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-47819f68bd0a98ce975bc52bfa4ae31f05ec31144e1167531bc3e777b6ce20a3 | A positive integer is called *happy* if: 1. All its digits are different and not 0, 2. One of its digits is equal to the sum of the other digits. How many *happy* numbers are there? | [
"2806"
] | [
"A positive integer is called *happy* if:\n1. All its digits are different and not 0,\n2. One of its digits is equal to the sum of the other digits.\n\nHow many *happy* numbers are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3502e0b86145f062c1a4cb74739996012c56fe011378691786d39e53a384960a | A positive integer is called *irregular* if it is not a multiple of any of its digits. For example, $203$ is irregular because $203$ is not a multiple of $2$, $0$, or $3$. Determine the largest possible value of $n$ such that a set of $n$ consecutive positive integers consists entirely of irregular numbers. | [
"3"
] | [
"A positive integer is called *irregular* if it is not a multiple of any of its digits. For example, $203$ is irregular because $203$ is not a multiple of $2$, $0$, or $3$. Determine the largest possible value of $n$ such that a set of $n$ consecutive positive integers consists entirely of irregular numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-bf1476b73f2b67baebcc20f0a563121152b8a636de4acb307d321e64f6948a8e | A positive integer is called *olympic* if it has $n$ digits, all different than zero, and if it is equal to the sum of the $n$-factorials of its digits. Find all positive olympic integers. (Note: The definition of $n$-factorial is provided in the problem statement for clarity.) | [
"1, 2"
] | [
"A positive integer is called *olympic* if it has $n$ digits, all different than zero, and if it is equal to the sum of the $n$-factorials of its digits. Find all positive olympic integers.\n\n(Note: The definition of $n$-factorial is provided in the problem statement for clarity.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c11f5c470389e171e300167fd855226d85c1a969596af72732b29bba2fff4834 | A positive integer is called *piola* if the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9,$ and $10$ is $9$, and its digits are all different and nonzero. How many *piolas* are there between $1$ and $100000$? | [
"7"
] | [
"A positive integer is called *piola* if the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9,$ and $10$ is $9$, and its digits are all different and nonzero. How many *piolas* are there between $1$ and $100000$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-39c863d14f08f2405269145af6c353ef5934b7ae7e13d02f3fc120f5e2a14533 | A positive integer is called Tiranian if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$? | [
"2"
] | [
"A positive integer is called Tiranian if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 46,
"genselect": 8
} | 54 | 0 | 0 |
omr-ws-e918105beeeb86f8fcafa8d70bfc1db9f3f7cff48a3b5ea22f9697d6d42fe715 | A positive integer is called [i]amazing[/i] if it has at most 3 digits when expressed in base 21 and its square, when expressed in base 21, ends in the same 3 digits. Compute the sum of all amazing numbers. Express your answer in base 21. | [
"1002"
] | [
"A positive integer is called [i]amazing[/i] if it has at most 3 digits when expressed in base 21 and its square, when expressed in base 21, ends in the same 3 digits. Compute the sum of all amazing numbers. Express your answer in base 21."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a1facccca033f4c28f2daefae46e3546f7acb7a6bab51548daab79f1dac31a7e | A positive integer is called [i]fancy[/i] if it can be expressed in the form $$2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}},$$ where $a_1,a_2, \cdots, a_{100}$ are non-negative integers that are not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is a [i]fancy[/i] number. | [
"\\( 2^{101} - 1 \\)"
] | [
"A positive integer is called [i]fancy[/i] if it can be expressed in the form $$2^{a_1}+2^{a_2}+ \\cdots+ 2^{a_{100}},$$ where $a_1,a_2, \\cdots, a_{100}$ are non-negative integers that are not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is a [i]fancy[/i] number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-164d584e9a40d106e16a6807cf648a0802a74c82695b5552ed39027c9caaaf08 | A positive integer is called \(\textbf{fantastic}\) if it is divisible by the sum of its digits. Let \( S(k) \) be the smallest value of \( n \) such that \( n, n + 2, n + 4, \dots, n + 2k \) are all fantastic. Find the sum of the digits of \( S(5) \). | [
"2"
] | [
"A positive integer is called \\(\\textbf{fantastic}\\) if it is divisible by the sum of its digits. Let \\( S(k) \\) be the smallest value of \\( n \\) such that \\( n, n + 2, n + 4, \\dots, n + 2k \\) are all fantastic. Find the sum of the digits of \\( S(5) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cdb01ffe84218ed587dfc803de30e231618b02a89fe93c57fd924eba2c6be48 | A positive integer is called a [i]dragon[/i] if it can be written as the sum of four positive integers $a, b, c,$ and $d$ such that $a+4=b-4=4c=d/4$. Find the smallest dragon. | [
"50"
] | [
"A positive integer is called a [i]dragon[/i] if it can be written as the sum of four positive integers $a, b, c,$ and $d$ such that $a+4=b-4=4c=d/4$. Find the smallest dragon."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e311a4163766499031966ed7407d9127f85e3e00ca883e5718ed3131f88da319 | A positive integer is called a [i]lucky number[/i] if it has a value 19 times the sum of its digits. How many different [i]lucky numbers[/i] are there? | [
"11"
] | [
"A positive integer is called a [i]lucky number[/i] if it has a value 19 times the sum of its digits. How many different [i]lucky numbers[/i] are there?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 11
} | 33 | 0 | 0 |
omr-ws-4c8dacbc6b97f71ad27774413d6d667603c38b2cae022fe5138f7aa4de2e56eb | A positive integer is called a straight number if its digits are consecutive integers in increasing order. For instance, 123 and 4567 are straight numbers. Let $S$ be the sum of all 7-digit straight numbers. Find the remainder when $S$ is divided by $1000$. | [
"34"
] | [
"A positive integer is called a straight number if its digits are consecutive integers in increasing order. For instance, 123 and 4567 are straight numbers. Let $S$ be the sum of all 7-digit straight numbers. Find the remainder when $S$ is divided by $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b352efeac5ad1c1459915c57fe259749b8580389781a0bafc6195574cc4f29f7 | A positive integer is called almost-kinda-semi-prime if it has a prime number of positive integer divisors. Given that there are 168 primes less than 1000, how many almost-kinda-semi-prime numbers are there less than 1000? | [
"177"
] | [
"A positive integer is called almost-kinda-semi-prime if it has a prime number of positive integer divisors. Given that there are 168 primes less than 1000, how many almost-kinda-semi-prime numbers are there less than 1000?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61d18a90e3fb67cfe8efd4e8cee4e773976859547e91b89ccebd79d8de39cadc | A positive integer is called balanced if the average of the first two digits is 2, the average of the first three digits is 3, the average of the first four digits is 4, and so on. What is the largest balanced number? | [
"40579"
] | [
"A positive integer is called balanced if the average of the first two digits is 2, the average of the first three digits is 3, the average of the first four digits is 4, and so on. What is the largest balanced number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 5,
