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-c0fa58c28da9b3c39a519921553ca8894a145584894387b9b58c6efc37c256a6 | A positive integer \( n \) is added to the numerator and denominator of \( S(2) \) such that the value of the resulting fraction is \( \frac{1}{2} \). Compute \( n \). | [
"5\n\nThe forum discussion provides a detailed solution to each of the five problems. For the problem we are looking at, which is the third problem in the sequence, the solution states that \\( n = 5 \\). Therefore, the answer is \\( n = 5 \\)."
] | [
"A positive integer \\( n \\) is added to the numerator and denominator of \\( S(2) \\) such that the value of the resulting fraction is \\( \\frac{1}{2} \\). Compute \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52d80064c1de0111dc65bf136bde527376c960f243ada7e7ef280fd5c481450d | A positive integer \( n \) is called "beautiful" if it satisfies both conditions: - \( n \) has at least 4 positive divisors. - For all divisors \( a, b \) of \( n \) (with \( 1 < a < b < n \)), \( b - a \) is also a divisor of \( n \). Find all "beautiful" numbers. | [
"\\(6, 8, 12\\)"
] | [
"A positive integer \\( n \\) is called \"beautiful\" if it satisfies both conditions:\n- \\( n \\) has at least 4 positive divisors.\n- For all divisors \\( a, b \\) of \\( n \\) (with \\( 1 < a < b < n \\)), \\( b - a \\) is also a divisor of \\( n \\).\nFind all \"beautiful\" numbers."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-21f223d07ffe6ed38888030bdcbd177e844586108dc7bdd942527b550e71a958 | A positive integer \( n \) is called "fine" if it has two prime factors \( p \) and \( q \) such that \( p + q \) is also a prime factor of \( n \). For example, \( 210 \) is a "fine" number since it has \( 2, 3 \), and \( 2 + 3 = 5 \) as prime factors. How many integers between \( 1 \) and \( 2012 \) are "fine"? | [
"97"
] | [
"A positive integer \\( n \\) is called \"fine\" if it has two prime factors \\( p \\) and \\( q \\) such that \\( p + q \\) is also a prime factor of \\( n \\). For example, \\( 210 \\) is a \"fine\" number since it has \\( 2, 3 \\), and \\( 2 + 3 = 5 \\) as prime factors. How many integers between \\( 1 \\) and \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-1e78fc323f463db1710675f5c8fab705dc6081130723deafa8361e2d7cc9352a | A positive integer \( n \) is called "good" if a unit square can be tiled with exactly \( n \) rectangles, each similar to a \( 2 \times 1 \) rectangle. Find the set of all good integers \( \ge 5 \). | [
"\\( n \\neq 1, 3, 4, 6 \\)"
] | [
"A positive integer \\( n \\) is called \"good\" if a unit square can be tiled with exactly \\( n \\) rectangles, each similar to a \\( 2 \\times 1 \\) rectangle. Find the set of all good integers \\( \\ge 5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-41da18f55980323f266a0b33f8b99370627e4acee3286bf782abc2054c0e4800 | A positive integer \( n \) is called "good" if for all \( a \in \mathbb{Z}^+ \) such that \( \gcd(a, n) = 1 \) and for all odd primes \( p \) where \( p \mid a^2 + n \), the equation \( x^2 + ny^2 = p \) has integer solutions. Find the largest "good" number. | [
"4"
] | [
"A positive integer \\( n \\) is called \"good\" if for all \\( a \\in \\mathbb{Z}^+ \\) such that \\( \\gcd(a, n) = 1 \\) and for all odd primes \\( p \\) where \\( p \\mid a^2 + n \\), the equation \\( x^2 + ny^2 = p \\) has integer solutions. Find the largest \"good\" number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e7b64b6a2c5a4511b9e91b1b6b2693300946ece499a4377ea75c14185c77487b | A positive integer \( n \) is called "good" if the sum of the squares of all its proper divisors equals \( 2n + 4 \). Find all good numbers less than \( 10000 \). | [
"8, 15, 35, 143, 323, 899, 1763, 3599, 5183"
] | [
"A positive integer \\( n \\) is called \"good\" if the sum of the squares of all its proper divisors equals \\( 2n + 4 \\). Find all good numbers less than \\( 10000 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8be5115c5e255da09bb90dab1daca12266f1b7c8231b30e3171910623e8bec57 | A positive integer \( n \) is called "just ups, no downs" if it satisfies the following conditions: \[ \lceil \frac{n}{2} \rceil > \frac{n}{2}, \] \[ \lceil \frac{\lceil \frac{n}{2} \rceil}{2} \rceil > \frac{\lceil \frac{n}{2} \rceil}{2}, \] \[ \lceil \frac{\lceil \frac{\lceil \frac{n}{2} \rceil}{2} \rceil}{2} \rceil >... | [
"232"
] | [
"A positive integer \\( n \\) is called \"just ups, no downs\" if it satisfies the following conditions:\n\\[\n\\lceil \\frac{n}{2} \\rceil > \\frac{n}{2},\n\\]\n\\[\n\\lceil \\frac{\\lceil \\frac{n}{2} \\rceil}{2} \\rceil > \\frac{\\lceil \\frac{n}{2} \\rceil}{2},\n\\]\n\\[\n\\lceil \\frac{\\lceil \\frac{\\lceil \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9434eaa630ce396a3836b9fb998b0f9f3d20fd641d630c23e2ef208e65749f4 | A positive integer \( n \) is called "nice" if there is a positive integer \( m \) with exactly four positive divisors (including \( 1 \) and \( m \)) such that the sum of the four divisors is equal to \( n \). How many numbers in the set \(\{2010, 2011, 2012, \ldots, 2019\}\) are nice? | [
"1"
] | [
"A positive integer \\( n \\) is called \"nice\" if there is a positive integer \\( m \\) with exactly four positive divisors (including \\( 1 \\) and \\( m \\)) such that the sum of the four divisors is equal to \\( n \\). How many numbers in the set \\(\\{2010, 2011, 2012, \\ldots, 2019\\}\\) are nice?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-b789f42f72aafd1941370872b8ef875c922738196d3c7140c144284233da8dde | A positive integer \( n \) is called "rioplatense" if there exists an integer \( m \) such that \( m + 2S(m) = n \), where \( S(m) \) is the sum of the digits of \( m \). Find all positive integers that are rioplatense. | [
"All positive multiples of 3."
