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-5c6021695c5a046827a9b6eaead8e4be7c9801dd522944fa83adea1aeb4418c2 | Add $10_7 + 163_7.$ Express your answer in base 7. | [
"203_7"
] | [
"Add $10_7 + 163_7.$ Express your answer in base 7."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ff2a87100ed9b83f28310fb264d6c8576140c88c30cd6105ece11aa2e27504b | Add $175_{9} + 714_{9} + 61_9$. Express your answer in base $9$. | [
"1061_{9}"
] | [
"Add $175_{9} + 714_{9} + 61_9$. Express your answer in base $9$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c5946a072a8207210a1e45acd845bc7bb2bf2db1ce7fa6ebcc8fa95797dd30d7 | Add $1_3 + 12_3 + 212_3 + 2121_3.$ Express your answer in base 3. | [
"10200_3"
] | [
"Add $1_3 + 12_3 + 212_3 + 2121_3.$ Express your answer in base 3."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a64b13111c2cbfe6b346674bc411f819caa6241d8f6369e0197a4abea33905b7 | Add $4_6 + 14_6.$ Express your answer in base $6.$ | [
"22_6"
] | [
"Add $4_6 + 14_6.$ Express your answer in base $6.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-638658a97faf858d38a6881ec6f6dfcfb82d57c556fdfee3ca53f6ea80c5e141 | Add $704_{12} + 159_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary. | [
"861_{12}"
] | [
"Add $704_{12} + 159_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8e22306a409f183f203e8cdd6a1e90316f66b853e44793ecc06e72e1ca57b1c0 | Add $81.76$ to $34.587$ and round your answer to the nearest tenth. | [
"116.3"
] | [
"Add $81.76$ to $34.587$ and round your answer to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e866f21d025110688071ff0e289693d2d23d2ffefe90c08c5498df163476c82 | Add 13 to the month you were born multiplied by 10. Multiply the result by 10. Now add the day of the month you were born. Explain how you can determine the day and month of birth from the number obtained. | [
"\\text{Subtract 130, then month is quotient and day is remainder when divided by 100.}"
] | [
"Add 13 to the month you were born multiplied by 10. Multiply the result by 10. Now add the day of the month you were born. Explain how you can determine the day and month of birth from the number obtained."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a78911265e3b2983c0e8f3567a2589b6dfe2881a7593a3d7f069cb9439cacf8 | Add 53.463 to 12.9873 and round to the nearest thousandth. | [
"66.450"
] | [
"Add 53.463 to 12.9873 and round to the nearest thousandth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9311931fd39bfbb936adbc1cf870ba875af783f4a327f336d4f1a7104921dd0b | Add the discriminant of the equation $3x^2 + 4x + 8 = 0$ to the sum of all perfect squares from 0 to 101. What is this sum? | [
"305"
] | [
"Add the discriminant of the equation $3x^2 + 4x + 8 = 0$ to the sum of all perfect squares from 0 to 101. What is this sum?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-a181fd5088fa34476ee3771c4ea843ad22a2f939fd1df34ff89de4176529408f | Add the fractions 2/3, 3/8, and 5/12 and reduce your answer to lowest terms. | [
"\\dfrac{35}{24}"
] | [
"Add the fractions 2/3, 3/8, and 5/12 and reduce your answer to lowest terms."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39aaf08c26961c556590801008df5fa45d7943404632b12842bb09323ffb05cc | Add the product of the digits 6 and 9 to the sum of the same numbers. Then, add the difference of the said numbers with the quotient of them. Finally, subtract that answer from the first equation. | [
"\\(\\frac{214}{3}\\)"
] | [
"Add the product of the digits 6 and 9 to the sum of the same numbers. Then, add the difference of the said numbers with the quotient of them. Finally, subtract that answer from the first equation."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-9ab6438f0d83ee1c49484cb3a080ed38bd1ab655b47a639ab26c6e067c3d44d3 | Add the three numbers (a nine-digit number, a single-digit number, and another nine-digit number) and obtain a palindromic number. What are the three numbers? | [
"\\( 555555555, 1, 555555555 \\)"
] | [
"Add the three numbers (a nine-digit number, a single-digit number, and another nine-digit number) and obtain a palindromic number. What are the three numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 22
} | 23 | 0 | 0 |
omr-ws-7b1f7f9a0bc1f073f27b0b8b14f03b2d309fae8f5779b4d98255f61ab1d3b523 | Add three digits to 523 so that the resultant six-digit number is divisible by 7, by 8 and by 9. | [
"523152 and 523656"
] | [
"Add three digits to 523 so that the resultant six-digit number is divisible by 7, by 8 and by 9."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a9939389e6a26af47eae040cb4613f0d3bd7fd3a2de5ad52636bb6c580fa451 | Addison and Olivia walk up an escalator with 100 steps. Addison reaches the top by walking 40 steps, while Olivia reaches the top by walking 60 steps. What is the ratio of Olivia's walking speed on the escalator to Addison's walking speed? | [
"\\(\\frac{2}{3}\\)"
] | [
"Addison and Olivia walk up an escalator with 100 steps. Addison reaches the top by walking 40 steps, while Olivia reaches the top by walking 60 steps. What is the ratio of Olivia's walking speed on the escalator to Addison's walking speed?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-693a90ea2c161b0e1b3afd960540314c767fde7b3b10a112def8285a520ad4a9 | Addison's age is three times Brenda's age. Janet is six years older than Brenda. Addison and Janet are twins. How old is Brenda? | [
"3"
] | [
"Addison's age is three times Brenda's age. Janet is six years older than Brenda. Addison and Janet are twins. How old is Brenda?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f10e7373410f5f9be5e2cd752749182bbde78471919f605d45043327f19cfc2b | Adi places 10 points on a Cartesian plane. Budi places circles of radius one such that no two circles intersect and all 10 points are either on or inside the circles. If Budi can always accomplish this task, find the maximum possible value of the sum of the x-coordinates of the centers of these circles. If Budi cannot ... | [
"-1"
] | [
"Adi places 10 points on a Cartesian plane. Budi places circles of radius one such that no two circles intersect and all 10 points are either on or inside the circles. If Budi can always accomplish this task, find the maximum possible value of the sum of the x-coordinates of the centers of these circles. If Budi ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-9214ff25ef957d3d50a73b5d1c324eac38db5b392cc76b8e15fc39cbec75eba7 | Adi the Baller is shooting hoops, making a shot with probability \( p \). He keeps shooting until he misses. The value of \( p \) that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as \( \frac{1}{\sqrt[b]{a}} \) for a prime \( a \) and positive integer \( b \). Find \( a + b ... | [
"37"
] | [
"Adi the Baller is shooting hoops, making a shot with probability \\( p \\). He keeps shooting until he misses. The value of \\( p \\) that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as \\( \\frac{1}{\\sqrt[b]{a}} \\) for a prime \\( a \\) and positive integer \\( b \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-86a52f8566a891ccbf3a19edc1adfb927c4b26a82b6e79addf96a294c13cb1e7 | Adithya and Bill are playing a game on a connected graph with \( n > 2 \) vertices, two of which are labeled \( A \) and \( B \), so that \( A \) and \( B \) are distinct and non-adjacent and known to both players. Adithya starts on vertex \( A \) and Bill starts on vertex \( B \). Each turn, both players move simultan... | [