"tir": 4
} | 46 | 0 | 0 |
omr-ws-6ffde88c8d7f309ab05d574c93e9f974c30a2ed8ef798883d2d810776ee22b2b | A positive integer is called beautiful if it has an even number of prime factors, and ugly otherwise. Find the smallest positive integer \( n \) such that \( n, n+1, \ldots, n+2016 \) are all beautiful, while \( n-1 \) and \( n+2017 \) are both ugly. Calculate the sum of the prime factors of \( n \). | [
"2018"
] | [
"A positive integer is called beautiful if it has an even number of prime factors, and ugly otherwise. Find the smallest positive integer \\( n \\) such that \\( n, n+1, \\ldots, n+2016 \\) are all beautiful, while \\( n-1 \\) and \\( n+2017 \\) are both ugly. Calculate the sum of the prime factors of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-33e9ca55efd3e8ff66658fdaee444778df075ba70a4747ac136d88cc9de673b2 | A positive integer is called cool if it can be expressed as the sum of two positive cubes in exactly two distinct ways. Let $S$ be the set of all cool numbers. Find the product of all distinct prime factors of the smallest element in $S$. | [
"1729"
] | [
"A positive integer is called cool if it can be expressed as the sum of two positive cubes in exactly two distinct ways. Let $S$ be the set of all cool numbers. Find the product of all distinct prime factors of the smallest element in $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-896ef079f5d029fe4eec3d4f676d0e3a494e4ef1ab2432dc69dbbfeaa8a71d1f | A positive integer is called good if it can be expressed as the sum of two consecutive positive integers. Let $P(n)$ denote the product of two good numbers, $a$ and $b$, where $a = n + (n+1)$ and $b = m + (m+1)$ for some positive integers $n$ and $m$. Find the smallest value of $P(n)$ for $n = 5$. | [
"33"
] | [
"A positive integer is called good if it can be expressed as the sum of two consecutive positive integers. Let $P(n)$ denote the product of two good numbers, $a$ and $b$, where $a = n + (n+1)$ and $b = m + (m+1)$ for some positive integers $n$ and $m$. Find the smallest value of $P(n)$ for $n = 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b337e5e3efc5f86cf069cb99d1fdecd6e965d0f934cecb7c8f7d50e376d0d18 | A positive integer is called happy if it is less than $2023$ and has all distinct digits. Find the number of happy integers. | [
"1249"
] | [
"A positive integer is called happy if it is less than $2023$ and has all distinct digits. Find the number of happy integers."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef4132119a8f7faded4d799264529c0648ff787d9fdfa2f497dc6d01f543beb2 | A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of 5-digit happy integers. | [
"1110"
] | [
"A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of 5-digit happy integers."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-b656e8d45a8acc21d58cc962dfeb718177b95953186a878380765d677eca628c | A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of 5-digit happy numbers. | [
"1100"
] | [
"A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of 5-digit happy numbers."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-230972882a5738c49aad7487345c3b710c9a7a41c304916247c9d547915b6a4f | A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. How many five-digit positive integers are happy? | [
"1110"
] | [
"A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. How many five-digit positive integers are happy?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-e47802fa4dd77a0df5bc0488f8e7c3a89a9bda7883545c09071a6d00e00bc2d7 | A positive integer is called homogeneous if all its digits are the same. Let $S(n)$ denote the sum of all homogeneous numbers less than $10^n$ that are divisible by 2008. Calculate $S(12)$. | [
"0"
] | [
"A positive integer is called homogeneous if all its digits are the same. Let $S(n)$ denote the sum of all homogeneous numbers less than $10^n$ that are divisible by 2008. Calculate $S(12)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5a40fbb17b64773082083e6a36e2c3fcf217944a0eddabea68149b4bdd98cbfb | A positive integer is called joli if it can be written as the arithmetic mean of two or more (not necessarily distinct) powers of two. Find the sum of all distinct joli integers less than $2^{10}$. | [
"523776"
] | [
"A positive integer is called joli if it can be written as the arithmetic mean of two or more (not necessarily distinct) powers of two. Find the sum of all distinct joli integers less than $2^{10}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-40872e0ee8de410595a3b7c5d538882fa14fa751904cd683568352850c3b1aed | A positive integer is called lucky if it can be written as the sum of positive integers whose reciprocals sum to $1$. Let $U(n)$ be the number of unlucky numbers less than or equal to $n$. Find $U(100)$. | [
"6"
] | [
"A positive integer is called lucky if it can be written as the sum of positive integers whose reciprocals sum to $1$. Let $U(n)$ be the number of unlucky numbers less than or equal to $n$. Find $U(100)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-10346285499c0f3f36f6b71d3a0d45e57dc0fcdfaf317b39a812739a465b224e | A positive integer is called lucky if the sum of its digits is 7. Let \( S \) be the set of all lucky integers. Define \( f(a) = 7a \) for any \( a \in S \). Find the sum of all distinct values of \( f(a) \) for \( a \in S \) such that \( f(a) \) is also a lucky integer. | [
"0"
] | [
"A positive integer is called lucky if the sum of its digits is 7. Let \\( S \\) be the set of all lucky integers. Define \\( f(a) = 7a \\) for any \\( a \\in S \\). Find the sum of all distinct values of \\( f(a) \\) for \\( a \\in S \\) such that \\( f(a) \\) is also a lucky integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a82ba0f29f5bca2597a12e9f6813f1a748094f4fd75ff784bee3a6ffa5e25f47 | A positive integer is called polite if it can be written as the sum of two or more consecutive positive integers. Let \( g(n) \) be the smallest positive integer \( k \) such that \( n \) can be expressed as the sum of \( k \) consecutive positive integers if \( n \) is polite, or \( 0 \) if \( n \) is a power of two. ... | [
"292"
] | [
"A positive integer is called polite if it can be written as the sum of two or more consecutive positive integers. Let \\( g(n) \\) be the smallest positive integer \\( k \\) such that \\( n \\) can be expressed as the sum of \\( k \\) consecutive positive integers if \\( n \\) is polite, or \\( 0 \\) if \\( n \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-c75170a1d5d46eb5f23f49b578512f70cbbf7af79115875042b005a30af4e3a3 | A positive integer is called sparkly if it has exactly 9 digits, and for any \( n \) between 1 and 9 (inclusive), the \( n \)-th digit is a positive multiple of \( n \). How many positive integers are sparkly? | [
"17280"
] | [
"A positive integer is called sparkly if it has exactly 9 digits, and for any \\( n \\) between 1 and 9 (inclusive), the \\( n \\)-th digit is a positive multiple of \\( n \\). How many positive integers are sparkly?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f32d2ff8f6febebfd05bc6fe81648dcaaf9138ca8873138afdfa50d8e13f4eee | A positive integer is called superjoli if it can be written as the arithmetic mean of two or more distinct powers of two. Let \( S(n) \) be the smallest superjoli number greater than \( 2^n \). Find \( S(10) \). | [
"1025"
] | [