] | [
"A positive integer \\( n \\) is called \"rioplatense\" if there exists an integer \\( m \\) such that \\( m + 2S(m) = n \\), where \\( S(m) \\) is the sum of the digits of \\( m \\). Find all positive integers that are rioplatense."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 19
} | 32 | 0 | 0 |
omr-ws-0d187d149366a43770f8c55b2a8c5c7569def1d78410282948d8fac0a0252d96 | A positive integer \( n \) is called $\textit{good}$ if \( 2 \mid \tau(n) \) and if its divisors are \[ 1 = d_1 < d_2 < \cdots < d_{2k-1} < d_{2k} = n, \] then \( d_{k+1} - d_k = 2 \) and \( d_{k+2} - d_{k-1} = 65 \). Find the smallest $\textit{good}$ number. | [
"2024"
] | [
"A positive integer \\( n \\) is called $\\textit{good}$ if \\( 2 \\mid \\tau(n) \\) and if its divisors are\n\n\\[\n1 = d_1 < d_2 < \\cdots < d_{2k-1} < d_{2k} = n,\n\\]\n\nthen \\( d_{k+1} - d_k = 2 \\) and \\( d_{k+2} - d_{k-1} = 65 \\). Find the smallest $\\textit{good}$ number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-88602fed47509e6ffaebd9e1d1932c0cb2d78f3304736d53fccfca66bf551f87 | A positive integer \( n \) is called 'divine' if there exists a positive integer \( m \) such that \[ \sum_{k=0}^{n-1} \frac{1}{m+k} \] is an integer. Find all divine numbers. | [
"\\( n = 1 \\)"
] | [
"A positive integer \\( n \\) is called 'divine' if there exists a positive integer \\( m \\) such that\n\\[\n\\sum_{k=0}^{n-1} \\frac{1}{m+k}\n\\]\nis an integer. Find all divine numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9619d01b4e7925fff54f1ed3ba6dcc8f9c26a6b0b004c8e9246af8c75f9b558b | A positive integer \( n \) is called 'good' if there exists an integer \( k \) such that \( n = \left\lfloor \frac{2011}{k} \right\rfloor \). What is the smallest positive integer that is not good? | [
"48"
] | [
"A positive integer \\( n \\) is called 'good' if there exists an integer \\( k \\) such that \\( n = \\left\\lfloor \\frac{2011}{k} \\right\\rfloor \\). What is the smallest positive integer that is not good?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-309c3f03a883feb16f7418adb6c3c0c5d5ba79d110e327a617e02a4bc46681d7 | A positive integer \( n \) is called *essential* if it satisfies the equation \( \tau(\phi(n)) = \phi(\tau(n)) \), where \( \tau(n) \) denotes the number of positive integer divisors of \( n \), and \( \phi(n) \) denotes the number of positive integers less than or equal to \( n \) that are relatively prime to \( n \).... | [
"510"
] | [
"A positive integer \\( n \\) is called *essential* if it satisfies the equation \\( \\tau(\\phi(n)) = \\phi(\\tau(n)) \\), where \\( \\tau(n) \\) denotes the number of positive integer divisors of \\( n \\), and \\( \\phi(n) \\) denotes the number of positive integers less than or equal to \\( n \\) that are relat... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94742f578a32aa6477a777da025f5e06ff50a27a9b52ff8403e601c94157c2b4 | A positive integer \( n \) is called *good* if it can be expressed as the sum of two positive integers in exactly one way, and the product of these two integers is also equal to \( n \). Find all good positive integers. | [
"4"
] | [
"A positive integer \\( n \\) is called *good* if it can be expressed as the sum of two positive integers in exactly one way, and the product of these two integers is also equal to \\( n \\). Find all good positive integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-00a989f7e5b46b91e2f6d66c1292b4d1ffa5e6998e521380c0edc82379f53a28 | A positive integer \( n \) is called *nice* if it has at least 3 proper divisors and it is equal to the sum of its three largest proper divisors. Find the number of *nice* integers not greater than 3000. | [
"200"
] | [
"A positive integer \\( n \\) is called *nice* if it has at least 3 proper divisors and it is equal to the sum of its three largest proper divisors. Find the number of *nice* integers not greater than 3000."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-4c50003586c4a388e5fe862553b25f2e17481a55ede87b9715f91b823524e4b2 | A positive integer \( n \) is called *pretty* if there exist two divisors \( d_1, d_2 \) of \( n \) such that \( d_2 - d_1 = d \) for each divisor \( d \) of \( n \) (where \( 1 < d < n \)). Find the smallest pretty number larger than 401 that is a multiple of 401. | [
"\\( 160400 \\)"
] | [
"A positive integer \\( n \\) is called *pretty* if there exist two divisors \\( d_1, d_2 \\) of \\( n \\) such that \\( d_2 - d_1 = d \\) for each divisor \\( d \\) of \\( n \\) (where \\( 1 < d < n \\)). Find the smallest pretty number larger than 401 that is a multiple of 401."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 5,
"tir": 4
} | 10 | 0 | 0 |
omr-ws-27a50149f774f6a5b9731e26a6919c9189c1cdd09f86b9780b5d436965b5d2d3 | A positive integer \( n \) is called GOOD if there does not exist a permutation \((a_1, a_2, \ldots, a_n)\) of the integers from 1 to \( n \) such that \( k + a_k \) is a perfect square for all \( k \). Find all GOOD numbers up to 12. | [
"{3, 5, 8, 10, 12}"
] | [
"A positive integer \\( n \\) is called GOOD if there does not exist a permutation \\((a_1, a_2, \\ldots, a_n)\\) of the integers from 1 to \\( n \\) such that \\( k + a_k \\) is a perfect square for all \\( k \\). Find all GOOD numbers up to 12."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5814fa5eb3b318b87d7f375b1840edf675591bba53da3f95e2e89a7ff586aabf | A positive integer \( n \) is called Olympic if there exists a quadratic trinomial with integer coefficients \( f(x) \) satisfying \( f(f(\sqrt{n})) = 0 \). Find the sum of all Olympic numbers not exceeding 2015. | [
"29370"
] | [
"A positive integer \\( n \\) is called Olympic if there exists a quadratic trinomial with integer coefficients \\( f(x) \\) satisfying \\( f(f(\\sqrt{n})) = 0 \\). Find the sum of all Olympic numbers not exceeding 2015."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0381043ea56893a3fce027740f097ff39acf632515e60f263274feb8a189bc25 | A positive integer \( n \) is called [i]singly writable[/i] if there is a unique way to write \( n \) as the sum of distinct Fibonacci numbers, where the Fibonacci sequence is defined by \( F_1 = F_2 = 1 \) and \( F_{k+2} = F_{k+1} + F_k \) for all \( k \ge 1 \). Let \( S \) be the sum of all singly writable numbers fr... | [
"567"
] | [
"A positive integer \\( n \\) is called [i]singly writable[/i] if there is a unique way to write \\( n \\) as the sum of distinct Fibonacci numbers, where the Fibonacci sequence is defined by \\( F_1 = F_2 = 1 \\) and \\( F_{k+2} = F_{k+1} + F_k \\) for all \\( k \\ge 1 \\). Let \\( S \\) be the sum of all singly w... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 10
} | 30 | 0 | 0 |
omr-ws-79bd788e7473d474712833ac24b26365a582cae052be3c3c9d36a7530139d200 | A positive integer \( n \) is called \( crunchy \) if there exist \( 2n \) real numbers \( x_1, x_2, \ldots, x_{2n} \), not all equal, such that the sum of any \( n \) of the \( x_i \)'s is equal to the product of the other \( n \) of the \( x_i \)'s. Determine all crunchy integers. | [
"Even integers"
] | [
"A positive integer \\( n \\) is called \\( crunchy \\) if there exist \\( 2n \\) real numbers \\( x_1, x_2, \\ldots, x_{2n} \\), not all equal, such that the sum of any \\( n \\) of the \\( x_i \\)'s is equal to the product of the other \\( n \\) of the \\( x_i \\)'s. Determine all crunchy integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-6813bd7cfbeabc85ea0aef3f588e057ccc477169e3a293c99a23e67a2b023297 | A positive integer \( n \) is called \(\textit{divicient}\) if the number of divisors of \( n \), divided by \( n \) itself, exceeds \(\frac{1}{4}\). For example, \( 10 \) is divicient since \(\frac{d(10)}{10} = \frac{4}{10} > \frac{1}{4}\). How many positive integers are divicient? | [
"18"
] | [
"A positive integer \\( n \\) is called \\(\\textit{divicient}\\) if the number of divisors of \\( n \\), divided by \\( n \\) itself, exceeds \\(\\frac{1}{4}\\). For example, \\( 10 \\) is divicient since \\(\\frac{d(10)}{10} = \\frac{4}{10} > \\frac{1}{4}\\). How many positive integers are divicient?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ad6c1139bc1e71fcef559296972a5c1adc8d820968db5bdb20a6ca89bc63d92 | A positive integer \( n \) is called a "good number" if \( n^3 + 7n - 133 = m^3 \) for some positive integer \( m \). What is the sum of all good numbers? | [
"30"
] | [
"A positive integer \\( n \\) is called a \"good number\" if \\( n^3 + 7n - 133 = m^3 \\) for some positive integer \\( m \\). What is the sum of all good numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 12
} | 35 | 0 | 0 |
omr-ws-060382f70b47a89b082eff0615acd9e6c1e4cd670b7974d9ea8d03a458bc59d2 | A positive integer \( n \) is called a "googoori" number if both \( n-1 \) and \( n^2 + n - 1 \) are terms of the Fibonacci sequence. How many "googoori" numbers exist? | [