"\\(\\binom{n-1}{2} + 1\\)"
] | [
"Adithya and Bill are playing a game on a connected graph with \\( n > 2 \\) vertices, two of which are labeled \\( A \\) and \\( B \\), so that \\( A \\) and \\( B \\) are distinct and non-adjacent and known to both players. Adithya starts on vertex \\( A \\) and Bill starts on vertex \\( B \\). Each turn, both pl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-9e44a06c00b47ed11ad4c9e5e3e11200c61ea5288071a38e735e12eea5bd89e5 | Aditya plays a game on the number line starting from $0$. He flips a fair coin. If it lands heads, he takes two steps (of length $1$ each) ahead (i.e., in the positive direction), and if it lands tails, he takes one step (of length $1$) backwards (i.e., in the negative direction). The game ends when he lands on a posit... | [
"\\(\\frac{1}{2} (3 - \\sqrt{5})\\)"
] | [
"Aditya plays a game on the number line starting from $0$. He flips a fair coin. If it lands heads, he takes two steps (of length $1$ each) ahead (i.e., in the positive direction), and if it lands tails, he takes one step (of length $1$) backwards (i.e., in the negative direction). The game ends when he lands on a ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 7
} | 11 | 0 | 0 |
omr-ws-80c529ebdb08c72e3ad734c16f944e62ea0297ea88b5399edaac0aed5b3582c2 | Adrian has $2n$ cards numbered from $1$ to $2n$. He discards $n$ consecutively numbered cards. The sum of the numbers on the remaining cards is $1615$. Find all possible values of $n$. | [
"\\( n = 34 \\) or \\( n = 38 \\)"
] | [
"Adrian has $2n$ cards numbered from $1$ to $2n$. He discards $n$ consecutively numbered cards. The sum of the numbers on the remaining cards is $1615$. Find all possible values of $n$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-db249afc86c2c8ad10856f8342d00eb826227c9c99fb6dade2cb4dd1805f0818 | Adrian has drawn a circle in the $xy$-plane with a radius that is a positive integer at most $2008$. The origin lies somewhere inside the circle. You can ask questions of the form “Is the point $(x, y)$ inside your circle?” and Adrian will answer truthfully with “yes” or “no”. Using a binary search strategy, determine ... | [
"11"
] | [
"Adrian has drawn a circle in the $xy$-plane with a radius that is a positive integer at most $2008$. The origin lies somewhere inside the circle. You can ask questions of the form “Is the point $(x, y)$ inside your circle?” and Adrian will answer truthfully with “yes” or “no”. Using a binary search strategy, deter... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cc560df7468a32b37369b5d3956bbb70e31fd0c391f472ae10be5da229f7c32 | Adrian has drawn a circle in the xy-plane with a radius that is a positive integer at most 2008. The origin lies somewhere inside the circle. You can ask Adrian questions of the form "Is the point (x, y) inside your circle?" and he will answer truthfully. Using a binary search strategy, determine the maximum number of ... | [
"22"
] | [
"Adrian has drawn a circle in the xy-plane with a radius that is a positive integer at most 2008. The origin lies somewhere inside the circle. You can ask Adrian questions of the form \"Is the point (x, y) inside your circle?\" and he will answer truthfully. Using a binary search strategy, determine the maximum num... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-c54f4689d5eb8cdd920c5ffead7c06fe84b86c267cbb8dc271acfbb3d7a46d9e | Adrian has drawn a circle in the xy-plane with a radius that is a positive integer at most 2008. The origin lies somewhere inside the circle. You can ask questions of the form “Is the point $(x, y)$ inside your circle?” and Adrian will answer truthfully with “yes” or “no”. Determine the maximum number of questions need... | [
"11"
] | [
"Adrian has drawn a circle in the xy-plane with a radius that is a positive integer at most 2008. The origin lies somewhere inside the circle. You can ask questions of the form “Is the point $(x, y)$ inside your circle?” and Adrian will answer truthfully with “yes” or “no”. Determine the maximum number of questions... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-b24794dd3913b45a35b3ab701ae305d98dde7338a79135f2066d0fb7b9a97b83 | Adrian is to arrange three identical red marbles, four identical green marbles, and five identical blue marbles in such a way that all marbles of the same color are not together. In how many ways can Adrian finish this task? | [
"25762"
] | [
"Adrian is to arrange three identical red marbles, four identical green marbles, and five identical blue marbles in such a way that all marbles of the same color are not together. In how many ways can Adrian finish this task?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-511a9bbb32397fc09f07c7b15c9f84ab451ec6d9771096dbdfab6f46846b9701 | Adrian teaches a class of six pairs of twins. He wishes to set up teams for a quiz, but wants to avoid putting any pair of twins into the same team. Subject to this condition: i) In how many ways can he split them into two teams of six? ii) In how many ways can he split them into three teams of four? | [
"960"
] | [
"Adrian teaches a class of six pairs of twins. He wishes to set up teams for a quiz, but wants to avoid putting any pair of twins into the same team. Subject to this condition:\ni) In how many ways can he split them into two teams of six?\nii) In how many ways can he split them into three teams of four?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5e542a069bc34ad0826323d069651aa415a9254ebb0f24a133f52fabbae92c10 | Adrian, Martin, and Carl played a series of table tennis matches. Each time one player lost, they were replaced by the other player who was not playing. The first match was played by Adrian and Martin. Adrian won 12 matches and Martin won 21 matches. How many times did Adrian and Martin face each other? | [
"17"
] | [
"Adrian, Martin, and Carl played a series of table tennis matches. Each time one player lost, they were replaced by the other player who was not playing. The first match was played by Adrian and Martin. Adrian won 12 matches and Martin won 21 matches. How many times did Adrian and Martin face each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-afcff814f5d30f37aee0593f2f84f597b56d6bb12ab4f80310f02045dbcc5209 | Adriel has a collection of blue marbles and green marbles. The number of blue marbles is either \( 2, 5, 6, 7, 11 \), or \( 13 \). If two marbles are chosen simultaneously and at random from his collection, then the probability they have different colors is \( \frac{1}{2} \). How many blue marbles are in Adriel's colle... | [
"6"
] | [
"Adriel has a collection of blue marbles and green marbles. The number of blue marbles is either \\( 2, 5, 6, 7, 11 \\), or \\( 13 \\). If two marbles are chosen simultaneously and at random from his collection, then the probability they have different colors is \\( \\frac{1}{2} \\). How many blue marbles are in Ad... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b9d893478d94bb05960b8c2221c8d9c33c2effaa9eab96ba2146c5e00657a4a | Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived. | [
"154"
] | [