"A positive integer is called superjoli if it can be written as the arithmetic mean of two or more distinct powers of two. Let \\( S(n) \\) be the smallest superjoli number greater than \\( 2^n \\). Find \\( S(10) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c03ac954746e02443c752dd5d1f62bb5acfaf7fe1c6584461c3d9c21e2feba99 | A positive integer is called superprime if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers. | [
"\\( 1 \\) and \\( 3 \\)"
] | [
"A positive integer is called superprime if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 12
} | 22 | 0 | 0 |
omr-ws-d19a3b43efbc73db33caacd2d8f32a57b53413149884c13bfa2dde176f385ea6 | A positive integer is called tricubic if it is the cube of a positive integer. Find the smallest positive integer $n$ such that exactly two of the three numbers $n, n+2,$ and $n+28$ are tricubic. | [
"26"
] | [
"A positive integer is called tricubic if it is the cube of a positive integer. Find the smallest positive integer $n$ such that exactly two of the three numbers $n, n+2,$ and $n+28$ are tricubic."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-331dad9116579541326927b643547aac4e2cf7f66fb8db21cc3f9e6c3590afa4 | A positive integer is considered cascading if its digits when expressed in base 10 are strictly decreasing from left to right. For example, 8420, 763, and 5 are cascading numbers, but 8843 and 123 are not. What is the sum of all cascading numbers? You may express your answer in base 11, where the letter \(A\) represent... | [
"A801548430"
] | [
"A positive integer is considered cascading if its digits when expressed in base 10 are strictly decreasing from left to right. For example, 8420, 763, and 5 are cascading numbers, but 8843 and 123 are not. What is the sum of all cascading numbers? You may express your answer in base 11, where the letter \\(A\\) re... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-7bbe68beb1fea98a39bd17c3f3f4e7b72ae039f5ac130b5f2f097af6e7878333 | A positive integer is defined as "bold" if it has 8 positive divisors that sum up to 3240. Find the smallest positive bold number. | [
"1614"
] | [
"A positive integer is defined as \"bold\" if it has 8 positive divisors that sum up to 3240. Find the smallest positive bold number."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-4289d99e11892e69918f56a57cfff40fb688f2d86e8d05860c1be6c3751b36fb | A positive integer is defined as Noble if it is composite, not divisible by any prime number greater than $20$, and not divisible by any perfect cube greater than $1$. How many different Noble numbers are there? | [
"6552"
] | [
"A positive integer is defined as Noble if it is composite, not divisible by any prime number greater than $20$, and not divisible by any perfect cube greater than $1$. How many different Noble numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-09c0e83ae936097850cb170e5168bcadb0c4bbf87887fcf835f0dadbbbf00317 | A positive integer is defined as [b]superb[/b] if it is the least common multiple of $1, 2, \ldots, n$ for some positive integer $n$. Find all superb integers $x, y, z$ such that $x + y = z$. | [
"\\((x,y,z)=(\\text{lcm}(1,2,\\dots,2^k-1),\\text{lcm}(1,2,\\dots,2^k-1),\\text{lcm}(1,2,\\dots,2^k)), k\\in \\mathbb{N}\\)"
] | [
"A positive integer is defined as [b]superb[/b] if it is the least common multiple of $1, 2, \\ldots, n$ for some positive integer $n$. Find all superb integers $x, y, z$ such that $x + y = z$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-38f1889bd2e4938617450609296818630e04ace6557541a5bfbbfd01babe70ea | A positive integer is defined as [i]four-happy[/i] if it is equal to the sum of the squares of its four smallest positive divisors. Find all positive integers that are four-happy. | [
"130"
] | [
"A positive integer is defined as [i]four-happy[/i] if it is equal to the sum of the squares of its four smallest positive divisors. Find all positive integers that are four-happy."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-021ca8594dac01eab5bebd597fd32290c050ccb2a4721cd428b111b4b81b6711 | A positive integer is equal to the sum of its four smallest positive divisors. What is the largest prime that divides this positive integer? | [
"13"
] | [
"A positive integer is equal to the sum of its four smallest positive divisors. What is the largest prime that divides this positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-8d7731d717601e2883a86fcf27fefe574fca92aa05eb9efbb6871aa069b2328a | A positive integer is equal to the sum of the squares of its first four positive divisors. What is the largest prime which divides this positive integer? | [
"13"
] | [
"A positive integer is equal to the sum of the squares of its first four positive divisors. What is the largest prime which divides this positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-51dc42a186b33abf251d05d564008370e5609b00b960daa67e9a5bfe81ff4cf9 | A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer? | [
"13"
] | [
"A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 79,
"tir": 11
} | 90 | 0 | 0 |
omr-ws-e000707164a2e389bc9b590645b617b954eaf02d5442a8063dd11c8b22e9f3cc | A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit while keeping the order of all the remaining digits intact, the resulting number is exactly double the original number. Find the smallest possible nefelibata number. | [
"105263157894736842"
] | [
"A positive integer is said to be \"nefelibata\" if, upon taking its last digit and placing it as the first digit while keeping the order of all the remaining digits intact, the resulting number is exactly double the original number. Find the smallest possible nefelibata number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-bcd85a49423bfd17d4d3c20dbe34c416ee236986d7ba13be5b5df4fbac030233 | A positive integer is said to be *friendly* if it is composed of the digits in the set $\{1,2\}$ and the sum of its digits is equal to $12$. How many friendly positive integers are there? | [
"233"
] | [
"A positive integer is said to be *friendly* if it is composed of the digits in the set $\\{1,2\\}$ and the sum of its digits is equal to $12$. How many friendly positive integers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-43943a50a81c491563b35b957fbe8db3adf97247e6286b33d3915ebe47895f7a | A positive integer is said to be binary-emulating if its base three representation consists of only 0s and 1s. Determine the sum of the first 15 binary-emulating numbers. | [
"320"
] | [
"A positive integer is said to be binary-emulating if its base three representation consists of only 0s and 1s. Determine the sum of the first 15 binary-emulating numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-436100e90f33f05158e942e3ff45400d97fbb8234b14004c9b19871e04dae446 | A positive integer is said to be special if the sum of the remainders obtained when it is divided by five consecutive positive integers is $32$. Find the smallest positive integer that is special. | [
"24"
] | [
"A positive integer is said to be special if the sum of the remainders obtained when it is divided by five consecutive positive integers is $32$. Find the smallest positive integer that is special."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-b8bd1a1fc78e34195e870b18d2f81ab7d29ef7a0375820fb8b0f408f63b4a5d7 | A positive integer is said to be square-containing if the squares of each of its digits appear somewhere in the number. For example, if a number has the digit 5, it must contain the digit 2 immediately followed by the digit 5. For example, the number 1648 is not square-containing because it does not contain 36 even tho... | [
"81649025367"
] | [