"3"
] | [
"A positive integer \\( n \\) is called a \"googoori\" number if both \\( n-1 \\) and \\( n^2 + n - 1 \\) are terms of the Fibonacci sequence. How many \"googoori\" numbers exist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-2f93ae1ede5d34f8bbe1bc16688eb576da2626b59748663506f76f196bbcbca8 | A positive integer \( n \) is called a "yo-yo" if the absolute value of the difference between any two consecutive digits of \( n \) is at least 7. Compute the number of 8-digit yo-yos. | [
"1233"
] | [
"A positive integer \\( n \\) is called a \"yo-yo\" if the absolute value of the difference between any two consecutive digits of \\( n \\) is at least 7. Compute the number of 8-digit yo-yos."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7d955968b1d35f98279ce1cf2516f87104acba8e0a6305769d8b8c14f8841e9 | A positive integer \( n \) is called a good number if \(\left\{ \frac{n}{10^k} \right\} > \frac{n}{10^{10}}\) for \( k = 1, 2, \ldots, 9 \). How many good numbers are there? Note: \(\left\{ x \right\} = x - \left[ x \right]\). | [
"\\( 9^{10} \\)"
] | [
"A positive integer \\( n \\) is called a good number if \\(\\left\\{ \\frac{n}{10^k} \\right\\} > \\frac{n}{10^{10}}\\) for \\( k = 1, 2, \\ldots, 9 \\). How many good numbers are there?\n\nNote: \\(\\left\\{ x \\right\\} = x - \\left[ x \\right]\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-adf41addb9c0c94ef7723607d3b4b8c82e8e0e03aebe88bc53d328e223ad7174 | A positive integer \( n \) is called a good number if there exist positive integers \( a, b, \) and \( c \) such that \( n = a^2 + b^3 + c^4 \). Find the sum of all distinct positive integers \( k \) such that \( 18 + k \) is a good number. | [
""
] | [
"A positive integer \\( n \\) is called a good number if there exist positive integers \\( a, b, \\) and \\( c \\) such that \\( n = a^2 + b^3 + c^4 \\). Find the sum of all distinct positive integers \\( k \\) such that \\( 18 + k \\) is a good number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-5b2dc85545cfa99c679df5889a489c84374091ea21c42d5e842fa63137c2c3bc | A positive integer \( n \) is called a primary divisor if for every positive divisor \( d \) of \( n \), at least one of the numbers \( d - 1 \) and \( d + 1 \) is prime. Determine the largest primary divisor number. | [
"96"
] | [
"A positive integer \\( n \\) is called a primary divisor if for every positive divisor \\( d \\) of \\( n \\), at least one of the numbers \\( d - 1 \\) and \\( d + 1 \\) is prime. Determine the largest primary divisor number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-1f321e08da2d50b8a53f1faa8c588819d22d0bb22030ee4305cd7f92f3fb02bd | A positive integer \( n \) is called amazing if given any infinite set of primes, one can choose \( n \) primes \( p_1, p_2, \ldots, p_n \) such that \( p_1 \times p_2 \times \cdots \times p_n - 1 \) is divisible by 2020. Compute the sum of all amazing numbers less than 2020. | [
"21000"
] | [
"A positive integer \\( n \\) is called amazing if given any infinite set of primes, one can choose \\( n \\) primes \\( p_1, p_2, \\ldots, p_n \\) such that \\( p_1 \\times p_2 \\times \\cdots \\times p_n - 1 \\) is divisible by 2020. Compute the sum of all amazing numbers less than 2020."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 8
} | 55 | 0 | 0 |
omr-ws-d09f206e2848f3414c805702fbd8b414d831f3d4477c984fadf7a9e93507be73 | A positive integer \( n \) is called amazing if the set \(\{1, 2, \ldots, n\}\) can be partitioned into nice subsets, where a set of positive integers is called nice if the number of its elements equals the average of its numbers. Find the sum of all perfect squares less than 1000 that are amazing numbers. | [
"10416"
] | [
"A positive integer \\( n \\) is called amazing if the set \\(\\{1, 2, \\ldots, n\\}\\) can be partitioned into nice subsets, where a set of positive integers is called nice if the number of its elements equals the average of its numbers. Find the sum of all perfect squares less than 1000 that are amazing numbers."... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-721ae8761dc50c812260f40914659651807f0c07038c08107aea2cc7eb5ab37d | A positive integer \( n \) is called an "interesting" number if it satisfies \[ \left\{ \frac{n}{10^k} \right\} > \frac{n}{10^{10}} \] for all \( k = 1, 2, \ldots, 9 \). Find the number of interesting numbers. | [
"} 999989991 \\]\n\nPost 10 and Post 11 also confirm this answer through mathematical reasoning.\n\nTherefore, the answer is:"
] | [
"A positive integer \\( n \\) is called an \"interesting\" number if it satisfies\n\\[ \\left\\{ \\frac{n}{10^k} \\right\\} > \\frac{n}{10^{10}} \\]\nfor all \\( k = 1, 2, \\ldots, 9 \\). Find the number of interesting numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-00541b8f3ff81d08820aee48a42f952701102da7814ec8076993933473c5ff09 | A positive integer \( n \) is called an \( MTRP \) number if it satisfies the following conditions: 1. For any positive integer \( m \), \( n \equiv 1 \pmod{m} \). 2. The sum of the digits of \( n^2 \) in decimal representation is greater than or equal to the sum of the digits of \( n \) in decimal representation. Find... | [
"0"
] | [
"A positive integer \\( n \\) is called an \\( MTRP \\) number if it satisfies the following conditions:\n1. For any positive integer \\( m \\), \\( n \\equiv 1 \\pmod{m} \\).\n2. The sum of the digits of \\( n^2 \\) in decimal representation is greater than or equal to the sum of the digits of \\( n \\) in decimal... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-3a1bdc5539168dfc76907fbf2317dc31b105ddf0cc25569700d0ea680e599678 | A positive integer \( n \) is called brainy if there exist coprime positive integers \( a \) and \( b \) satisfying \[ n \mid a^{p(n)} + b^{p(n)}, \] but \[ n \nmid a + b, \] where \( p(n) \) denotes the largest prime divisor of \( n \). Find the number of integers \( 1 \le m \le 100 \) such that \( 2023m \) is brainy. | [
"100"
] | [
"A positive integer \\( n \\) is called brainy if there exist coprime positive integers \\( a \\) and \\( b \\) satisfying \n\\[ n \\mid a^{p(n)} + b^{p(n)}, \\]\nbut \n\\[ n \\nmid a + b, \\]\nwhere \\( p(n) \\) denotes the largest prime divisor of \\( n \\). Find the number of integers \\( 1 \\le m \\le 100 \\) s... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9d9634ce9c1824e2f3150d299b0889ecd0fdcf647f427c25777b7e8213492d1 | A positive integer \( n \) is called delightful if there exists an integer \( k \) such that \( 1 < k < n \) and \[ 1 + 2 + \cdots + (k-1) = (k+1) + (k+2) + \cdots + n. \] Does there exist a delightful number \( N \) satisfying the inequalities \[ 2013^{2013} < \frac{N}{2013^{2013}} < 2013^{2013} + 4? \] | [
"No solution."
] | [
"A positive integer \\( n \\) is called delightful if there exists an integer \\( k \\) such that \\( 1 < k < n \\) and \n\\[ 1 + 2 + \\cdots + (k-1) = (k+1) + (k+2) + \\cdots + n. \\]\nDoes there exist a delightful number \\( N \\) satisfying the inequalities \n\\[ 2013^{2013} < \\frac{N}{2013^{2013}} < 2013^{2013... | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4e4247b0dbb04e6ae9f9ece9a8b99b329398f41977a3702c6529ee88d2b2536a | A positive integer \( n \) is called foolish if the sum of the digits of \( 2^n \) is one less than the sum of the digits of \( 3^n \). It is known that 1 and 3 are foolish. Is it possible to find all foolish numbers? | [
"\\text{No}"
] | [
"A positive integer \\( n \\) is called foolish if the sum of the digits of \\( 2^n \\) is one less than the sum of the digits of \\( 3^n \\). It is known that 1 and 3 are foolish. Is it possible to find all foolish numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-9bd95a97c1d5c9cd95f09adf1ef6609fc7b4bd1f36e926727cfdbdc33fd426b7 | A positive integer \( n \) is called fragile if there exist positive integers \( a, b, x, y \) such that \( a + b = n \) and \( \frac{x}{a} + \frac{y}{b} = 1 \). Determine the set of fragile numbers. | [
"The set of fragile numbers is the set of all non-prime positive integers."
] | [
"A positive integer \\( n \\) is called fragile if there exist positive integers \\( a, b, x, y \\) such that \\( a + b = n \\) and \\( \\frac{x}{a} + \\frac{y}{b} = 1 \\). Determine the set of fragile numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-26bc565fccb9999df4f1eb0121947ebbe9227e2293c2492c94044164cd2440c9 | A positive integer \( n \) is called good if for all positive integers \( a \) which can be written as \( a = n^2 \sum_{i=1}^n x_i^2 \) where \( x_1, x_2, \ldots, x_n \) are integers, it is possible to express \( a \) as \( a = \sum_{i=1}^n y_i^2 \) where \( y_1, y_2, \ldots, y_n \) are integers with none of them divis... | [
"All positive integers except \\( 1, 2, \\) and \\( 4 \\)."
