | [
"14"
] | [
"Anya and Borya are playing a game with a \"pyramid\" consisting of 10 rows, where row $i$ has exactly $i$ coins arranged in a line. On each turn, a player chooses any number of contiguous coins within a single row and removes them. The player who removes the last coin wins. Anya goes first and then the players alt... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-fea33ee9865edac030e8b87ccfc88b0eee1e07c3b5bd05e6b5a7457d646faeee | Anya and Borya are playing a game with a pyramid consisting of 10 rows, where row \( i \) has exactly \( i \) coins arranged in a line. On each turn, a player chooses any number of contiguous coins within a single row and removes them. The player who removes the last coin wins. Anya goes first and then the players alte... | [
"Anya"
] | [
"Anya and Borya are playing a game with a pyramid consisting of 10 rows, where row \\( i \\) has exactly \\( i \\) coins arranged in a line. On each turn, a player chooses any number of contiguous coins within a single row and removes them. The player who removes the last coin wins. Anya goes first and then the pla... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9b6a8028538541043f217a3bf2244cf12f4ef71935e12c8fe4ed2d9b1979fb93 | AoPS Administrators have a \(\frac{1}{A}\) chance to delete a spam topic. After deleting spam topics, \(\frac{1}{B}\) of all topics are spam. Without AoPS Administrators, what is the expected fraction of all topics that would be spam? | [
"\\dfrac{A}{AB - B + 1}"
] | [
"AoPS Administrators have a \\(\\frac{1}{A}\\) chance to delete a spam topic. After deleting spam topics, \\(\\frac{1}{B}\\) of all topics are spam. Without AoPS Administrators, what is the expected fraction of all topics that would be spam?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a95096164f08eb956d819a30a4a4e30b74ada17c65698c4b912dbc7b65bef7d9 | Apart from \( n = 14 \), what is the next smallest even integer such that \( \phi(x) = n \) has no solution? | [
"26"
] | [
"Apart from \\( n = 14 \\), what is the next smallest even integer such that \\( \\phi(x) = n \\) has no solution?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-e3759709433b6bfb6aa0e7a42a134fcc74bfb91194169f8870e49fb40d26c748 | Apartment rentals in Fairview run $\$$0.90 per square foot. Jillian has determined that she can afford $\$$630 per month for rent. What is the largest apartment, in square feet, she should consider at the given rate? | [
"700"
] | [
"Apartment rentals in Fairview run $\\$$0.90 per square foot. Jillian has determined that she can afford $\\$$630 per month for rent. What is the largest apartment, in square feet, she should consider at the given rate?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-49ae0a31497018dece6d5e195fbccba0c3bf8f6257822230874a9a65a07bb629 | Apple Inc. has 500 barrels of wine, one of which is poisoned. The poison causes death if 50 ml is consumed and loss of ears if 25 ml is consumed, with effects manifesting between 4 and 6 months after consumption. Apple Inc. plans to use anti-globalists to test the barrels by giving them up to 1 liter per day in 25 ml o... | [
"6"
] | [
"Apple Inc. has 500 barrels of wine, one of which is poisoned. The poison causes death if 50 ml is consumed and loss of ears if 25 ml is consumed, with effects manifesting between 4 and 6 months after consumption. Apple Inc. plans to use anti-globalists to test the barrels by giving them up to 1 liter per day in 25... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6674451ecb1982745c96930f298e81512cc73c0a24e1bfab0c31565bf939efb | Apples are sold at the farmer's market at a rate of $\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples? | [
"12"
] | [
"Apples are sold at the farmer's market at a rate of $\\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d1aaf74c8f618af1523ec6588708f6f27785b1b4a5624330620753851cd9120 | Apply Aitken's method to the sequence defined by \( x_{n+1} = 2e^{-x_n} \) with the initial value \( x_0 = 0.8 \). | [
"0.8532"
] | [
"Apply Aitken's method to the sequence defined by \\( x_{n+1} = 2e^{-x_n} \\) with the initial value \\( x_0 = 0.8 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-6abbc10e848f574ac782a792cee2ec9acf5415207ba42c1a8f07c1491782cc3e | Apply Ceva's theorem to the case where \(AD\), \(BE\), and \(CF\) are concurrent cevians of \(\Delta ABC\) with \(AD\) parallel to \(BC\) and \(CF\) parallel to \(AB\). | [
"1"
] | [
"Apply Ceva's theorem to the case where \\(AD\\), \\(BE\\), and \\(CF\\) are concurrent cevians of \\(\\Delta ABC\\) with \\(AD\\) parallel to \\(BC\\) and \\(CF\\) parallel to \\(AB\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c75c8c2f29a2d4502caa87e376352e41049dfb28bafd2290fd09569ed91b5e5d | Apply Taylor's theorem to find an upper bound on the error in the approximation of the square root of 7.9 using a Taylor polynomial of degree three based at 8. | [
"\\(\\frac{15}{384} (7.9)^{-7/2} (1.1)^4\\)"
] | [