"A positive integer is said to be square-containing if the squares of each of its digits appear somewhere in the number. For example, if a number has the digit 5, it must contain the digit 2 immediately followed by the digit 5. For example, the number 1648 is not square-containing because it does not contain 36 eve... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-21a8781a100fc518b076c6898017a8fadb52f349a3eb7362748d016dc0db6c9c | A positive integer is such that both adding $2000$ and subtracting $17$ results in a perfect square. Determine that number. | [
"1016081"
] | [
"A positive integer is such that both adding $2000$ and subtracting $17$ results in a perfect square. Determine that number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52f5a6cccce6ed4de24c2d15ee8720bfd729cfe772fa3e0339186d9d7d911f42 | A positive integer is the product of \( n \) distinct primes. In how many ways can it be expressed as the difference of two squares? | [
"\\(2^{n-1}\\)"
] | [
"A positive integer is the product of \\( n \\) distinct primes. In how many ways can it be expressed as the difference of two squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-8b70a5b4b0c98cb705c11847e6f3bd9f2aba4d2993353095a2a1ae163f2b8a48 | A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron, the greatest common divisor (GCD) of its endpoints is written. If the numbers written on the edges are six consecutive integers, find the smallest possible value of the largest integer written on a vertex of the tetrah... | [
"12"
] | [
"A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron, the greatest common divisor (GCD) of its endpoints is written. If the numbers written on the edges are six consecutive integers, find the smallest possible value of the largest integer written on a vertex of the t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-68003b22463de9443426ce80f4f5b08c312cec6f678b97300a432f396a5cfc8c | A positive integer is written in each square of a $2010$ by $2010$ chessboard. The difference between the numbers in any two adjacent squares (sharing the same edge) is less than or equal to $2010$. Let $N$ be the minimum number of squares that must contain the same number for any such arrangement. Find the value of $N... | [
"2010"
] | [
"A positive integer is written in each square of a $2010$ by $2010$ chessboard. The difference between the numbers in any two adjacent squares (sharing the same edge) is less than or equal to $2010$. Let $N$ be the minimum number of squares that must contain the same number for any such arrangement. Find the value ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-5bdb15ce0840ced836b020be6f6578785222fa26c786f4904286c44f679f7453 | A positive integer is written in each square of a $2010^2 \times 2010^2$ chessboard. The difference between the numbers in any two adjacent squares (sharing the same edge) is less than or equal to $2010$. Find the minimum number of squares that must contain the same number. | [
"1005"
] | [
"A positive integer is written in each square of a $2010^2 \\times 2010^2$ chessboard. The difference between the numbers in any two adjacent squares (sharing the same edge) is less than or equal to $2010$. Find the minimum number of squares that must contain the same number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-7811cb15eef9639dee289a47a0053c5d51fbafb6cc3fbb6be4c0988c27170ba5 | A positive integer is written in each square of a $2010^2 \times 2010^2$ chessboard. The difference between the numbers in any two adjacent squares (sharing the same edge) is less than or equal to $2010$. Find the minimum possible value of $k$ such that at least $k$ squares contain the same number. | [
"1005"
] | [
"A positive integer is written in each square of a $2010^2 \\times 2010^2$ chessboard. The difference between the numbers in any two adjacent squares (sharing the same edge) is less than or equal to $2010$. Find the minimum possible value of $k$ such that at least $k$ squares contain the same number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-32a4719a4e8bcfd96f6e343a9be214744e1685cd53b2a117ada8b4b427de9154 | A positive integer is written on the back of each of three puzzle pieces. The numbers are not necessarily different but the total of the three numbers is 14. Alpha, Beta, and Gamma each select one of the pieces without seeing the numbers. Alpha says, "I know that Beta and Gamma have different numbers." Beta then says, ... | [
"\\( (A, B, C) = (1, 7, 6) \\)"
] | [
"A positive integer is written on the back of each of three puzzle pieces. The numbers are not necessarily different but the total of the three numbers is 14. Alpha, Beta, and Gamma each select one of the pieces without seeing the numbers. Alpha says, \"I know that Beta and Gamma have different numbers.\" Beta then... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 23
} | 36 | 0 | 0 |
omr-ws-3971e4fb0c7ceee562d8d1b0995610028f3ef2890f6cf143660ad1054d550855 | A positive integer leaves a remainder of 2 when divided by 9, a remainder of 5 when divided by 11, and a remainder of 8 when divided by 12. What is the remainder when the number is divided by 66? | [
"38"
] | [
"A positive integer leaves a remainder of 2 when divided by 9, a remainder of 5 when divided by 11, and a remainder of 8 when divided by 12. What is the remainder when the number is divided by 66?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-f23d355616c7a894f0ed4f176e5f406b65b635ddc31cb808c0182c07461a40dd | A positive integer less than \( 10^{10^{100}} \) is chosen randomly. To the nearest centillionth, what is the probability that this chosen number is less than or equal to \( \lfloor 10^{10^{99.99999999}} \rfloor \)? | [
"0"
] | [
"A positive integer less than \\( 10^{10^{100}} \\) is chosen randomly. To the nearest centillionth, what is the probability that this chosen number is less than or equal to \\( \\lfloor 10^{10^{99.99999999}} \\rfloor \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7efd7c7dc3e426e209cf15ef34175e61a44882a676d1b60152afec97466e6fb2 | A positive integer m "covers" the number 1998 if 1, 9, 9, 8 appear in this order as digits of m. Let k(n) be the number of positive integers that cover 1998 and have exactly n digits (n > 4), all different from 0. What is the remainder of k(n) when divided by 8? | [
"1"
] | [
"A positive integer m \"covers\" the number 1998 if 1, 9, 9, 8 appear in this order as digits of m. Let k(n) be the number of positive integers that cover 1998 and have exactly n digits (n > 4), all different from 0. What is the remainder of k(n) when divided by 8?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-2e7ebef6a2f3432a54fba69fb22da7c79b255e75aa582530b9b099ca5569a2d7 | A positive integer number is called "almost square" if it is the product of two consecutive numbers. A positive integer number is called "can show" if it is the quotient of one "almost square" number divided by another "almost square" number. Let $g(n)$ be the number of distinct "can show" numbers that can be formed by... | [
"3"
] | [
"A positive integer number is called \"almost square\" if it is the product of two consecutive numbers. A positive integer number is called \"can show\" if it is the quotient of one \"almost square\" number divided by another \"almost square\" number. Let $g(n)$ be the number of distinct \"can show\" numbers that c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff82e221725fa4178ddb91ea195f5eeaf2460f3a1c40aac9bbbfee15f1cb1f3e | A positive integer that cannot be expressed as a sum of two or more consecutive positive integers is called $\textit{interesting}$. What are all the $\textit{interesting integers}$? | [
"\\(1, 2, 4, 8, 16, \\ldots\\) or more formally, \\(2^s\\) where \\(s \\geq 0\\)."