] | [
"A positive integer \\( n \\) is called good if for all positive integers \\( a \\) which can be written as \\( a = n^2 \\sum_{i=1}^n x_i^2 \\) where \\( x_1, x_2, \\ldots, x_n \\) are integers, it is possible to express \\( a \\) as \\( a = \\sum_{i=1}^n y_i^2 \\) where \\( y_1, y_2, \\ldots, y_n \\) are integers ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-02096abee0f8d3394d554d415c9f169280edcf41ead0eeb4bb6929bb840f1c51 | A positive integer \( n \) is called good if it can be uniquely written simultaneously as \( a_1 + a_2 + \cdots + a_k \) and as \( a_1 a_2 \cdots a_k \), where \( a_i \) are positive integers and \( k \ge 2 \). Find, in terms of prime numbers, all good natural numbers. | [
"\\( n = p^2 \\) or \\( n = pq \\) where \\( p \\) and \\( q \\) are prime numbers."
] | [
"A positive integer \\( n \\) is called good if it can be uniquely written simultaneously as \\( a_1 + a_2 + \\cdots + a_k \\) and as \\( a_1 a_2 \\cdots a_k \\), where \\( a_i \\) are positive integers and \\( k \\ge 2 \\). Find, in terms of prime numbers, all good natural numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-5f502440dab8a31ae7d81006588114897343ce43c978be62474231031a60b80a | A positive integer \( n \) is called good if the remainder of any perfect square when divided by \( n \) is a perfect square. Let \( S \) be the set of all good integers. Find the sum of all elements in \( S \) that are less than 500. | [
"51"
] | [
"A positive integer \\( n \\) is called good if the remainder of any perfect square when divided by \\( n \\) is a perfect square. Let \\( S \\) be the set of all good integers. Find the sum of all elements in \\( S \\) that are less than 500."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-83a6a0b1f656a32179b1ef2f498b96e9cf824a8eee024bb961137729ee039674 | A positive integer \( n \) is called guayaquilean if the sum of the digits of \( n \) is equal to the sum of the digits of \( n^2 \). Find all possible values that the sum of the digits of a guayaquilean number can take. | [
"\\(9k\\) or \\(9k+1\\) for any non-negative integer \\(k\\)."
] | [
"A positive integer \\( n \\) is called guayaquilean if the sum of the digits of \\( n \\) is equal to the sum of the digits of \\( n^2 \\). Find all possible values that the sum of the digits of a guayaquilean number can take."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 22,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-d2d6ebba36ba1e7a61e0d9d57fb8010b01031026353c4d93383eb89eae260319 | A positive integer \( n \) is called highly divisible if it has more divisors than any smaller positive integer. Let \( f(p) \) be the smallest highly divisible integer \( r \) such that \( pr \) is also highly divisible for a given prime \( p \). Find the value of \( f(7) \). | [
"120"
] | [
"A positive integer \\( n \\) is called highly divisible if it has more divisors than any smaller positive integer. Let \\( f(p) \\) be the smallest highly divisible integer \\( r \\) such that \\( pr \\) is also highly divisible for a given prime \\( p \\). Find the value of \\( f(7) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fed7b3eed9d5291ad77b2fe242d9a52fd25ba4b28e2aa2f0caa4702a3d432ffc | A positive integer \( n \) is called lucky if \( n^2 \) can be written as the sum of \( n \) consecutive positive integers. Find the number of lucky numbers strictly less than 2015. | [
"1007"
] | [
"A positive integer \\( n \\) is called lucky if \\( n^2 \\) can be written as the sum of \\( n \\) consecutive positive integers. Find the number of lucky numbers strictly less than 2015."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70d07012a6cc9497555205d3cf00d322d6c4f90d3af0655488db3ab121c1fc19 | A positive integer \( n \) is called nice if it has at least 3 proper divisors and it is equal to the sum of its three largest proper divisors. Find the number of nice integers not greater than 3000. | [
"200"
] | [
"A positive integer \\( n \\) is called nice if it has at least 3 proper divisors and it is equal to the sum of its three largest proper divisors. Find the number of nice integers not greater than 3000."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 9
} | 18 | 0 | 0 |
omr-ws-c452b5e1aba0889ee0920c7b2b85500d308235db5b38d1358a56cc2861255554 | A positive integer \( n \) is called nice if there exist distinct divisors \( d_1 \), \( d_2 \), \( d_3 \) of \( n \) such that \( d_1 + d_2 + d_3 = n \). How many positive integers less than 1000 are nice? | [
"166"
] | [
"A positive integer \\( n \\) is called nice if there exist distinct divisors \\( d_1 \\), \\( d_2 \\), \\( d_3 \\) of \\( n \\) such that \\( d_1 + d_2 + d_3 = n \\). How many positive integers less than 1000 are nice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-6fc1eb1ecd5a7eab1c4f1f09ce76a5e5855d0ecf137b467e0564fd80ff5eeb6e | A positive integer \( n \) is called smooth if there exist integers \( a_1, a_2, \dotsc, a_n \) satisfying \[ a_1 + a_2 + \dotsb + a_n = a_1 \cdot a_2 \cdot \dotsb \cdot a_n = n. \] Find all smooth numbers. | [
"All \\( n \\equiv 0,1 \\mod 4 \\) except for \\( n=4 \\)."
] | [
"A positive integer \\( n \\) is called smooth if there exist integers \\( a_1, a_2, \\dotsc, a_n \\) satisfying\n\\[ a_1 + a_2 + \\dotsb + a_n = a_1 \\cdot a_2 \\cdot \\dotsb \\cdot a_n = n. \\]\nFind all smooth numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-abf303e53df67bab323477e937554d49934ba0b8a138f4b267c4c4dd26cd4955 | A positive integer \( n \) is called special if \( n \) is divisible by 4, \( n+1 \) is divisible by 5, and \( n+2 \) is divisible by 6. How many special integers smaller than 1000 are there? | [
"17"
] | [
"A positive integer \\( n \\) is called special if \\( n \\) is divisible by 4, \\( n+1 \\) is divisible by 5, and \\( n+2 \\) is divisible by 6. How many special integers smaller than 1000 are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-982f21d8eed9a3fced73a8ebea5f53875bea788be072725beb43f7493227aa93 | A positive integer \( n \) is called squarefull if for every prime \( p \) that divides \( n \), \( p^2 \) also divides \( n \). Let \( S \) be the set of all squarefull numbers less than \( 1000 \) that can be expressed as \( a^2b^3 \) where \( b \) is squarefree and \( a, b \in \mathbb{N} \). Calculate the sum of all... | [
"18969"
] | [
"A positive integer \\( n \\) is called squarefull if for every prime \\( p \\) that divides \\( n \\), \\( p^2 \\) also divides \\( n \\). Let \\( S \\) be the set of all squarefull numbers less than \\( 1000 \\) that can be expressed as \\( a^2b^3 \\) where \\( b \\) is squarefree and \\( a, b \\in \\mathbb{N} \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-4b2b8b47c6d23712e498130fba8eb08a651155f7a3da007f0883448356e8db48 | A positive integer \( n \) is called tripairable if it is possible to partition the set \(\{1, 2, \ldots, n\}\) into disjoint pairs such that the sum of the two elements in each pair is a power of 3. For example, 6 is tripairable because \(\{1, 2, 3, 4, 5, 6\} = \{1, 2\} \cup \{3, 6\} \cup \{4, 5\}\) and \(1+2=3^1\), \... | [
"225"
] | [
"A positive integer \\( n \\) is called tripairable if it is possible to partition the set \\(\\{1, 2, \\ldots, n\\}\\) into disjoint pairs such that the sum of the two elements in each pair is a power of 3. For example, 6 is tripairable because \\(\\{1, 2, 3, 4, 5, 6\\} = \\{1, 2\\} \\cup \\{3, 6\\} \\cup \\{4, 5\... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60ecf997e67419c98dc3e19df0d7f6c044dedccf748a5915244c665371bd0407 | A positive integer \( n \) is called tripariable if it is possible to partition the set \( \{1, 2, \dots, n\} \) into disjoint pairs such that the sum of two elements in each pair is a power of 3. How many positive integers less than or equal to 2024 are tripariable? | [
"111"
] | [
"A positive integer \\( n \\) is called tripariable if it is possible to partition the set \\( \\{1, 2, \\dots, n\\} \\) into disjoint pairs such that the sum of two elements in each pair is a power of 3. How many positive integers less than or equal to 2024 are tripariable?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 3
} | 13 | 0 | 0 |