"Adults made up $\\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-b00dccd08f976b1cd358fb6b41ef46579db49a8da266e42a194571f6c2389ae6 | Afkar selects the terms of an infinite geometric sequence $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$ to create a new infinite geometric sequence whose sum is $\frac{1}{7}$. The first three terms Afkar chooses are ... | [
"\\(\\frac{1}{8}, \\frac{1}{64}, \\frac{1}{512}\\)"
] | [
"Afkar selects the terms of an infinite geometric sequence $1, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{8}, \\ldots$ to create a new infinite geometric sequence whose sum is $\\frac{1}{7}$. The first three terms Afkar chooses are ..."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 19
} | 23 | 0 | 0 |
omr-ws-ba55afa857e64947424f6e1b58a94772235e149ff9060e0358ce5d6a0bd005a2 | After $k$ balls have been pulled, what's the expected number of boards with fewer than or equal to $b$ balls, given there are $n$ boards? | [
"n \\sum_{m=0}^{b} \\binom{k}{m} \\left( \\frac{1}{n} \\right)^m \\left( 1 - \\frac{1}{n} \\right)^{k - m}"
] | [
"After $k$ balls have been pulled, what's the expected number of boards with fewer than or equal to $b$ balls, given there are $n$ boards?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e67bd49d8c96370074dbe56b0f2583d852494349fd16be886a312af0efd0e868 | After $n$ moves, what is the expected distance of an ant from the origin, given that at each step, the ant walks either forward or backward with equal probability? | [
"\\dfrac{n}{2^{n-1}} \\dbinom{n-1}{\\left\\lfloor \\dfrac{n-1}{2} \\right\\rfloor}"
] | [
"After $n$ moves, what is the expected distance of an ant from the origin, given that at each step, the ant walks either forward or backward with equal probability?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-bfeef09974b418d3947ec3ef26f51f5b8ca3fea69038169079e0ae1089b8d785 | After 2k coin tosses, Charikaar is back where he started. Calculate the probability of this event occurring and express your answer in the form \( A \times 2^{Bk} \binom{2k}{k} \). Find the ordered pair (A, B). | [
"(1, -2)"
] | [
"After 2k coin tosses, Charikaar is back where he started. Calculate the probability of this event occurring and express your answer in the form \\( A \\times 2^{Bk} \\binom{2k}{k} \\). Find the ordered pair (A, B)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a0c78eb9684a634eefe370d5e6fdf62d67e1dd26287156f27c732ec33b4fa0d | After 3 months, a club has 20 members. After 6 months, there are 200 members. What was the club's original membership? | [
"2"
] | [
"After 3 months, a club has 20 members. After 6 months, there are 200 members. What was the club's original membership?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ba34bda8334fcf9507f7d0022eb33eb104cdc360a76be97b0466fb23caaf0664 | After Caitlin has moved her marker four times on a $3 \times 3$ board, starting from the upper-left square and moving randomly to adjacent squares (not diagonally), the probability that she has fought the dragon exactly once can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are positive, coprime integers. W... | [
"3"
] | [
"After Caitlin has moved her marker four times on a $3 \\times 3$ board, starting from the upper-left square and moving randomly to adjacent squares (not diagonally), the probability that she has fought the dragon exactly once can be expressed in the form $\\frac{a}{b}$, where $a$ and $b$ are positive, coprime inte... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-dccc7022b3705ab511c7f93b5f8be02ad1f033425061eec3fcc792331d929fc3 | After Caitlin has moved her marker four times on a $3 \times 3$ board, starting from the upper-left square and moving randomly to adjacent squares (not diagonally), the probability that she has never encountered the dragon but has found the treasure can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are posi... | [
"25"
] | [
"After Caitlin has moved her marker four times on a $3 \\times 3$ board, starting from the upper-left square and moving randomly to adjacent squares (not diagonally), the probability that she has never encountered the dragon but has found the treasure can be expressed in the form $\\frac{a}{b}$, where $a$ and $b$ a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-38a451f1bae2f4a767cefe939039e8b556373d40b26a4dfd7d90beb3eb3925b0 | After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team ... | [
"11"
] | [
"After Euclid High School's last basketball game, it was determined that $\\frac{1}{4}$ of the team's points were scored by Alexa and $\\frac{2}{7}$ were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-43332c8f556351e9dac3c0eb7479b2468398827ef1e19ae49f67a0ccf0b59538 | After Taemin and Younghee had been painting a wall for five hours, Jaejoon helped them and they finished 3 hours later. If Jaejoon had not helped them, it would have taken Taemin and Younghee 5 more hours to complete the job. How long would it take for Jaejoon to paint the wall alone? | [
"15"
] | [
"After Taemin and Younghee had been painting a wall for five hours, Jaejoon helped them and they finished 3 hours later. If Jaejoon had not helped them, it would have taken Taemin and Younghee 5 more hours to complete the job. How long would it take for Jaejoon to paint the wall alone?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 8
} | 30 | 0 | 0 |
omr-ws-970ab6d285bfaf2efd759b7c3d2e82d974a24d5a299b3b9cab6a660dbb6e2364 | After \( n \) seconds, what is the probability that the number of stones in each of two buckets is divisible by 3, given that every second Fatty McButterpants throws a stone into the first bucket, the second bucket, or no buckets, each with probability \(\frac{1}{3}\)? | [
"\\dfrac{1}{9} \\left(1 + \\dfrac{4 \\cos \\dfrac{n\\pi}{6} + 2 \\cos \\dfrac{n\\pi}{2}}{3^{n/2}} \\right)"
] | [
"After \\( n \\) seconds, what is the probability that the number of stones in each of two buckets is divisible by 3, given that every second Fatty McButterpants throws a stone into the first bucket, the second bucket, or no buckets, each with probability \\(\\frac{1}{3}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-2afbb8fe21eb86e480f63292efec43ce18980d53f27636b46a5523482b4c79d9 | After \( t \) years, \( 50e^{-0.015t} \) pounds of a deposit of a radioactive substance remain. What is the average amount per year not lost by radioactive decay during the second hundred years? | [
"5.778"
] | [
"After \\( t \\) years, \\( 50e^{-0.015t} \\) pounds of a deposit of a radioactive substance remain. What is the average amount per year not lost by radioactive decay during the second hundred years?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-ab9a93aa317ef1f081d2d94d23bcd9099f9889fd3b5994d31d1bffd9a5f6e6bf | After \( t \) years, \( 50e^{-0.015t} \) pounds of a radioactive substance remain. Calculate the average amount per year not lost by radioactive decay during the second hundred years. Note: The second hundred years refers to the time interval from \( t = 100 \) to \( t = 200 \). | [
"5.776"
] | [
"After \\( t \\) years, \\( 50e^{-0.015t} \\) pounds of a radioactive substance remain. Calculate the average amount per year not lost by radioactive decay during the second hundred years.\n\nNote: The second hundred years refers to the time interval from \\( t = 100 \\) to \\( t = 200 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 6
} | 8 | 0 | 0 |