] | [
"A positive integer that cannot be expressed as a sum of two or more consecutive positive integers is called $\\textit{interesting}$. What are all the $\\textit{interesting integers}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-9277b6874265b7440cd4b184e7e9c29997c344ebb31a30783077a9c56fb34d08 | A positive integer was written into every cell of a 10×10 table. Each number that is either smaller than all of its neighbors or bigger than all of its neighbors was colored. It turned out that just two numbers in the entire table were not colored, and neither of them is located at a corner. Find the smallest possible ... | [
"4"
] | [
"A positive integer was written into every cell of a 10×10 table. Each number that is either smaller than all of its neighbors or bigger than all of its neighbors was colored. It turned out that just two numbers in the entire table were not colored, and neither of them is located at a corner. Find the smallest poss... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 23,
"genselect": 8
} | 31 | 0 | 0 |
omr-ws-8813841ab3ea6f4c5370aeff2577444349c0be469939d89d511be2c139fe2fff | A positive integer which contains no prime factors except 2 and 3 can be divided by exactly 24 different numbers and its square can be divided by exactly 77 different numbers. What is the integer? | [
"864"
] | [
"A positive integer which contains no prime factors except 2 and 3 can be divided by exactly 24 different numbers and its square can be divided by exactly 77 different numbers. What is the integer?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-2fe6fb7b4f201c9aef18cbc7174660cba6926b2d44922d7dcf3c9af3f1abf910 | A positive integer which contains no prime factors except 2 and 3 can be divided by exactly 24 different numbers, and its square can be divided by exactly 77 different numbers. What is the integer? | [
"864 or 1944"
] | [
"A positive integer which contains no prime factors except 2 and 3 can be divided by exactly 24 different numbers, and its square can be divided by exactly 77 different numbers. What is the integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 4
} | 26 | 0 | 0 |
omr-ws-aca95065391bd0f114e82ee4faa7bd39359ee2945ca6107764ad3c09f3354376 | A positive integer with 3 digits $\overline{ABC}$ is called $Lusophon$ if $\overline{ABC}+\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers. | [
"\\(\\{110, 143, 242, 341, 440, 164, 263, 362, 461, 560, 198, 297, 396, 495, 594, 693, 792, 891, 990\\}\\)"
] | [
"A positive integer with 3 digits $\\overline{ABC}$ is called $Lusophon$ if $\\overline{ABC}+\\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-afd6de758fcda5f83bf0ba94bb6b01445eaecd1a62378a23ed3f32a600f1a0eb | A positive natural number \( n \) is said to be *comico* if its prime factorization is \( n = p_1 p_2 \cdots p_k \) with \( k \ge 3 \), and the primes \( p_1, p_2, \ldots, p_k \) satisfy the conditions: \[ p_1 + p_2 = c_1^2, \] \[ p_1 + p_2 + p_3 = c_2^2, \] \[ \vdots \] \[ p_1 + p_2 + \cdots + p_k = c_{k-1}^2, \] wher... | [
"9911"
] | [
"A positive natural number \\( n \\) is said to be *comico* if its prime factorization is \\( n = p_1 p_2 \\cdots p_k \\) with \\( k \\ge 3 \\), and the primes \\( p_1, p_2, \\ldots, p_k \\) satisfy the conditions:\n\\[ p_1 + p_2 = c_1^2, \\]\n\\[ p_1 + p_2 + p_3 = c_2^2, \\]\n\\[ \\vdots \\]\n\\[ p_1 + p_2 + \\cdo... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-8a88e4e263fbccb60213c0cd111a9219b57e9c9deecedbeb47ffaa217b6d8a6a | A positive number \(N\) has 3 digits when expressed in base 9. When \(N\) is expressed in base 6, the digits are reversed in order. Find the middle digit. | [
"5"
] | [
"A positive number \\(N\\) has 3 digits when expressed in base 9. When \\(N\\) is expressed in base 6, the digits are reversed in order. Find the middle digit."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2adcd984203960ed3fd585bd51d1a33681c213ff2c1d26c6548f9f0a35c99fdb | A positive number has a decimal part and an integer part. The decimal part, the integer part, and the positive number itself form a geometric sequence. Find this positive number. | [
"\\(\\frac{1+\\sqrt{5}}{2}\\)"
] | [
"A positive number has a decimal part and an integer part. The decimal part, the integer part, and the positive number itself form a geometric sequence. Find this positive number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-a5267bd6cfa0d9040ed0d5809c56a64bf68e03a9d6d01307a6318a8ecca7d4f8 | A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000? | [
"28"
] | [
"A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60c93c6787a407a8fa2b596918d0bd0a70066b08e5da959053e6427a3ed63345 | A positive number is said to be [i]twelvish[/i] if the sum of its digits in its decimal representation is equal to $12$. What is the total number of [i]twelvish[/i] numbers from $1$ to $999$? | [
"79"
] | [
"A positive number is said to be [i]twelvish[/i] if the sum of its digits in its decimal representation is equal to $12$. What is the total number of [i]twelvish[/i] numbers from $1$ to $999$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ccaa221cd4000b72ccaa8d7f0c862ced92f27b2db4651b9b027467a417050b8 | A positive number was rounded up to the nearest integer and became a number that is 28% larger than the original number. Find the original number (find all solutions). | [
"\\(\\frac{25}{32}, \\frac{25}{16}, \\frac{75}{32}, \\frac{25}{8}\\)"
] | [
"A positive number was rounded up to the nearest integer and became a number that is 28% larger than the original number. Find the original number (find all solutions)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-a4f06fa54c4a744c0cec4016784550553bc58a08e9203eb529cb652e86eafaaf | A positive rational number $r_0$ is written on the blackboard. Every minute, Vasya replaces the number $r_n$ with $r_{n+1} = \sqrt{r_n + 1}$. After $k$ minutes, Vasya obtains the number $r_k$. Find the smallest positive integer $k$ such that $r_k$ is guaranteed to be irrational, given that $r_0 = \frac{1}{2}$. | [
"1"
] | [
"A positive rational number $r_0$ is written on the blackboard. Every minute, Vasya replaces the number $r_n$ with $r_{n+1} = \\sqrt{r_n + 1}$. After $k$ minutes, Vasya obtains the number $r_k$. Find the smallest positive integer $k$ such that $r_k$ is guaranteed to be irrational, given that $r_0 = \\frac{1}{2}$."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-814b79dc54166d55eb8e311b2e2683c94496c51920edd98a90d1dfcc30315fc1 | A positive rational number \( x \) is called *banzai* if the following conditions are met: - \( x = \frac{p}{q} > 1 \) where \( p \) and \( q \) are coprime natural numbers. - There exist constants \( \alpha \) and \( N \) such that for all integers \( n \geq N \), \[ \left| \left\{ x^n \right\} - \alpha \right| \leq \... | [
"\\infty"
] | [
"A positive rational number \\( x \\) is called *banzai* if the following conditions are met:\n- \\( x = \\frac{p}{q} > 1 \\) where \\( p \\) and \\( q \\) are coprime natural numbers.\n- There exist constants \\( \\alpha \\) and \\( N \\) such that for all integers \\( n \\geq N \\),\n\\[ \\left| \\left\\{ x^n \\r... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11c612b1af98760e1125210ab3ab8468c76df798d19d2fd8bb17afaa05a84c7d | A positive real number $k$, a triangle $ABC$ with circumcircle $\omega$, and a point $M$ on its side $AB$ are fixed. The point $P$ moves along $\omega$, and $Q$ on segment $CP$ is such that $CQ : QP = k$. The line through $P$, parallel to $CM$, intersects the line $MQ$ at point $N$. If the radius of the circle on which... | [