omr-ws-ddb1d3a649012493ad2206de8517f7b04632b3af6212923d7e4bb5873f0aab07 | A positive integer \( n \) is called triprime if any three consecutive digits of \( n \) form a three-digit prime. Let \( S \) be the largest such triprime number with distinct digits. What are the last three digits of \( S \)? | [
"179"
] | [
"A positive integer \\( n \\) is called triprime if any three consecutive digits of \\( n \\) form a three-digit prime. Let \\( S \\) be the largest such triprime number with distinct digits. What are the last three digits of \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-a682e2efa70491ba9dbc470c73f7e9efaf288246b028a2f91faae38b6862fa9c | A positive integer \( n \) is called yummy if for any quintuples of integers \((a, b, c, d, k)\) that satisfy \[ a^2 - b^2 \equiv b^2 - c^2 \equiv c^2 - d^2 \equiv k \pmod{n}, \] we have \( k \equiv 0 \pmod{n} \). Find the sum of all yummy integers. | [
"15"
] | [
"A positive integer \\( n \\) is called yummy if for any quintuples of integers \\((a, b, c, d, k)\\) that satisfy\n\\[ a^2 - b^2 \\equiv b^2 - c^2 \\equiv c^2 - d^2 \\equiv k \\pmod{n}, \\]\nwe have \\( k \\equiv 0 \\pmod{n} \\). Find the sum of all yummy integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6bcab3328ef46fd3004bc738ca8c28110e84615b50b20f113f3e93c32a244ad1 | A positive integer \( n \) is chosen randomly with the probability of choosing \( i \) being \( \left(\frac{1}{2}\right)^i \). What is the best strategy for finding out the number \( n \) in the least expected number of questions if only questions in the form "Is \( n \) greater than \( i \)?" are allowed? | [
"2"
] | [
"A positive integer \\( n \\) is chosen randomly with the probability of choosing \\( i \\) being \\( \\left(\\frac{1}{2}\\right)^i \\). What is the best strategy for finding out the number \\( n \\) in the least expected number of questions if only questions in the form \"Is \\( n \\) greater than \\( i \\)?\" are... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-a2bf8374cf008c3994acec547f6da67d420e7755947149316850d8d51af52089 | A positive integer \( n \) is considered [color=#c00][i]good[/i][/color] if there exist positive integers \( a \), \( b \), and \( c \) such that \[ \frac{a^2 + b^2 + c^2}{ab + bc + ca} = n. \] Let \( g(n) \) be the smallest possible value of \( a + b + c \) for which \( n \) is [color=#c00][i]good[/i][/color]. Find th... | [
"9"
] | [
"A positive integer \\( n \\) is considered [color=#c00][i]good[/i][/color] if there exist positive integers \\( a \\), \\( b \\), and \\( c \\) such that\n\\[\n\\frac{a^2 + b^2 + c^2}{ab + bc + ca} = n.\n\\]\nLet \\( g(n) \\) be the smallest possible value of \\( a + b + c \\) for which \\( n \\) is [color=#c00][i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-59cc1192ee4d061f321faae050423697986844bf07ceaad501e401d691365178 | A positive integer \( n \) is considered \textit{squarable} if the last two digits of \( n^2 \) form a perfect square. Find the number of positive integers \( n \) less than \( 1000 \) such that \( f(n^{(2^k)}) \) is \textit{squarable} for all nonnegative integers \( k \), where \( f(n) \) denotes the remainder when \(... | [
"39"
] | [
"A positive integer \\( n \\) is considered \\textit{squarable} if the last two digits of \\( n^2 \\) form a perfect square. Find the number of positive integers \\( n \\) less than \\( 1000 \\) such that \\( f(n^{(2^k)}) \\) is \\textit{squarable} for all nonnegative integers \\( k \\), where \\( f(n) \\) denotes ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-78e112c5e8f6b63243af6c04c62f542f86fe27913f2b42faa9b29ccdc4ca6244 | A positive integer \( n \) is defined as "nice" if there exists a positive integer \( m \) with exactly four positive divisors (including 1 and \( m \)) such that the sum of these four divisors equals \( n \). How many numbers in the set \(\{2010, 2011, 2012, \ldots, 2019\}\) are nice? | [
"1"
] | [
"A positive integer \\( n \\) is defined as \"nice\" if there exists a positive integer \\( m \\) with exactly four positive divisors (including 1 and \\( m \\)) such that the sum of these four divisors equals \\( n \\). How many numbers in the set \\(\\{2010, 2011, 2012, \\ldots, 2019\\}\\) are nice?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da7ede27b78cd2081b8a848bdbe6de007f0804c6dcb06f4f3f29c2125e23bf2e | A positive integer \( n \) is defined as [i]friendly[/i] if the difference between each pair of neighboring digits of \( n \), when written in base 10, is exactly 1. For example, 6787 is friendly, but 211 and 901 are not. Find all odd natural numbers \( m \) for which there exists a friendly integer divisible by \( 64m... | [
"\\( m \\) such that \\( (m, 5) = 1 \\)"
] | [
"A positive integer \\( n \\) is defined as [i]friendly[/i] if the difference between each pair of neighboring digits of \\( n \\), when written in base 10, is exactly 1. For example, 6787 is friendly, but 211 and 901 are not. Find all odd natural numbers \\( m \\) for which there exists a friendly integer divisibl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 7,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-1da3fee20d3ed818df3116e4ec2e21fda3e636d663167b6232cf0c88a880da36 | A positive integer \( n \) is defined as \(\textit{strange-ish}\) if the last four digits of \( n^2 \) are \( 1001 \). Let \( a \) and \( b \) be the two smallest strange-ish numbers. If \( a = 10^4 \cdot \alpha + 1001 \) and \( b = 10^4 \cdot \beta + 1001 \), compute \( \alpha + \beta \). | [
"10000"
] | [
"A positive integer \\( n \\) is defined as \\(\\textit{strange-ish}\\) if the last four digits of \\( n^2 \\) are \\( 1001 \\). Let \\( a \\) and \\( b \\) be the two smallest strange-ish numbers. If \\( a = 10^4 \\cdot \\alpha + 1001 \\) and \\( b = 10^4 \\cdot \\beta + 1001 \\), compute \\( \\alpha + \\beta \\).... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b4467b057b851630f4f413e913dccd0f50b3f665135f51b33528163759dd29b | A positive integer \( n \) is defined as memorable if its binary representation contains strictly more \( 1 \)'s than \( 0 \)'s. Let \( S(k) \) be the sum of the first \( k \) memorable perfect squares. Find the remainder when \( S(10) \) is divided by \( 1000 \). | [
"277"
] | [
"A positive integer \\( n \\) is defined as memorable if its binary representation contains strictly more \\( 1 \\)'s than \\( 0 \\)'s. Let \\( S(k) \\) be the sum of the first \\( k \\) memorable perfect squares. Find the remainder when \\( S(10) \\) is divided by \\( 1000 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-8ef07763221f483bfb42f2f9fea1ff17d8dc496844002201ee756ec7599ed0b2 | A positive integer \( n \) is friendly if the difference of each pair of neighbouring digits of \( n \), written in base 10, is exactly 1. For example, 6787 is friendly, but 211 and 901 are not. Find all odd natural numbers \( m \) for which there exists a friendly integer divisible by \( 64m \). | [
"1929"
] | [
"A positive integer \\( n \\) is friendly if the difference of each pair of neighbouring digits of \\( n \\), written in base 10, is exactly 1. For example, 6787 is friendly, but 211 and 901 are not. Find all odd natural numbers \\( m \\) for which there exists a friendly integer divisible by \\( 64m \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-7981b1902be07111077d568004747adf39b3f0247cd8091918a7c1e2cc2c5b2e | A positive integer \( n \) is known as an interesting number if it satisfies \[ \left\{\frac{n}{10^k}\right\} > \frac{n}{10^{10}} \] for all \( k = 1, 2, \ldots, 9 \), where \( \{ \} \) denotes the decimal part of the number. Find the number of interesting numbers. | [
"900000000"
] | [
"A positive integer \\( n \\) is known as an interesting number if it satisfies \n\\[\n\\left\\{\\frac{n}{10^k}\\right\\} > \\frac{n}{10^{10}}\n\\]\nfor all \\( k = 1, 2, \\ldots, 9 \\), where \\( \\{ \\} \\) denotes the decimal part of the number. Find the number of interesting numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-512e91466fff3e9cbfbdec962a03ad78f7cff2bd938741923e6525fb4b77c568 | A positive integer \( n \) is known as an interesting number if it satisfies \[ \{n/10^k\} > \frac{n}{10^{10}} \] for all \( k = 1, 2, \ldots, 9 \), where \( \{x\} \) denotes the decimal part of \( x \). Find the number of interesting numbers. | [
"900000000"
] | [