omr-ws-fd871aed2eaf02b718cb3241c4964ef53ce4798d5f4d032bb950ab3175de9d4b | After \(\frac{2}{3}\) of the boys left a school dance, the ratio of girls to boys was 9:5. Then 21 girls left, and the ratio became 3:4. How many students total remained at the dance? | [
"45"
] | [
"After \\(\\frac{2}{3}\\) of the boys left a school dance, the ratio of girls to boys was 9:5. Then 21 girls left, and the ratio became 3:4. How many students total remained at the dance?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f77f3ffe03a2852fcf680edaf4b405417fef9ff1a98d7067402f0d5c57ffe7e3 | After a basketball game, it was determined that 1/4 of the team's points were scored by Alexa and 2/7 by Brittany. Chelsea scored 15 points, and none of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members? | [
"11"
] | [
"After a basketball game, it was determined that 1/4 of the team's points were scored by Alexa and 2/7 by Brittany. Chelsea scored 15 points, and none of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3956f5d0b55e1c0a0551dde47c731d52279c4029ec7b57a0e4312bccba6c7dcf | After a break, 16 children sit back down at a round table such that each child either sits in their original seat or in one of the two neighboring seats. How many seating arrangements are possible? | [
"43691"
] | [
"After a break, 16 children sit back down at a round table such that each child either sits in their original seat or in one of the two neighboring seats. How many seating arrangements are possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7be3878980ea7fb745672122ab3a4dc6e9558d0f104a71d6dc22912a095167d | After a few hours, the party has thinned out. The Venusian attendance taker counts \( BC \) attendees in base 7, and the Martian attendance taker counts \( BA \) attendees in base 8. What is \( C \)? | [
"6"
] | [
"After a few hours, the party has thinned out. The Venusian attendance taker counts \\( BC \\) attendees in base 7, and the Martian attendance taker counts \\( BA \\) attendees in base 8. What is \\( C \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8cf79ebcbddad6b5aff2dfb28bb3496823e7127cd8d9dec8928be802f48db54 | After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the coach could have participated in? | [
"5"
] | [
"After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the coach could have participated in?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ca75ba4481ea74a6ce40e4aa26c8e924ec626f86c9cafa4d2126b95e8931328 | After a huge party, a cake is to be split among 2020 people in the following manner: - The 1st person in line gets $1/2020$ of the cake. - The 2nd person in line gets $2/2020$ of the remaining cake. - The 3rd person in line gets $3/2020$ of the remaining cake. - ... - The 2019th person in line gets $2019/2020$ of the r... | [
"2019"
] | [
"After a huge party, a cake is to be split among 2020 people in the following manner:\n- The 1st person in line gets $1/2020$ of the cake.\n- The 2nd person in line gets $2/2020$ of the remaining cake.\n- The 3rd person in line gets $3/2020$ of the remaining cake.\n- ...\n- The 2019th person in line gets $2019/2020... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb904a0af8efe66b5bfe105fc95701bd86bf46936ff0bac7021fd36f3d0a3653 | After a rainstorm, the depth of water in a cylindrical rain gauge with a base diameter of 20 cm and a height of 40 cm is 10 mm. What is the depth of water in a conical rain gauge with the same base diameter and height? | [
"16.87 cm"
] | [
"After a rainstorm, the depth of water in a cylindrical rain gauge with a base diameter of 20 cm and a height of 40 cm is 10 mm. What is the depth of water in a conical rain gauge with the same base diameter and height?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 21
} | 22 | 0 | 0 |
omr-ws-0856fe3055210d82713de0c035aaffee5843f1299dd9fd0ca810da070b9eedfe | After a rainstorm, the depth of water in a cylindrical rain gauge with a base diameter of 20 cm and a height of 40 cm is 10 mm. What is the depth of water in a double-cone rain gauge, where each cone has the same base diameter and height as the conical rain gauge, and the total height of the double-cone is 80 cm? | [
"1.026 cm"
] | [
"After a rainstorm, the depth of water in a cylindrical rain gauge with a base diameter of 20 cm and a height of 40 cm is 10 mm. What is the depth of water in a double-cone rain gauge, where each cone has the same base diameter and height as the conical rain gauge, and the total height of the double-cone is 80 cm?"... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-e6d68ed536a45e72549aed5c24036a5c043d02f03a9d7cbe4ca2460a04147083 | After a raise, your hourly wage is increased from $7.62 to $12. What is the percentage increase in your hourly wage? | [
"57.48\\%"
] | [
"After a raise, your hourly wage is increased from $7.62 to $12. What is the percentage increase in your hourly wage?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a39d0458e57a47bddba3e114355bf3452fc3ab2e2646bb53a79a960131a9d377 | After a single riffle shuffle of a deck of 52 cards, let $N$ be the minimum number of cards you need to flip over to guarantee finding a specific card. Calculate $N$. | [
"27"
] | [
"After a single riffle shuffle of a deck of 52 cards, let $N$ be the minimum number of cards you need to flip over to guarantee finding a specific card. Calculate $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-d0f89676f3b1cf53eff68b67515d5a1183efd64d9acf88a4fe78b3cfe52117e7 | After a tournament where each pair of six tennis players plays one game with a declared winner and loser, let the players be labeled $A, B, C, D, E, F$. Define a "cyclic triplet" as a set of three players where each player has won against one and lost to another within the triplet. Find the number of ways to divide the... | [
"2"
] | [
"After a tournament where each pair of six tennis players plays one game with a declared winner and loser, let the players be labeled $A, B, C, D, E, F$. Define a \"cyclic triplet\" as a set of three players where each player has won against one and lost to another within the triplet. Find the number of ways to div... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-c8a6cb373d46710ff7453fee6696e0524832d4f6173ffe80d3db6575cb2f6e0b | After collecting the salaries of the last 5 years, make a table and find the relationship between the salaries of consecutive years. What is the conclusion you make? Find the sum of salaries for 5 years. - Year 1 (2003): P - Year 2 (2004): P - Year 3 (2005): P \((1 + \frac{r}{100})^2\) - Year 4 (2006): P \((1 + \frac{r... | [
"\\( P \\left(2 + \\left(1 + \\frac{r}{100}\\right)^2 + \\left(1 + \\frac{r}{100}\\right)^3 + \\left(1 + \\frac{r}{100}\\right)^4 \\right) \\)"
] | [
"After collecting the salaries of the last 5 years, make a table and find the relationship between the salaries of consecutive years. What is the conclusion you make? Find the sum of salaries for 5 years.\n\n- Year 1 (2003): P\n- Year 2 (2004): P\n- Year 3 (2005): P \\((1 + \\frac{r}{100})^2\\)\n- Year 4 (2006): P ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 23
} | 44 | 0 | 0 |
omr-ws-bf448308078d71ea89def7372b00c84208e0fce0ab1710bf1ec64d2ebe00ca25 | After dinner, Rick asks his father for 1000 scoops of ice cream as dessert. Rick’s father responds, “I will give you 2 scoops of ice cream, plus 1 additional scoop for every ordered pair (a, b) of real numbers satisfying 1/(a + b) = 1/a + 1/b you can find.” If Rick finds every solution to the equation, how many scoops ... | [