"R^2"
] | [
"A positive real number $k$, a triangle $ABC$ with circumcircle $\\omega$, and a point $M$ on its side $AB$ are fixed. The point $P$ moves along $\\omega$, and $Q$ on segment $CP$ is such that $CQ : QP = k$. The line through $P$, parallel to $CM$, intersects the line $MQ$ at point $N$. If the radius of the circle o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b47a61b6b2124bd524bd9ce5ed91e888b5dbf1d9a6c3fa01ef54171fbfdde106 | A positive real number $x$ is such that \[ \sqrt[3]{1-x^3} + \sqrt[3]{1+x^3} = 1. \]Find $x^6.$ | [
"\\frac{28}{27}"
] | [
"A positive real number $x$ is such that \\[\n\\sqrt[3]{1-x^3} + \\sqrt[3]{1+x^3} = 1.\n\\]Find $x^6.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a7a64ba4c1b0c9291721b41d1bab0478aa24220fd06706ea82e68bacc1728ee | A positive real number is written to each vertex of a regular 2019-gon. The square of the number written at any vertex is equal to the sum of the two numbers written at its two adjacent vertices. If the number at one vertex is 1, find the sum of all the numbers written at the vertices. | [
"4038"
] | [
"A positive real number is written to each vertex of a regular 2019-gon. The square of the number written at any vertex is equal to the sum of the two numbers written at its two adjacent vertices. If the number at one vertex is 1, find the sum of all the numbers written at the vertices."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-34c77ff43363abb5565eaab1e59e6b00f40f1639d6f98bd70b58573f1b6d56e2 | A positive two-digit number is even and is a multiple of 11. The product of its digits is a perfect cube. What is this two-digit number? | [
"88"
] | [
"A positive two-digit number is even and is a multiple of 11. The product of its digits is a perfect cube. What is this two-digit number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a752936dbf686e6e2100cb858506614d1ae639f44ca2a8975eb7c6de929b110 | A post office is staffed by two clerks, and the time each clerk spends with a customer is exponentially distributed with parameter $\lambda$. Mr. Smith enters the post office and finds each clerk is busy with a customer. No other customers are waiting. What is the probability that Mr. Smith is the last of the three cus... | [
"$\\frac{1}{2}$"
] | [
"A post office is staffed by two clerks, and the time each clerk spends with a customer is exponentially distributed with parameter $\\lambda$. Mr. Smith enters the post office and finds each clerk is busy with a customer. No other customers are waiting. What is the probability that Mr. Smith is the last of the thr... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 24
} | 38 | 0 | 0 |
omr-ws-2ab312b970693e5a5263993db18ccbef64011c311d80b95f8ac1f791faa10bfe | A poster is taped flush on a wall that is leaning toward the observer at an angle of 15 degrees off vertical. From a point on the floor 6 feet from the base of the wall, the line of sight to the top of the poster makes an angle of 45 degrees with the floor, and the line of sight to the bottom of the poster makes an ang... | [
"\\( 3\\sqrt{2} - \\sqrt{6} \\)"
] | [
"A poster is taped flush on a wall that is leaning toward the observer at an angle of 15 degrees off vertical. From a point on the floor 6 feet from the base of the wall, the line of sight to the top of the poster makes an angle of 45 degrees with the floor, and the line of sight to the bottom of the poster makes a... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-31e08a475f8146b30d820352a3c767da3e5731727f82582adda6e1324564f74f | A potter forms a piece of clay into a cylinder. As he rolls it, the length, $L$, of the cylinder increases and the radius, $r$, decreases. If the length of the cylinder is increasing at $0.1$ cm per second, find the rate at which the radius is changing when the radius is $1$ cm and the length is $5$ cm. | [
"\\(-0.01\\) cm/s"
] | [
"A potter forms a piece of clay into a cylinder. As he rolls it, the length, $L$, of the cylinder increases and the radius, $r$, decreases. If the length of the cylinder is increasing at $0.1$ cm per second, find the rate at which the radius is changing when the radius is $1$ cm and the length is $5$ cm."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-f5ba0ef0135dcfaa77d942e98dda6e0aff0afea83e41fbb28c2353ab4553e188 | A potter makes a bowl, beginning with a sphere of clay and then cutting away the top and bottom of the sphere with two parallel cuts that are equidistant from the center. Finally, he hollows out the remaining shape, leaving a base at one end. Assume that the thickness of the bowl is negligible. If the potter wants the ... | [
"\\(\\frac{1}{3}\\)"
] | [
"A potter makes a bowl, beginning with a sphere of clay and then cutting away the top and bottom of the sphere with two parallel cuts that are equidistant from the center. Finally, he hollows out the remaining shape, leaving a base at one end. Assume that the thickness of the bowl is negligible. If the potter wants... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-ae7e42c388173a27d4b03c008d7a04870f10f480d760f896121a795b68cce141 | A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft ... | [
"4.5"
] | [
"A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-f9a2c253bf986b66c5bc02644848bea5603c586ad03cfacb6c4508b047ed7308 | A power of two is written. One of its digits is changed into a different digit, and the resulting number's digits can be rearranged to form a power of 3. Find the sum of all such powers of two. Do there exist an infinite amount? For example, $2^5=32 \implies 72 \implies 27=3^3$. Inspiration for this problem: $2^{14}=16... | [
"24638"
] | [
"A power of two is written. One of its digits is changed into a different digit, and the resulting number's digits can be rearranged to form a power of 3. Find the sum of all such powers of two. Do there exist an infinite amount? \n\nFor example, $2^5=32 \\implies 72 \\implies 27=3^3$.\n\nInspiration for this probl... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-4beb46144511bf0fb63b167ef9ee3c11ab277e24323e431afe1368cc7b4f848f | A power station and a factory are on opposite sides of a river 60 meters wide. A cable must be run from the power station to the factory. It costs 25 dollars per meter to run the cable in the river and 20 dollars per meter on land. a. Express the total cost \( C \) as a function of \( x \), the distance downstream from... | [
"4900"
] | [
"A power station and a factory are on opposite sides of a river 60 meters wide. A cable must be run from the power station to the factory. It costs 25 dollars per meter to run the cable in the river and 20 dollars per meter on land.\na. Express the total cost \\( C \\) as a function of \\( x \\), the distance downs... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b680d4dac7e5c4dc3f9509505ac24c7b2b4fe43b418eb3a319b699034cc864b6 | A power station is on one side of a river that is 0.5 miles wide, and a factory is 6 miles downstream on the other side of the river. It costs $6 per foot to run overland power lines and $8 per foot to run underwater power lines. Write a cost function for running the power line from the power station to the factory. | [