"Apply Taylor's theorem to find an upper bound on the error in the approximation of the square root of 7.9 using a Taylor polynomial of degree three based at 8."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4977f8678e295af5692bdbe1d53c7f16d15ac8076ef6fa84a324577ead07d082 | Apply elementary operations to the matrix \[ \begin{pmatrix} A & 0\\ I & B \end{pmatrix} \] to transform it into \[ \begin{pmatrix} AB & 0\\ B & I \end{pmatrix}. \] Deduce that \(\det(AB) = \det A \det B\). | [
"\\det(AB) = \\det A \\det B"
] | [
"Apply elementary operations to the matrix \\[\n\\begin{pmatrix}\nA & 0\\\\\nI & B\n\\end{pmatrix}\n\\] to transform it into \\[\n\\begin{pmatrix}\nAB & 0\\\\\nB & I\n\\end{pmatrix}.\n\\] Deduce that \\(\\det(AB) = \\det A \\det B\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-80e3a7ef628791f97217606dada2238b2ba9e1cc83260f47f4be363437bbacb2 | Apply partial fraction decomposition to $\frac{1}{x^3 + 5x^2 + 8x + 4}$. | [
"\\(-\\frac{1}{x+2} - \\frac{1}{(x+2)^2} + \\frac{1}{x+1}\\)"
] | [
"Apply partial fraction decomposition to $\\frac{1}{x^3 + 5x^2 + 8x + 4}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-3eea395cdcd43cfbb6f34a3d819e44b1765872cf782b76df4ebb1bcc58bb9c8d | Apply the Gram-Schmidt orthogonalization procedure to the canonical basis $\{1, x, x^2, x^3, x^4\}$ in order to find an orthonormal basis for the space $P_4([0, 1])$ with respect to the inner product $\langle p(x), q(x) \rangle = \int_0^1 p(x)q(x) \, dx$. | [
"\\[ \\{1, x - \\frac{1}{2}, x^2 + \\frac{1}{6} - x, x^3 - \\frac{1}{20} + \\frac{3}{5}x - \\frac{3}{2}x^2, x^4 + \\frac{1}{70} - \\frac{2}{7}x + \\frac{9}{7}x^2 - 2x^3\\} \\]"
] | [
"Apply the Gram-Schmidt orthogonalization procedure to the canonical basis $\\{1, x, x^2, x^3, x^4\\}$ in order to find an orthonormal basis for the space $P_4([0, 1])$ with respect to the inner product $\\langle p(x), q(x) \\rangle = \\int_0^1 p(x)q(x) \\, dx$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1fe1d06d9d8e31b3328bb14cddd25d31b63eec014deeddf707a44170f37ba24f | Apply the Gram-Schmidt process to the ordered basis $\{i + j + k, i + j, i\}$ in $\mathbb{R}^3$ and find the resulting orthonormal basis. | [
"\\(\\left\\{ \\frac{\\bold{i}+\\bold{j}+\\bold{k}}{\\sqrt{3}} , \\frac{\\bold{i}+\\bold{j}-2\\bold{k}}{\\sqrt{6}} , \\frac{\\bold{i}-\\bold{j}}{\\sqrt{2}} \\right\\}\\)"
] | [
"Apply the Gram-Schmidt process to the ordered basis $\\{i + j + k, i + j, i\\}$ in $\\mathbb{R}^3$ and find the resulting orthonormal basis."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-3df0a452a67484da3e37ce4bee0669a98613ee74dfcbce1c2dd06fa82bdfdabd | Apply the Gram-Schmidt process to the standard monomial basis vectors in the polynomial space \( P_3 \), with the inner product defined on the interval \([-1,1]\). Choose basis vectors so that \( p(1) = 1 \) and draw a sketch plot showing all four of them on the same axes over the interval \([-1,1]\). | [
"\\( p_{0}(x) = 1 \\), \\( p_{1}(x) = x \\), \\( p_{2}(x) = \\frac{3}{2}x^2 - \\frac{1}{2} \\), \\( p_{3}(x) = \\frac{5}{2}x^3 - \\frac{3}{2}x \\)"
] | [
"Apply the Gram-Schmidt process to the standard monomial basis vectors in the polynomial space \\( P_3 \\), with the inner product defined on the interval \\([-1,1]\\). Choose basis vectors so that \\( p(1) = 1 \\) and draw a sketch plot showing all four of them on the same axes over the interval \\([-1,1]\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 9
} | 15 | 0 | 0 |
omr-ws-a581b7ca738f7f0d11dccdc2215e08777641f85cad4ab436e38a19d4fbf35878 | Apply the Gram-Schmidt process to the vectors $\beta_1 = (3,0,4)$, $\beta_2 = (-1,0,7)$, and $\beta_3 = (2,9,11)$ in $\mathbb{R}^3$ with the standard inner product to obtain an orthonormal basis for $\mathbb{R}^3$. | [
"\\left\\{ \\left( \\frac{3}{5}, 0, \\frac{4}{5} \\right), \\left( -\\frac{4}{5}, 0, \\frac{3}{5} \\right), (0, 1, 0) \\right\\}"
] | [
"Apply the Gram-Schmidt process to the vectors $\\beta_1 = (3,0,4)$, $\\beta_2 = (-1,0,7)$, and $\\beta_3 = (2,9,11)$ in $\\mathbb{R}^3$ with the standard inner product to obtain an orthonormal basis for $\\mathbb{R}^3$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5ce4b6e90d509d1f7cb72089c9482e0054b4f460d6037b7197e14603c69b3171 | Apply the Lagrange mean value theorem to the function \( f(x) = \frac{x^3 \ln{x}}{3} - \frac{x^3}{9} \) on the interval \([k, k+1]\) for \( 1 \le k \le n \). Calculate the sum of all possible values of \( c \) in the interval \((k, k+1)\) that satisfy the mean value theorem for \( k = 1 \) and \( n = 5 \). | [
""
] | [
"Apply the Lagrange mean value theorem to the function \\( f(x) = \\frac{x^3 \\ln{x}}{3} - \\frac{x^3}{9} \\) on the interval \\([k, k+1]\\) for \\( 1 \\le k \\le n \\). Calculate the sum of all possible values of \\( c \\) in the interval \\((k, k+1)\\) that satisfy the mean value theorem for \\( k = 1 \\) and \\(... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 1 |
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