"2"
] | [
"After dinner, Rick asks his father for 1000 scoops of ice cream as dessert. Rick’s father responds, “I will give you 2 scoops of ice cream, plus 1 additional scoop for every ordered pair (a, b) of real numbers satisfying 1/(a + b) = 1/a + 1/b you can find.” If Rick finds every solution to the equation, how many sc... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5f78937652c70c8a832a6af28fcfedf5a82e65d62f114643c962bb2aac9ed15 | After dividing 30 marbles into 5 boxes, the student paints the marbles using different colors such that no two marbles in the same box have the same color and from any two boxes, it is impossible to choose 8 marbles painted in 4 colors. Find the minimum number of colors the student must use to paint the marbles for any... | [
"no_valid_answer_found"
] | [
"After dividing 30 marbles into 5 boxes, the student paints the marbles using different colors such that no two marbles in the same box have the same color and from any two boxes, it is impossible to choose 8 marbles painted in 4 colors. Find the minimum number of colors the student must use to paint the marbles fo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea24e0176a4c8b36ff8cf0ed48e5c1f7326dc30f15663f556c8379797d3cf2d5 | After drawing $m$ lines on a plane, exactly 200 different intersection points are formed. What is the lowest value of $m$? | [
"15"
] | [
"After drawing $m$ lines on a plane, exactly 200 different intersection points are formed. What is the lowest value of $m$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd23027d74344f2c3059c32e9bbf9cbca10320a0dc2232081e733608f51c34ab | After each play of a game, a player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$. It is observed that every integer $x \geq 1986$ can be obtained as the total score, whereas $1985$ and $663$ cannot. Determine the values of $a$ and $b$. | [
"\\( a = 332, b = 7 \\)"
] | [
"After each play of a game, a player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$. It is observed that every integer $x \\geq 1986$ can be obtained as the total score, whereas $1985$ and $663$ cannot. Determine the values of $a$ and $b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 20
} | 26 | 0 | 0 |
omr-ws-b18f0d0d30bc0f2004f0193b7d2b4fcf1ca618b65b26ef52c95f34e5b3c5747e | After eating lunch together, Jay and Paul start walking in opposite directions. Jay walks 0.75 miles every 15 minutes and Paul power walks 2.5 miles every 30 minutes. In miles, how far apart are they after 1.5 hours? | [
"12"
] | [
"After eating lunch together, Jay and Paul start walking in opposite directions. Jay walks 0.75 miles every 15 minutes and Paul power walks 2.5 miles every 30 minutes. In miles, how far apart are they after 1.5 hours?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55671012a9c4d2e72e7ae46e53ced4da4555f5525e7650b8ff4920808864fbd1 | After expanding \((x^2 + 3x + 1)^7\), let the coefficient of \(x^n\) be \(a_n\) for \(n = 1, 2, 3, \ldots, 14\). Find the value of \(-\frac{1}{9}(a_1 - a_3 + a_5 - a_7 + a_9 - a_{11} + a_{13})\). | [
"243"
] | [
"After expanding \\((x^2 + 3x + 1)^7\\), let the coefficient of \\(x^n\\) be \\(a_n\\) for \\(n = 1, 2, 3, \\ldots, 14\\). Find the value of \\(-\\frac{1}{9}(a_1 - a_3 + a_5 - a_7 + a_9 - a_{11} + a_{13})\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-a58fe8222007b9e67ecb82aa72b6552b1f5b4b13ad89f8e81ffa472a0d5297f4 | After expanding the polynomial $(1+x+y)^{2023}$ and collecting like terms, find the total number of coefficients $a_i$ that are divisible by $5$. | [
"1439219"
] | [
"After expanding the polynomial $(1+x+y)^{2023}$ and collecting like terms, find the total number of coefficients $a_i$ that are divisible by $5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18dba25545a6b545ff960cc60b7f4423e85b7cf7a41c665d354b8e1dbcba3271 | After expanding the polynomial \( (1 + x + y)^{2023} \) and collecting the like terms, we obtain the expression \[ a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + \cdots + a_ky^{2023} \] Find the total number of \( a_i \)'s which are divisible by 5. | [
"2044800"
] | [
"After expanding the polynomial \\( (1 + x + y)^{2023} \\) and collecting the like terms, we obtain the expression \n\\[ a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + \\cdots + a_ky^{2023} \\]\nFind the total number of \\( a_i \\)'s which are divisible by 5."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-890bc7cf43754a9ba829503d4aa5320f2f08c2eea5a690e231b923e47e6e2735 | After extending the columns to 65536 and copying the 4096 unique numbers 16 times, how many rows are needed to arrange them in all possible combinations? The result should be smaller than 65536! but larger than (4096!)^16. How to calculate it correctly? | [
"\\(\\frac{65536!}{(16!)^{4096}}\\)"
] | [
"After extending the columns to 65536 and copying the 4096 unique numbers 16 times, how many rows are needed to arrange them in all possible combinations? The result should be smaller than 65536! but larger than (4096!)^16. How to calculate it correctly?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-59a7acb20db65ad161e22f1b756ff02235d9fe660949331181f567f3bebaca0d | After finding the average of $35$ scores, an absent-minded professor carelessly included the average with the $35$ scores and found the average of these $36$ numbers. What was the ratio of the second average to the true average? | [
"1"
] | [
"After finding the average of $35$ scores, an absent-minded professor carelessly included the average with the $35$ scores and found the average of these $36$ numbers. What was the ratio of the second average to the true average?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0decf1b95700fc9191c4c62a131c572b8ceab2aedfe7364fb9c3b7072f65a68b | After finding the average of 35 scores, a student carelessly included the average with the 35 scores and found the average of these 36 numbers. Find the ratio of the second average to the true average. | [
"1"
] | [
"After finding the average of 35 scores, a student carelessly included the average with the 35 scores and found the average of these 36 numbers. Find the ratio of the second average to the true average."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc7fcd8d72380aa0a4178b848c10a4c05515566f9df34a5302d9013c29d02b6f | After finding the average of 35 scores, a student carelessly included the average with the 35 scores and found the average of these 36 numbers. What is the ratio of the second average to the true average? | [
"1:1"
] | [
"After finding the average of 35 scores, a student carelessly included the average with the 35 scores and found the average of these 36 numbers. What is the ratio of the second average to the true average?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc4a0cd81b1d10437c9a410fe76feed6ee71c6d77e5f56ef4e6b07234c6b02ec | After finding the average of 35 scores, a student carelessly included the average with the 35 scores and found the average of these 36 numbers. What was the ratio of the second average to the true average? | [
"1"
] | [