"A positive integer \\( n \\) is known as an interesting number if it satisfies \n\\[\n\\{n/10^k\\} > \\frac{n}{10^{10}}\n\\]\nfor all \\( k = 1, 2, \\ldots, 9 \\), where \\( \\{x\\} \\) denotes the decimal part of \\( x \\). Find the number of interesting numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-ca31a3e9362dd3be3c81e2c88f4af3e24a7d5dc2077a8ec757a2d2aa51d3ff4a | A positive integer \( n \) is orz if \( \sqrt{n} \) is an integer and has at least 2 distinct digits, and each digit of \( \sqrt{n} \) is also a digit of \( n \). Find the smallest orz number. | [
"100"
] | [
"A positive integer \\( n \\) is orz if \\( \\sqrt{n} \\) is an integer and has at least 2 distinct digits, and each digit of \\( \\sqrt{n} \\) is also a digit of \\( n \\). Find the smallest orz number."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-b43ca7d1f1cd7f36ca4ae9dbf58e7c263e85734bafe00b2bd2b39f8909c3b482 | A positive integer \( n \) is said to be a "good number" if both \( n+1 \) and \( 8n+1 \) are perfect squares. Find all integers \( k \) satisfying \( |k| \leq 10 \) such that \( 4n + k \) is composite for all "good numbers" \( n \). | [
"-8, -6, -4, -3, -2, 0, 2, 3, 4, 6, 8, 9, 10"
] | [
"A positive integer \\( n \\) is said to be a \"good number\" if both \\( n+1 \\) and \\( 8n+1 \\) are perfect squares. Find all integers \\( k \\) satisfying \\( |k| \\leq 10 \\) such that \\( 4n + k \\) is composite for all \"good numbers\" \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-8fecb29d301901bcb51f6b8a44d78e0f233d6ba503445407bc6cda7ee90242ec | A positive integer \( n \) is said to be good if whenever \( n \) divides \( a + b + c + ab + bc + ca + abc \) for some positive integers \( a, b, \) and \( c \), then \( n \) must divide at least one of \( a, b, \) or \( c \). Determine all good integers. | [
"\\( n \\in \\{1, 2, 3, 4, 6\\} \\)"
] | [
"A positive integer \\( n \\) is said to be good if whenever \\( n \\) divides \\( a + b + c + ab + bc + ca + abc \\) for some positive integers \\( a, b, \\) and \\( c \\), then \\( n \\) must divide at least one of \\( a, b, \\) or \\( c \\). Determine all good integers."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-24196b007783fef31ca63c6e89ac0915a663311159659c67b64fc2ae9b464753 | A positive integer \( n \) is written on a blackboard. Every minute, the number \( a \) on the blackboard is erased and replaced with the sum of \( a \) and the largest power of two that divides \( a+1 \). After 2007 minutes, the number on the blackboard is \( N \). If the initial number \( n \) is 1, find the remainde... | [
"1023"
] | [
"A positive integer \\( n \\) is written on a blackboard. Every minute, the number \\( a \\) on the blackboard is erased and replaced with the sum of \\( a \\) and the largest power of two that divides \\( a+1 \\). After 2007 minutes, the number on the blackboard is \\( N \\). If the initial number \\( n \\) is 1, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17d7ea0288806b3960a7d22f1d9516cce08ea101c53df58cbb71273de543698e | A positive integer \( n \) not exceeding 100 is chosen in such a way that if \( n \le 50 \), then the probability of choosing \( n \) is \( p \), and if \( n > 50 \), then the probability of choosing \( n \) is \( 3p \). The probability that a perfect square is chosen is given. | [
"\\dfrac{2}{25}"
] | [
"A positive integer \\( n \\) not exceeding 100 is chosen in such a way that if \\( n \\le 50 \\), then the probability of choosing \\( n \\) is \\( p \\), and if \\( n > 50 \\), then the probability of choosing \\( n \\) is \\( 3p \\). The probability that a perfect square is chosen is given."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-418d8b037c7deea569ba8ca3cb7d983d6562dd84dde37e47e0d66745a6c93f17 | A positive integer \( n \) not exceeding 100 is chosen such that if \( n \le 50 \), the probability of choosing \( n \) is \( p \), and if \( n > 50 \), the probability of choosing \( n \) is \( 3p \). What is the probability that a perfect square is chosen? | [
".065"
] | [
"A positive integer \\( n \\) not exceeding 100 is chosen such that if \\( n \\le 50 \\), the probability of choosing \\( n \\) is \\( p \\), and if \\( n > 50 \\), the probability of choosing \\( n \\) is \\( 3p \\). What is the probability that a perfect square is chosen?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-b5afa1394359c0dd60dd0a768ac03c752cb5c164c1649d65c8d0ab41a9f348d9 | A positive integer \( n \) not exceeding 100 is chosen such that if \( n \leq 50 \), the probability of choosing \( n \) is \( p \), and if \( n > 50 \), the probability of choosing \( n \) is \( 3p \). What is the probability that a perfect square is chosen? | [
"0.08"
] | [
"A positive integer \\( n \\) not exceeding 100 is chosen such that if \\( n \\leq 50 \\), the probability of choosing \\( n \\) is \\( p \\), and if \\( n > 50 \\), the probability of choosing \\( n \\) is \\( 3p \\). What is the probability that a perfect square is chosen?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-4d72968641ec19230afa82c172bc719835ec5a0b80963aeb4c68d07611056cbb | A positive integer \( n \), when written in base \( b \), has the form 211. When \( n \) is written in base \( b+2 \), it has the form 110. Determine \( n \) and \( b \) (in base 10). | [
"\\( n = 56 \\), \\( b = 5 \\)"
] | [
"A positive integer \\( n \\), when written in base \\( b \\), has the form 211. When \\( n \\) is written in base \\( b+2 \\), it has the form 110. Determine \\( n \\) and \\( b \\) (in base 10)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 27
} | 30 | 0 | 0 |
omr-ws-137e9428effe1640b2c28a6c8573ceff2606a2fe67b56aa926c3cb5f9bc4b498 | A positive integer \( n \ge 4 \) is called [i]triangulable[/i] if every regular \( n \)-gon has a bicoloured triangulation such that for each vertex \( A \) of the \( n \)-gon, the number of black triangles of which \( A \) is a vertex is greater than the number of white triangles of which \( A \) is a vertex. Find all... | [
"\\( n \\) such that \\( 3 \\mid n \\) and \\( n \\ge 4 \\)"
] | [
"A positive integer \\( n \\ge 4 \\) is called [i]triangulable[/i] if every regular \\( n \\)-gon has a bicoloured triangulation such that for each vertex \\( A \\) of the \\( n \\)-gon, the number of black triangles of which \\( A \\) is a vertex is greater than the number of white triangles of which \\( A \\) is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c098763911c968f5e236957cd383bcc8215f0a7a0d9565f2cdba7161f56fee2f | A positive integer \( n \geq 2 \) is called "junk" if there exist two distinct \( n \)-digit binary strings \( a_1a_2\cdots a_n \) and \( b_1b_2\cdots b_n \) such that: 1. \( a_1 + a_2 = b_1 + b_2 \), 2. \( a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1} \) for all \( 2 \leq i \leq n-1 \), 3. \( a_{n-1} + a_n = b_{n-... | [
"672"
] | [
"A positive integer \\( n \\geq 2 \\) is called \"junk\" if there exist two distinct \\( n \\)-digit binary strings \\( a_1a_2\\cdots a_n \\) and \\( b_1b_2\\cdots b_n \\) such that:\n1. \\( a_1 + a_2 = b_1 + b_2 \\),\n2. \\( a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1} \\) for all \\( 2 \\leq i \\leq n-1 \\),... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-bbe291aefd5e9006a074d1280aba19610d3baf5f827049102f4d15167eb0cbb6 | A positive integer \( n \geq 2 \) is called a "good" number if for all integers \( 2 \leq k \leq n \), there exist positive integers \( a_1, a_2, \ldots, a_k \) such that \( \gcd(a_i, n) = 1 \) and \( n = a_1 + a_2 + \cdots + a_k \). Find the sum of all "good" numbers not exceeding 2016. | [
"1016065"
] | [
"A positive integer \\( n \\geq 2 \\) is called a \"good\" number if for all integers \\( 2 \\leq k \\leq n \\), there exist positive integers \\( a_1, a_2, \\ldots, a_k \\) such that \\( \\gcd(a_i, n) = 1 \\) and \\( n = a_1 + a_2 + \\cdots + a_k \\). Find the sum of all \"good\" numbers not exceeding 2016."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0ef9833e88b1344679b9c8376f25da461fd463ddc84846f8f452076684f60b1 | A positive integer \( n \geq 2 \) is called peculiar if the number \[ \binom{n}{i} + \binom{n}{j} - i - j \] is even for all integers \( i \) and \( j \) such that \( 0 \leq i \leq j \leq n \). Determine all peculiar numbers. | [
"\\(2^k - 2\\) for integer \\(k\\)."