"\\( 5280\\left[6(6-x)+8\\sqrt{x^2+\\frac{1}{4}}\\right] \\)"
] | [
"A power station is on one side of a river that is 0.5 miles wide, and a factory is 6 miles downstream on the other side of the river. It costs $6 per foot to run overland power lines and $8 per foot to run underwater power lines. Write a cost function for running the power line from the power station to the factor... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-1880bc418f0903c0565e119eb8db9d18c991eec0ed76fc5d4f6ee61c1626a7c6 | A praying mantis is located at (3, 3) in the coordinate plane. On each hop, it moves from (x, y) to either (x + 1, y), (x, y + 1), or (x + 1, y + 1). How many different paths can the praying mantis take to get to (7, 7)? | [
"321"
] | [
"A praying mantis is located at (3, 3) in the coordinate plane. On each hop, it moves from (x, y) to either (x + 1, y), (x, y + 1), or (x + 1, y + 1). How many different paths can the praying mantis take to get to (7, 7)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-15c7cbe93689ad1ad89758297d8a49a928ef8eec466008e51c2de0ff9aa77679 | A pre-release booster pack for a trading card game contains 1 of 5 different new cards. How many such packs, on average, do you need to purchase in order to obtain at least one of each of the 5 new cards? | [
"11.4165"
] | [
"A pre-release booster pack for a trading card game contains 1 of 5 different new cards. How many such packs, on average, do you need to purchase in order to obtain at least one of each of the 5 new cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-a8aa745ef75d9b78a9a83e9c7aa8413f009a3c8efc95dd167c7099d2b6bedbd5 | A prim number is a number that is prime if its last digit is removed. A rime number is a number that is prime if its first digit is removed. Determine how many numbers between \(100\) and \(999\) inclusive are both prim and rime numbers. | [
"52"
] | [
"A prim number is a number that is prime if its last digit is removed. A rime number is a number that is prime if its first digit is removed. Determine how many numbers between \\(100\\) and \\(999\\) inclusive are both prim and rime numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-7a619da4c9c4897918a6d21317558ad84b0d8c6587f79731802ceb2d7512240d | A primary water tank is a sphere with a 4.5 metre radius. It needs to be filled so that the water is exactly 6.5 metres deep at the center. If one bucket of water holds exactly 10 litres, how many bucketfuls of water are required to fill the tank to the required depth? Round up to the nearest whole number. | [
"254406"
] | [
"A primary water tank is a sphere with a 4.5 metre radius. It needs to be filled so that the water is exactly 6.5 metres deep at the center. If one bucket of water holds exactly 10 litres, how many bucketfuls of water are required to fill the tank to the required depth? Round up to the nearest whole number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f038ee20d6ba9b07b3e6048c6b18f6e58655810ed68df3d1f4e155e76f80849 | A prime $p$ is called $\textit{philé}$ if there exists a polynomial $P(x) = ax^3 + bx^2 + cx + d$ with non-negative integer coefficients $a, b, c, d < p$ such that the set $\{P(n) \mid 1 \leq n \leq p\}$ forms a complete residue system modulo $p$. Find all $\textit{philé}$ primes. Note: A set $A$ is a complete residue ... | [
"3 and primes of the form $3k+2$"
] | [
"A prime $p$ is called $\\textit{philé}$ if there exists a polynomial $P(x) = ax^3 + bx^2 + cx + d$ with non-negative integer coefficients $a, b, c, d < p$ such that the set $\\{P(n) \\mid 1 \\leq n \\leq p\\}$ forms a complete residue system modulo $p$. Find all $\\textit{philé}$ primes.\n\nNote: A set $A$ is a co... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 6,
"tir": 9
} | 53 | 0 | 0 |
omr-ws-ed063198c0342a45db84a1f434a1a7d5c5a10308a0b4be82ce9a037bb72ee899 | A prime $p$ is called $n$-$\textit{rephinado}$ if $n \mid p - 1$ and all integers from $1$ to $\lfloor \sqrt[\delta]{p} \rfloor$ are $n$-th residues modulo $p$, where $\delta = \varphi + 1$ and $\varphi = \frac{1 + \sqrt{5}}{2}$. Determine if there are infinitely many $n$ for which there are infinitely many $n$-$\texti... | [
"No"
] | [
"A prime $p$ is called $n$-$\\textit{rephinado}$ if $n \\mid p - 1$ and all integers from $1$ to $\\lfloor \\sqrt[\\delta]{p} \\rfloor$ are $n$-th residues modulo $p$, where $\\delta = \\varphi + 1$ and $\\varphi = \\frac{1 + \\sqrt{5}}{2}$. Determine if there are infinitely many $n$ for which there are infinitely ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 7,
"tir": 25
} | 60 | 0 | 0 |
omr-ws-020418653183a0857262ad59370d45a46d957b8a2035a6e233158c57c0c1c968 | A prime $p$ is such that $\frac{1}{p}$ when represented in octal (base $8$) notation equals $0.\overline{00331}$. What is the order of $2$ in the group of units of the ring of integers modulo $p$? | [
"15"
] | [
"A prime $p$ is such that $\\frac{1}{p}$ when represented in octal (base $8$) notation equals $0.\\overline{00331}$. What is the order of $2$ in the group of units of the ring of integers modulo $p$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-5c8d2e25914904e6cf0e3f1de7e855867161ff3259939a017f535aa472508a75 | A prime \( p \) is said to be awesome if there exist positive integers \( a \) and \( b \) such that the average of the divisors of \( 31^a p^b \) is an integer. Determine the number of awesome primes that are less than 100. | [
"24"
] | [
"A prime \\( p \\) is said to be awesome if there exist positive integers \\( a \\) and \\( b \\) such that the average of the divisors of \\( 31^a p^b \\) is an integer. Determine the number of awesome primes that are less than 100."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-24b7e0f50c309da37e14c335189d0563643f05d7fe0462f1f6ea36b97ced90d1 | A prime \(p\) is "good" if it divides the number of the form \((x+y)^6 - 3xy(x+y)^4 - x^3y^3\) for \(\gcd(x, y) = 1\). Determine which primes are "good" based on this condition. | [
"5"
] | [
"A prime \\(p\\) is \"good\" if it divides the number of the form \\((x+y)^6 - 3xy(x+y)^4 - x^3y^3\\) for \\(\\gcd(x, y) = 1\\). Determine which primes are \"good\" based on this condition."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-dcd16b2f69081fbcefb6ae7579081f24af44ba06feca7b15d5c789da30ceaf76 | A prime \(p\) is "good" if it divides the number of the form \((xy)^3 + (yz)^3 + (xz)^3\) for \(x + y + z = 0\) and \(\gcd(x, y) = \gcd(y, z) = \gcd(x, z) = 1\). Determine which primes are "good" based on this condition. | [
"3, 5, 47, 151, 353"
] | [
"A prime \\(p\\) is \"good\" if it divides the number of the form \\((xy)^3 + (yz)^3 + (xz)^3\\) for \\(x + y + z = 0\\) and \\(\\gcd(x, y) = \\gcd(y, z) = \\gcd(x, z) = 1\\). Determine which primes are \"good\" based on this condition."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b24f37274df988c376ad6f1504c09922860d7eafa3b4bf700687332df1b98c5d | A prime number $p$ divides $a^2 + 2$ for a natural number $a$. Let $f(p)$ be the smallest positive integer such that either $p$ or $2p$ can be expressed as $x^2 + 2y^2$ for some natural numbers $x$ and $y$ with $x + y = f(p)$. Find $f(11)$. | [