"After finding the average of 35 scores, a student carelessly included the average with the 35 scores and found the average of these 36 numbers. What was the ratio of the second average to the true average?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-353f412a9baae03559a4a29010e2ba13299fe64547a84fb9810187c01701a4ff | After finishing the procedure \( T \), denote by \( c_n \) the probability that the state will become state \( C \) (a total of five balls are put on the triangle) for the first time. Express \( c_n \) in terms of \( n \). | [
"\\dbinom{n-1}{4} \\left( \\dfrac{1}{2} \\right)^n"
] | [
"After finishing the procedure \\( T \\), denote by \\( c_n \\) the probability that the state will become state \\( C \\) (a total of five balls are put on the triangle) for the first time. Express \\( c_n \\) in terms of \\( n \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-d26fb87faf6133b26b15b4235995c3bc2d67ba33eb22b6b5304912ad6165dc31 | After finishing the procedure \( T \), denote by \( d_n \) the probability that the state will become state \( D \) (a total of six balls are put on the triangle) for the first time. For \( n \geq 3 \), express \( d_n \) in terms of \( n \). | [
"\\dfrac{(n-1)(n-2)}{2^{n+1}}"
] | [
"After finishing the procedure \\( T \\), denote by \\( d_n \\) the probability that the state will become state \\( D \\) (a total of six balls are put on the triangle) for the first time. For \\( n \\geq 3 \\), express \\( d_n \\) in terms of \\( n \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-393828bd7f049b31a1df886ebe5733afa68c6dc0bb50735c4de37b9dcf8625e1 | After flipping over how many cards from the top of a standard deck of 52 cards is the probability greater than 0.5 that Hubert has already flipped over all 13 hearts? | [
"50"
] | [
"After flipping over how many cards from the top of a standard deck of 52 cards is the probability greater than 0.5 that Hubert has already flipped over all 13 hearts?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-ad40f22c5c354721027f9fc90f38f23b3af6d01d3a047433ab3686ef71e42496 | After folding a strip of paper consisting of 2048 unit squares in half 11 times (each time folding the right edge over to coincide with the left edge), how many squares lie underneath the square that was originally the 2018th square from the left? | [
"127"
] | [
"After folding a strip of paper consisting of 2048 unit squares in half 11 times (each time folding the right edge over to coincide with the left edge), how many squares lie underneath the square that was originally the 2018th square from the left?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-291a4d103e81e65a01ea372178f2f6229e84191bfff07b7fb794ec5ef0bc4e2b | After how many batches is the probability that Miya and Harshini will be successful in all of them less than 1/2, given that the probability of success for any given batch is 90%? | [
"7"
] | [
"After how many batches is the probability that Miya and Harshini will be successful in all of them less than 1/2, given that the probability of success for any given batch is 90%?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be31ecc518808a83aad6b1060891d868bb8fc680ee0a700899f1c42caa606aae | After how many digits does the decimal representation of $\frac{1}{2007}$ repeat? | [
"222"
] | [
"After how many digits does the decimal representation of $\\frac{1}{2007}$ repeat?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-da6e0c738ea479d720f2ad3fd32adf91def681205797ba254b003c87bbcf1695 | After how many hours does the maximum average BAC occur for the function $C(t) = 0.135t \cdot e^{-2.802t}$ during the first 5 hours? (Round your answer to two decimal places.) | [
"0.36"
] | [
"After how many hours does the maximum average BAC occur for the function $C(t) = 0.135t \\cdot e^{-2.802t}$ during the first 5 hours? (Round your answer to two decimal places.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-ef77ffa0702965469867c0371f514758c036e35b0dd7b81a3d39a780d5fefa92 | After how many numbers written on the paper can we be sure that there are three numbers that start with the same digit and also end with the same digit? (The first and last digits do not have to be the same.) | [
"181"
] | [
"After how many numbers written on the paper can we be sure that there are three numbers that start with the same digit and also end with the same digit? (The first and last digits do not have to be the same.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f2f6fd62209c6f71512e97d6e1ee3a3a173fb67dc6c306b46a43ae8e85b9acbe | After how many steps do we get three numbers with the sum of 2017, starting from the numbers 2, 3, and 5, where in each step one number is increased by 1, another by 2, and the third remains unchanged? | [
"669"
] | [
"After how many steps do we get three numbers with the sum of 2017, starting from the numbers 2, 3, and 5, where in each step one number is increased by 1, another by 2, and the third remains unchanged?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb5c09057a58cbffe63e307d0d96fb9cd172184c54ad3b90adc31e66a6068c3d | After how many terms of the summation $4\sum^{\infty}_{n=0}\frac{(-1)^{n}}{2n+1}$ will the sum be within 0.01 of $\pi$? | [
"200"
] | [
"After how many terms of the summation $4\\sum^{\\infty}_{n=0}\\frac{(-1)^{n}}{2n+1}$ will the sum be within 0.01 of $\\pi$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-46ee04c73083db9d668433bf497ae30ca22fc75031b7e5fc887e219874949175 | After how many tries are you expected to get a head followed by a tail when flipping a coin? | [
"6"
] | [
"After how many tries are you expected to get a head followed by a tail when flipping a coin?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23db99c1dda1262b895e2a0acf633fe655a8db7dcc9761c3c45aa83de880e12b | After how many whole days will Crocodile's total earnings exceed Peter Pan's total earnings, given that Peter Pan wants to be paid $6.40 daily and Crocodile demands to be paid 10 cents on day 1, 20 cents on day 2, 40 cents on day 3, and so on, doubling each day? | [
"9"
] | [
"After how many whole days will Crocodile's total earnings exceed Peter Pan's total earnings, given that Peter Pan wants to be paid $6.40 daily and Crocodile demands to be paid 10 cents on day 1, 20 cents on day 2, 40 cents on day 3, and so on, doubling each day?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2d99abbac05ff3d2d8c73d274cd7a2d056df3cd95cb3e6461a7913a28b6c39b | After identifying the heaviest weight, determine the number of additional weighings necessary to find the second heaviest weight from the remaining 7 weights using a balance without a scale. | [
"2"
] | [
"After identifying the heaviest weight, determine the number of additional weighings necessary to find the second heaviest weight from the remaining 7 weights using a balance without a scale."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b449d8ae9ea5e86be831ddb78d7162c8ce69040e18dc5377a9f676f24bbf19a | After listing all the powers of $2$ from $2^0$ to $2^{2017}$, Vfire observes that $2^{2017}$ has a leading digit of $1$, $250$ numbers have a leading digit of $3$, and $161$ numbers have a leading digit of $5$. Find the amount of numbers in his list that have a leading digit of $1$ or $2$. | [
"965"
] | [