] | [
"A positive integer \\( n \\geq 2 \\) is called peculiar if the number \n\\[\n\\binom{n}{i} + \\binom{n}{j} - i - j\n\\]\nis even for all integers \\( i \\) and \\( j \\) such that \\( 0 \\leq i \\leq j \\leq n \\). Determine all peculiar numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 28
} | 75 | 0 | 0 |
omr-ws-5bccd026ab534f72782ef5edb57a766650ae59782edac84d2d44dd9163892c05 | A positive integer \( n \geq 4 \) is called interesting if there exists a complex number \( z \) such that \( |z| = 1 \) and \[ 1 + z + z^2 + z^{n-1} + z^n = 0. \] Find how many interesting numbers are smaller than 2022. | [
"404"
] | [
"A positive integer \\( n \\geq 4 \\) is called interesting if there exists a complex number \\( z \\) such that \\( |z| = 1 \\) and \n\\[ 1 + z + z^2 + z^{n-1} + z^n = 0. \\]\nFind how many interesting numbers are smaller than 2022."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-f39c0d08cb6b78c627d362d0a260f4c1f78f416097c83b64da12ac808e55f7b8 | A positive integer \( u \) is called [i]boring[/i] if there are only finitely many triples of positive integers \( (n, a, b) \) such that \( n! = u^a - u^b \). Determine all the [i]boring[/i] integers. | [
"All positive integers \\( u \\) are boring."
] | [
"A positive integer \\( u \\) is called [i]boring[/i] if there are only finitely many triples of positive integers \\( (n, a, b) \\) such that \\( n! = u^a - u^b \\). Determine all the [i]boring[/i] integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 13
} | 20 | 0 | 0 |
omr-ws-2e8907d7d7dc7f0133b365a87b4b9568ca705599cfd0746fcaa8d7cc5cf12c7d | A positive integer \( u \) is called boring if there are only finitely many triples of positive integers \((n, a, b)\) such that \( n! = u^a - u^b \). Determine all the boring integers. | [
"1"
] | [
"A positive integer \\( u \\) is called boring if there are only finitely many triples of positive integers \\((n, a, b)\\) such that \\( n! = u^a - u^b \\). Determine all the boring integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b091e940c82e6398f263ae50e69c25c58af5296bd796e313c0be06335ad0d095 | A positive integer \( x \) is defined as [i]tivisible[/i] by a positive integer \( y \) if the number of factors of \( x \) (including 1 and itself) is divisible by the number of factors of \( y \) (including 1 and itself). How many positive integers less than 1000 are 1000 tivisible by? | [
"543"
] | [
"A positive integer \\( x \\) is defined as [i]tivisible[/i] by a positive integer \\( y \\) if the number of factors of \\( x \\) (including 1 and itself) is divisible by the number of factors of \\( y \\) (including 1 and itself). How many positive integers less than 1000 are 1000 tivisible by?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4fdbf3e348fd0a4128a57968ff1a3d1416dba9ae63c729de2c0cc850a00ff426 | A positive integer \( x \) is sunny if \( 3x \) has more digits than \( x \). If all sunny numbers are written in increasing order, what is the 50th number written? | [
"77"
] | [
"A positive integer \\( x \\) is sunny if \\( 3x \\) has more digits than \\( x \\). If all sunny numbers are written in increasing order, what is the 50th number written?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8fe9839b2c25b0010f637a3c1bcc81d23d8162008130015bfe18319556722aea | A positive integer \( x \) satisfies the equation: \[ \left\{\frac{x}{3}\right\} + \left\{\frac{x}{5}\right\} + \left\{\frac{x}{7}\right\} + \left\{\frac{x}{11}\right\} = \frac{248}{165} \] Find all possible values of: \[ \left\{\frac{2x}{3}\right\} + \left\{\frac{2x}{5}\right\} + \left\{\frac{2x}{7}\right\} + \left\{\... | [
"\\(\\frac{166}{165}\\)"
] | [
"A positive integer \\( x \\) satisfies the equation:\n\\[\n\\left\\{\\frac{x}{3}\\right\\} + \\left\\{\\frac{x}{5}\\right\\} + \\left\\{\\frac{x}{7}\\right\\} + \\left\\{\\frac{x}{11}\\right\\} = \\frac{248}{165}\n\\]\nFind all possible values of:\n\\[\n\\left\\{\\frac{2x}{3}\\right\\} + \\left\\{\\frac{2x}{5}\\ri... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 28
} | 74 | 0 | 0 |
omr-ws-2e98e34640994dd75e4a952cc04ae43b6d1b2f77308c4b0bca831d8f94e6fe58 | A positive integer \(a > 1\) is given (in decimal notation). We copy it twice and obtain a number \(b = \overline{aa}\) which happens to be a multiple of \(a^2\). Find all possible values of \(b/a^2\). | [
"1, 7, 8, 9, 10"
] | [
"A positive integer \\(a > 1\\) is given (in decimal notation). We copy it twice and obtain a number \\(b = \\overline{aa}\\) which happens to be a multiple of \\(a^2\\). Find all possible values of \\(b/a^2\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63aba7d637eb60bd1bacb8d6d27a0680acb5cc0bb140c15361f453832a79288b | A positive integer \(m\) is palindromic in base \(\ell\) if its digits are the same front-to-back and back-to-front in base \(\ell\). For \(j, k \in \mathbb{N}\), let \(\mu(j, k)\) be the smallest base-10 integer that is palindromic in both base \(j\) and base \(k\), and let \(\nu(j, k) := (j + k) \cdot \mu(j, k)\). Fi... | [
"14"
] | [
"A positive integer \\(m\\) is palindromic in base \\(\\ell\\) if its digits are the same front-to-back and back-to-front in base \\(\\ell\\). For \\(j, k \\in \\mathbb{N}\\), let \\(\\mu(j, k)\\) be the smallest base-10 integer that is palindromic in both base \\(j\\) and base \\(k\\), and let \\(\\nu(j, k) := (j ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-ce4c5105b298f3bcab2932b564aa389d3ba1a0df71ced9caefd7bab9bcebb42d | A positive integer \(n\) is called a good number if every integer multiple of \(n\) is divisible by \(n\) however its digits are rearranged. How many good numbers are there? | [
"3"
] | [
"A positive integer \\(n\\) is called a good number if every integer multiple of \\(n\\) is divisible by \\(n\\) however its digits are rearranged. How many good numbers are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cc5a75c76da89151787cc51eb54da123e6b25b75ecf7ffcb36a751cd1bdae342 | A positive integer \(n\) is such that \(n^k\) has initial digit \(k\) for all \(1 \le k \le m\). What is the maximum \(m\) across all \(n\)? | [
"4"
] | [
"A positive integer \\(n\\) is such that \\(n^k\\) has initial digit \\(k\\) for all \\(1 \\le k \\le m\\). What is the maximum \\(m\\) across all \\(n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-9e403a8c4cb22bc7aa850f77313b91d80dffd18808262d3ddc4a1dc0ccbf05b5 | A positive integer belongs to the set \( S \) if the sum of all its integer factors is 256. Find the sum of all distinct elements of \( S \). It is given that this sum is a non-negative integer under 1000. | [
"217"
] | [
"A positive integer belongs to the set \\( S \\) if the sum of all its integer factors is 256. Find the sum of all distinct elements of \\( S \\). It is given that this sum is a non-negative integer under 1000."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-3fea0d4603e4705d023733a3e5c273ee5c4cd5b3164707d5f7df05de0b26e936 | A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | [
"23"
] | [
"A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-fd3146bd4a6d7f1d3f805e029ff1c37ee6fda7a8c7524a8a10560f07edb65bc2 | A positive integer gets reduced by 9 times when one of its digits is deleted, and the resultant number is divisible by 9. Prove that to find the resultant number by 9, it is again sufficient to delete one of its digits. Find all such numbers. | [
"\\( 108 \\times 10^n \\) for \\( n \\geq 0 \\)"
] | [
"A positive integer gets reduced by 9 times when one of its digits is deleted, and the resultant number is divisible by 9. Prove that to find the resultant number by 9, it is again sufficient to delete one of its digits. Find all such numbers."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d73755a4ba6a8430ca0ca38bafd939edb50216b91536934d74e6a257a838c1c | A positive integer gets reduced by 9 times when one of its digits is deleted, and the resulting number is divisible by 9. Prove that to divide the resulting number by 9, it is sufficient to delete one of its digits. Find all such numbers. | [
"108, 198, 207, 288, 297, 306, 378, 387, 396, 405, 468, 477, 486, 495, 504, 558, 567, 576, 585, 594, 603, 648, 657, 666, 675, 684, 693, 702, 738, 747, 756, 765, 774, 783, 792, 801, 828, 837, 846, 855, 864, 873, 882, 891, 900, 909, 918, 927, 936, 945, 954, 963, 972, 981, 990"
] | [