"4"
] | [
"A prime number $p$ divides $a^2 + 2$ for a natural number $a$. Let $f(p)$ be the smallest positive integer such that either $p$ or $2p$ can be expressed as $x^2 + 2y^2$ for some natural numbers $x$ and $y$ with $x + y = f(p)$. Find $f(11)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02a3b376d0a6b64e6c401369d324ea8399732d2ae5927ac4458a99646893496c | A prime number \( P \) greater than 100 leaves a remainder \( Q \) when divided by 28. How many values of \( Q \) exist? | [
"12"
] | [
"A prime number \\( P \\) greater than 100 leaves a remainder \\( Q \\) when divided by 28. How many values of \\( Q \\) exist?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-1c18be8fd4421c60186e420ef3aabde2cd6a128dc1e9d44c4dd136e9350aea19 | A prime number \( p \) divides \( a^2 + 2 \) for a natural number \( a \). Let \( g(p) \) be the smallest positive integer such that \( p \) or \( 2p \) can be expressed as \( x^2 + 2y^2 \) for some natural numbers \( x \) and \( y \) with \( x + y = g(p) \). Find \( g(11) \). | [
"4"
] | [
"A prime number \\( p \\) divides \\( a^2 + 2 \\) for a natural number \\( a \\). Let \\( g(p) \\) be the smallest positive integer such that \\( p \\) or \\( 2p \\) can be expressed as \\( x^2 + 2y^2 \\) for some natural numbers \\( x \\) and \\( y \\) with \\( x + y = g(p) \\). Find \\( g(11) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b13cfd1fc183131270b3c20fa369a7d8f22e86a54fed48f9ad004dede07d6d37 | A prime number \( p \) is called a "moderate" number if for every pair of positive integers \( k > 1 \) and \( m \), there exist \( k \) positive integers \( n_1, n_2, \ldots, n_k \) such that \[ n_1^2 + n_2^2 + \cdots + n_k^2 = p^{k+m}. \] If \( q \) is the smallest "moderate" number, determine the smallest prime \( r... | [
"7"
] | [
"A prime number \\( p \\) is called a \"moderate\" number if for every pair of positive integers \\( k > 1 \\) and \\( m \\), there exist \\( k \\) positive integers \\( n_1, n_2, \\ldots, n_k \\) such that \\[ n_1^2 + n_2^2 + \\cdots + n_k^2 = p^{k+m}. \\] If \\( q \\) is the smallest \"moderate\" number, determin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 43
} | 43 | 0 | 0 |
omr-ws-f14f476746053ba5a2f5e14045bced3a6f0bbe4dc067b4f13fd3d1d9b18b82bc | A prime number \( p \) is called cubic if every integer can be represented as \( x^3 + y^3 + pz \), where \( x \), \( y \), and \( z \) are integers. Find all cubic prime numbers. | [
"p \\equiv 2 \\pmod{3} \\text{ or } p = 3"
] | [
"A prime number \\( p \\) is called cubic if every integer can be represented as \\( x^3 + y^3 + pz \\), where \\( x \\), \\( y \\), and \\( z \\) are integers. Find all cubic prime numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-e6503cbbff355063c94ee73da640de7c7421494f1ff4b8f9f759f74988aa879f | A prime number \( p \) is called cubic if every integer can be represented as \( x^3 + y^3 + pz \), where \( x, y, z \) are integers. Find all cubic primes. | [
"3"
] | [
"A prime number \\( p \\) is called cubic if every integer can be represented as \\( x^3 + y^3 + pz \\), where \\( x, y, z \\) are integers. Find all cubic primes."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-655631189dcccba7e02a7a0145ebd1b1a6c412a092859319ec3db0a43088a8d5 | A prime number \( p \) is given. On the left side of the blackboard, numbers \( 1, 2, \ldots, p-1 \) are written, on the right just one \( 0 \). We consider a sequence of \( p-1 \) moves, each of them as follows: we choose a number from the left side, add it to all the numbers on the blackboard, and erase it. Determine... | [
"p \\neq 2 \\text{ and } p \\neq 3"
] | [
"A prime number \\( p \\) is given. On the left side of the blackboard, numbers \\( 1, 2, \\ldots, p-1 \\) are written, on the right just one \\( 0 \\). We consider a sequence of \\( p-1 \\) moves, each of them as follows: we choose a number from the left side, add it to all the numbers on the blackboard, and erase... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-b971faebd160048064ed06aafcac8380016250fdfa3caa842edd4a6e58198169 | A prime number \( p_i \) is called "simple" if \( p_i^{(n^2)} > p_{i-1} (n!)^4 \) for all positive integers \( n \). Find all the "simple" primes! | [
"All prime numbers \\( p_i \\) are \"simple\" primes. \n\nSince the problem asks for a numerical or mathematical expression and the answer is a statement about all primes, the closest concise form would be:"
] | [
"A prime number \\( p_i \\) is called \"simple\" if \\( p_i^{(n^2)} > p_{i-1} (n!)^4 \\) for all positive integers \\( n \\). Find all the \"simple\" primes!"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-234df6b9b3f01a9fa4e6918faaf51a37a5dfde5df37ff32b5dc73602eb71ec77 | A princess must choose from among \( r \) suitors who are presented to her one by one in random order. At each stage, she must either accept the current suitor or reject him and proceed to the next. What strategy will maximize her chance of stopping with the best suitor of all? | [
"Reject the first \\( \\sim r/e \\) suitors, then accept the first suitor who is ranked higher than all of the previous suitors. This strategy approaches a \\( 1/e \\) probability of success as \\( r \\) increases."
] | [
"A princess must choose from among \\( r \\) suitors who are presented to her one by one in random order. At each stage, she must either accept the current suitor or reject him and proceed to the next. What strategy will maximize her chance of stopping with the best suitor of all?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-20b47b5d36f1647185aafdb216d5147f6f79b30a44f29269ccdaf623123f640b | A principal of $4600 is invested at 7.25% interest, compounded annually. How much will the investment be worth after 12 years? Round your answer to the nearest dollar. | [
"11495"
] | [
"A principal of $4600 is invested at 7.25% interest, compounded annually. How much will the investment be worth after 12 years? Round your answer to the nearest dollar."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbf798b7cf86593839aa71adf871fec364e58eda22f17e091cc53a48a1096d4c | A printer numbers the pages of a book starting with $1$ and uses $3189$ digits in total. How many pages does the book have? | [
"1074"
] | [
"A printer numbers the pages of a book starting with $1$ and uses $3189$ digits in total. How many pages does the book have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-bebec59676f02345ff7ff5e202bfce07cdece7bc9872f6adddaec0eebbe4baf3 | A printer numbers the pages of a book starting with $1$. If the printer uses a total of $3,189$ digits, how many pages does the book have? | [
"1074"
] | [
"A printer numbers the pages of a book starting with $1$. If the printer uses a total of $3,189$ digits, how many pages does the book have?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 9
} | 25 | 0 | 0 |
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