"After listing all the powers of $2$ from $2^0$ to $2^{2017}$, Vfire observes that $2^{2017}$ has a leading digit of $1$, $250$ numbers have a leading digit of $3$, and $161$ numbers have a leading digit of $5$. Find the amount of numbers in his list that have a leading digit of $1$ or $2$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-f724d4f0ae9b7cb286f1a15a4f04eded2eada84ff44a5220f20cd4d0a56ab207 | After listing all the powers of $3$ from $3^0$ to $3^{2014}$, it is observed that $3^{2014}$ has a leading digit of $8$ and $93$ numbers in the list have a leading digit of $9$. Let $A$ be the number of times $1$ appears as the leading digit and let $B$ be the number of times $2$ appears as the leading digit. Calculate... | [
"961"
] | [
"After listing all the powers of $3$ from $3^0$ to $3^{2014}$, it is observed that $3^{2014}$ has a leading digit of $8$ and $93$ numbers in the list have a leading digit of $9$. Let $A$ be the number of times $1$ appears as the leading digit and let $B$ be the number of times $2$ appears as the leading digit. Calc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-19c75773e9448a755d1faff343ec175edaf753d63a53a9838ce028a84935b7bf | After meeting him in the afterlife, Gauss challenges Fermat to a boxing match. Each mathematician is wearing glasses, and Gauss has a $1/3$ probability of knocking off Fermat’s glasses during the match, whereas Fermat has a $1/5$ chance of knocking off Gauss’s glasses. Each mathematician has a $1/2$ chance of losing wi... | [
"\\dfrac{5}{37}"
] | [
"After meeting him in the afterlife, Gauss challenges Fermat to a boxing match. Each mathematician is wearing glasses, and Gauss has a $1/3$ probability of knocking off Fermat’s glasses during the match, whereas Fermat has a $1/5$ chance of knocking off Gauss’s glasses. Each mathematician has a $1/2$ chance of losi... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81ba0f9e50cceaec539d0ea409f8b9f806991e34c3fc2f670dd66b366de9f483 | After multiplying out and simplifying the polynomial \((x - 1)(x^2 - 1)(x^3 - 1) \cdots (x^{2007} - 1)\) and discarding all terms with powers greater than 2007, we obtain a new polynomial \( f(x) \). Find the degree of \( f(x) \) and the coefficient of the term with the highest power. | [
"1962"
] | [
"After multiplying out and simplifying the polynomial \\((x - 1)(x^2 - 1)(x^3 - 1) \\cdots (x^{2007} - 1)\\) and discarding all terms with powers greater than 2007, we obtain a new polynomial \\( f(x) \\). Find the degree of \\( f(x) \\) and the coefficient of the term with the highest power."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c41083ea9ebbfc1ed36d987130ccabee90318b94ce0c05ed44a5b3fd4541df3c | After omitting three consecutive numbers from the sequence of positive integers from \(1\) to \(1993\), it was found that the arithmetic mean of the remaining terms was still an integer. What three numbers were left out? | [
"\\(806, 807, 808\\)"
] | [
"After omitting three consecutive numbers from the sequence of positive integers from \\(1\\) to \\(1993\\), it was found that the arithmetic mean of the remaining terms was still an integer. What three numbers were left out?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fb40261017d19132130212e7716e42a32820496f1a92d2e8e913130fd28c523 | After one of the 4 men leaves, how many ways can Doris split the remaining 7 people into 3 groups, with each group consisting of at least 1 man and at least 1 woman? | [
"36"
] | [
"After one of the 4 men leaves, how many ways can Doris split the remaining 7 people into 3 groups, with each group consisting of at least 1 man and at least 1 woman?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-d924b57ecdafc64c4f7459abcff7c06a4d3e252473d921de4d9e1eb1ba0d505a | After opening the parachute, the new resistance constant is \( k = 1.5 \). Determine the correct differential equation for the new velocity \( v_2(t) \) and the initial condition for this equation. The initial condition should be based on the velocity at the moment the parachute is released, \( v_2(21.7) = 62.8768 \). | [
"\\(\\frac{dv_2}{dt} + 1.5v_2(t) = 9.81\\) with \\( v_2(21.7) = 62.8768 \\)"
] | [
"After opening the parachute, the new resistance constant is \\( k = 1.5 \\). Determine the correct differential equation for the new velocity \\( v_2(t) \\) and the initial condition for this equation. The initial condition should be based on the velocity at the moment the parachute is released, \\( v_2(21.7) = 62... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 24
} | 42 | 0 | 0 |
omr-ws-1fda788c9b67b5f7fec7e11f392c7205899c33d668c6c6005c4a4439aa2ee16f | After passing his license test, Victor is getting a car! He will only buy a car with license plates that use only the digits 1-9, such that each license plate contains exactly 7 digits. On each plate, all digits are distinct, and for all \( k \le 7 \), the kth digit is at least \( k \). Compute the number of valid lice... | [
"36"
] | [
"After passing his license test, Victor is getting a car! He will only buy a car with license plates that use only the digits 1-9, such that each license plate contains exactly 7 digits. On each plate, all digits are distinct, and for all \\( k \\le 7 \\), the kth digit is at least \\( k \\). Compute the number of ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c39b52603fc5e4fea444e6551739f16209d83e018ca5e862dd48de7ee51c3c8 | After simple interest for two months at 5% per annum was credited, a Boy Scout Troop had a total of $255.31 in the Council Treasury. The interest credited was a number of dollars plus the following number of cents. Determine the number of cents. | [
"11"
] | [
"After simple interest for two months at 5% per annum was credited, a Boy Scout Troop had a total of $255.31 in the Council Treasury. The interest credited was a number of dollars plus the following number of cents. Determine the number of cents."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-60eba7daef8f49985f8b63d19ab4ccdcbba4d006b92024b52facda986f567f86 | After simplification, each term of the sequence \( \left( \frac{3}{4} \right), \left( \frac{3}{4} \cdot \frac{8}{9} \right), \left( \frac{3}{4} \cdot \frac{8}{9} \cdot \frac{15}{16} \right), \left( \frac{3}{4} \cdot \frac{8}{9} \cdot \frac{15}{16} \cdot \frac{24}{25} \right), \ldots \) can be written in the form \( \fr... | [
"a = \\left\\lfloor \\dfrac{b}{2} \\right\\rfloor + 1"
] | [
"After simplification, each term of the sequence \\( \\left( \\frac{3}{4} \\right), \\left( \\frac{3}{4} \\cdot \\frac{8}{9} \\right), \\left( \\frac{3}{4} \\cdot \\frac{8}{9} \\cdot \\frac{15}{16} \\right), \\left( \\frac{3}{4} \\cdot \\frac{8}{9} \\cdot \\frac{15}{16} \\cdot \\frac{24}{25} \\right), \\ldots \\) c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 24,
"genselect": 3
} | 27 | 0 | 0 |
omr-ws-b184f792b99a23b05bbfc5764bc25e7ffe37afc0a743e0f6f76104269ba18a6e | After the addition of $n$ layers to an initially circular snowflake, let $x_n$ denote the number of all circles with exactly $3$ neighbors and $y_n$ the number of all circles with exactly $4$ neighbors. Determine all numbers $n$ for which the ratio $\frac{x_n}{y_n}$ has the value $\frac{5}{3}$. Conditions for the snowf... | [
"3"
] | [
"After the addition of $n$ layers to an initially circular snowflake, let $x_n$ denote the number of all circles with exactly $3$ neighbors and $y_n$ the number of all circles with exactly $4$ neighbors. Determine all numbers $n$ for which the ratio $\\frac{x_n}{y_n}$ has the value $\\frac{5}{3}$. \n\nConditions fo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-7f38e19482725b77883cd1ae4fcd038c7cd7a2fa33171f2c33c59d59c9f29800 | After the cards from 1 to 50 are shuffled and laid out in 5 rows of 10 cards each, the cards in each row are rearranged to increase from left to right, and then the cards in each column are rearranged to increase from top to bottom. If the final arrangement is denoted by a 5x10 matrix $M$, calculate the sum of all entr... | [
"55"
] | [
"After the cards from 1 to 50 are shuffled and laid out in 5 rows of 10 cards each, the cards in each row are rearranged to increase from left to right, and then the cards in each column are rearranged to increase from top to bottom. If the final arrangement is denoted by a 5x10 matrix $M$, calculate the sum of all... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-f6085d9bdddb957c4e83bd71730b8b84a2b51e0efa14e224fa4f8ce9181215db | After the first rotation, the sphere is rotated about the line in the \( x-y \) plane perpendicular to \( OP_{1} \), so that the line \( OP \) turns directly towards the positive \( z \)-axis through an angle \( \lambda \). Find the coordinates of \( P_{2} \). Also, find the coordinates of the points \( Q_{2} \) and \(... | [
"\\( P_2(\\cos\\phi\\cos\\lambda, \\sin\\phi, \\cos\\phi\\sin\\lambda) \\), \\( Q_2(-\\sin\\phi\\cos\\lambda, \\cos\\phi, -\\sin\\phi\\sin\\lambda) \\), \\( R_2(-\\sin\\lambda, 0, \\cos\\lambda) \\)"
] | [
"After the first rotation, the sphere is rotated about the line in the \\( x-y \\) plane perpendicular to \\( OP_{1} \\), so that the line \\( OP \\) turns directly towards the positive \\( z \\)-axis through an angle \\( \\lambda \\). Find the coordinates of \\( P_{2} \\). Also, find the coordinates of the points ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a470f76a7829decc42fcd1f29dc6a6a0eef0ba742d8f95a55e9347e8b46b7a4b | After the first round of a mathematical competition, the director of a school asks all students who have participated about their results. The first student reports her score. Each subsequent student answers with the difference in score compared to the student who answered just before them. All answers given are differ... | [
"5"
] | [
"After the first round of a mathematical competition, the director of a school asks all students who have participated about their results. The first student reports her score. Each subsequent student answers with the difference in score compared to the student who answered just before them. All answers given are d... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-c30ed35d5cd6f2517f0d34be20f1008327fdea607487d85aeab69928a1d4cb4c | After the first year, Mr. Money's bank account decreased by 25%, during the second year it increased by 20%, during the third year it decreased by 10%, and during the fourth year it increased by 20%. Does the account of Mr. Money increase or decrease during these four years, and by how much? | [
"decreased by 2.8%"
] | [
"After the first year, Mr. Money's bank account decreased by 25%, during the second year it increased by 20%, during the third year it decreased by 10%, and during the fourth year it increased by 20%. Does the account of Mr. Money increase or decrease during these four years, and by how much?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-c02efbab2831fcd08a9872d11b933cae89b627ab49894cd3fae6ee651c4d8796 | After the graduation exercises at a school the students exchanged photographs. How many students were there, if a total of 870 photographs were exchanged? | [
"30 students"
] | [
"After the graduation exercises at a school the students exchanged photographs. How many students were there, if a total of 870 photographs were exchanged?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97f1fb5253f73892afdc1efe7aeafa0a94ada67d444fca9b8af4181af6eb3859 | After the heater switches off, the temperature of the water decreases at a variable rate $k_2(\theta - 20)$ degrees C per second, where $k_2$ is a constant. Write down a differential equation to represent the situation as the temperature is decreasing. | [
"\\(\\frac{d\\theta}{dt} = -k_{2}(\\theta - 20)\\)"
] | [
"After the heater switches off, the temperature of the water decreases at a variable rate $k_2(\\theta - 20)$ degrees C per second, where $k_2$ is a constant. Write down a differential equation to represent the situation as the temperature is decreasing."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-956b1a926e87020513c9a4b08a9c3d6af7deb900fc76c0677f784c392f900770 | After the letter in one cell was deleted, only $525$ ways to read the word SARAJEVO remained. Find all possible positions of that cell. | [
"4"
] | [
"After the letter in one cell was deleted, only $525$ ways to read the word SARAJEVO remained. Find all possible positions of that cell."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-37221f88227e7b3ea99845883926d5c42ec432d456a5462abbdf0a8782007d76 | After the pie is removed from the oven and placed in a room at 80 °F, it cools to 185 °F in 10 minutes. Determine how much longer it will take for the pie to cool to 125 °F, to the nearest minute. | [
"28"
] | [
"After the pie is removed from the oven and placed in a room at 80 °F, it cools to 185 °F in 10 minutes. Determine how much longer it will take for the pie to cool to 125 °F, to the nearest minute."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35588a211aab4b69dd3d51fd254ae759a1b494259198f8d601840377a6f28912 | After the round of the student $16$, determine the configuration of the doors $1, 2, \dots, 16$ in a game where each student $k$ changes the state of the doors $k, 2k, 4k, 8k, \dots, 2^pk$ with $2^pk \leq 2021 \leq 2^{p+1}k$. | [
"Open, Closed, Open, Open, Closed, Closed, Open, Closed, Open, Open, Open, Open, Open, Open, Open, Open"
] | [
"After the round of the student $16$, determine the configuration of the doors $1, 2, \\dots, 16$ in a game where each student $k$ changes the state of the doors $k, 2k, 4k, 8k, \\dots, 2^pk$ with $2^pk \\leq 2021 \\leq 2^{p+1}k$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 28
} | 47 | 0 | 0 |
omr-ws-4119e0c9e1d95959b9c050679f8854ac17d65dbf2f39eb5bd10fd470bc49d512 | After the round of the student $2021$, determine how many doors are closed in a game where each student $k$ changes the state of the doors $k, 2k, 4k, 8k, \dots, 2^pk$ with $2^pk \leq 2021 \leq 2^{p+1}k$. | [
"673"
] | [
"After the round of the student $2021$, determine how many doors are closed in a game where each student $k$ changes the state of the doors $k, 2k, 4k, 8k, \\dots, 2^pk$ with $2^pk \\leq 2021 \\leq 2^{p+1}k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea466b901089d8a246c92d7fff1898e966b08d7049b19c22bdf7c537c04eb835 | After the teams have been chosen, each team selects a captain, who is equally likely to be any member of that team. Find the conditional distribution of the size of the team containing Player $A$, given that Player $A$ is the captain. | [
"\\frac{m_k}{m}"
] | [
"After the teams have been chosen, each team selects a captain, who is equally likely to be any member of that team. Find the conditional distribution of the size of the team containing Player $A$, given that Player $A$ is the captain."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
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