"A positive integer gets reduced by 9 times when one of its digits is deleted, and the resulting number is divisible by 9. Prove that to divide the resulting number by 9, it is sufficient to delete one of its digits. Find all such numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a17f43c57e34078259d3a9708aa53c9bea86019ebfdf8e5682f819ea2b60efbb | A positive integer has 2 terminal zeroes when written in base twelve and 1 terminal zero when written in base eighteen. In how many other positive integral bases greater than one must the representation of this same positive integer end in at least 2 terminal zeroes? | [
"5"
] | [
"A positive integer has 2 terminal zeroes when written in base twelve and 1 terminal zero when written in base eighteen. In how many other positive integral bases greater than one must the representation of this same positive integer end in at least 2 terminal zeroes?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9178583e26be98b6e00f2bf3a6a6ddad1a585a516e5416aca76412e8438ca2b | A positive integer has exactly $81$ divisors, which are arranged in a $9 \times 9$ table such that for any two numbers in the same row or column, one of them is divisible by the other. Find the maximum possible number of distinct prime divisors of this integer. | [
"2"
] | [
"A positive integer has exactly $81$ divisors, which are arranged in a $9 \\times 9$ table such that for any two numbers in the same row or column, one of them is divisible by the other. Find the maximum possible number of distinct prime divisors of this integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-4c74b3311155fb359d259557b968e6239dda9934d735fe3ff363b7ebf7ad03c9 | A positive integer has one terminal zero when written in base 12 and two terminal zeroes when written in base 5. In how many other positive integral bases greater than one must the representation of this same positive integer have at least one terminal zero? | [
"15"
] | [
"A positive integer has one terminal zero when written in base 12 and two terminal zeroes when written in base 5. In how many other positive integral bases greater than one must the representation of this same positive integer have at least one terminal zero?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-1c73b7075b112f460293d39b46312221839d5b000101319d607e86b160210f63 | A positive integer is called "almost prime" if it is not divisible by any prime from the interval $[3,19]$. A number is called "very non-prime" if it has at least 2 primes from the interval $[3,19]$ dividing it. What is the greatest number of almost prime numbers that can be selected such that the sum of any two of the... | [
"4"
] | [
"A positive integer is called \"almost prime\" if it is not divisible by any prime from the interval $[3,19]$. A number is called \"very non-prime\" if it has at least 2 primes from the interval $[3,19]$ dividing it. What is the greatest number of almost prime numbers that can be selected such that the sum of any t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-f0fc1e03a19f8db4fa86484cfd23916d8fc913f7ff34c60484b5981c8a0b36c5 | A positive integer is called "funny" if for some \( k \), the sum of its digits equals \( k \) and the number itself is divisible by \( k+1 \). What is the maximum number of consecutive integers that can be funny? | [
"2"
] | [
"A positive integer is called \"funny\" if for some \\( k \\), the sum of its digits equals \\( k \\) and the number itself is divisible by \\( k+1 \\). What is the maximum number of consecutive integers that can be funny?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-f188c60434a14dfa5f44bb175ce1ff0aefd7bee8dac73ad702a1ccf300ee3540 | A positive integer is called "good" if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Let $P$ be the product of two distinct "good" numbers. Find the smallest possible value of $P$ that is also "good". | [
"135"
] | [
"A positive integer is called \"good\" if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Let $P$ be the product of two distinct \"good\" numbers. Find the smallest possible value of $P$ that is also \"good\"."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d895bef8006a95987deaf0f79a90c63d17a752f8654b9a86e45e295a262be466 | A positive integer is called "lucky" if 31 divides the sum of its digits. What is the maximum possible difference between two consecutive "lucky" numbers? | [
"1899"
] | [
"A positive integer is called \"lucky\" if 31 divides the sum of its digits. What is the maximum possible difference between two consecutive \"lucky\" numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bf32508e3e883109c3e44e94463b1a6afd992958986f3ec8b033deb18c3acfa | A positive integer is called "lucky" if the sum of its digits is divisible by $31$. What is the maximum possible difference between two consecutive "lucky" numbers? | [
"4000"
] | [
"A positive integer is called \"lucky\" if the sum of its digits is divisible by $31$. What is the maximum possible difference between two consecutive \"lucky\" numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-ac5eaa83dd86e887698d871e975928424f65db33d7d446081beb505e73dbf439 | A positive integer is called "tico" if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. | [
"2014 is tico."
] | [
"A positive integer is called \"tico\" if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-cb5ea08b76b47afcc2c12dcdde3944ff83231ce911bd45fffc1bacc463bcc7bc | A positive integer is called $\textit{cheesy}$ if the average of its digits can be obtained by placing a decimal separator after the leftmost digit. For example, $123$ is cheesy because $\frac{1+2+3}{3} = 2.0$, which can be obtained by placing a decimal separator after the leftmost digit to get $1.23$. Find the sum of ... | [
"90"
] | [
"A positive integer is called $\\textit{cheesy}$ if the average of its digits can be obtained by placing a decimal separator after the leftmost digit. For example, $123$ is cheesy because $\\frac{1+2+3}{3} = 2.0$, which can be obtained by placing a decimal separator after the leftmost digit to get $1.23$. Find the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2421c4660747ccef4b079eb9e8523f9c8bac0bdfe94adb595f751bf2ea9ed8e5 | A positive integer is called $cyclic$ if it is not divisible by the square of any prime, and whenever $p<q$ are primes that divide it, $q$ does not leave a remainder of $1$ when divided by $p$. Compute the number of cyclic numbers less than or equal to $1000$. | [
"325"
] | [
"A positive integer is called $cyclic$ if it is not divisible by the square of any prime, and whenever $p<q$ are primes that divide it, $q$ does not leave a remainder of $1$ when divided by $p$. Compute the number of cyclic numbers less than or equal to $1000$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 11
} | 16 | 0 | 0 |
omr-ws-7d68ed18edea848313f91d9cb0e6bc540a301ce92edfa09bdf0b03c7a431adc1 | A positive integer is called **bold** if it has 8 positive divisors that sum up to 3240. For example, 2006 is bold because its 8 positive divisors, 1, 2, 17, 34, 59, 118, 1003, and 2006, sum up to 3240. Find the smallest positive bold number. | [
"1614"
] | [
"A positive integer is called **bold** if it has 8 positive divisors that sum up to 3240. For example, 2006 is bold because its 8 positive divisors, 1, 2, 17, 34, 59, 118, 1003, and 2006, sum up to 3240. Find the smallest positive bold number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-48c4389319a1aca962fd46b01dacf26382211f456eb6aad097066f95724d4a73 | A positive integer is called *good* if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Let $S(n)$ be the sum of all *good* integers less than $n$. Calculate $S(5000)$. | [
"2081664"
] | [
"A positive integer is called *good* if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Let $S(n)$ be the sum of all *good* integers less than $n$. Calculate $S(5000)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d799f13a880964c7e59ced46fae0365634863dba9fd04b61b33fd7327fbc655a | A positive integer is called *good* if it has exactly four positive divisors. Let $S(n)$ denote the number of sequences of seven consecutive good integers that end with $n$. Calculate the smallest positive integer $N$ such that $S(N) \geq 2$. | [
"no_valid_answer_found"
] | [
"A positive integer is called *good* if it has exactly four positive divisors. Let $S(n)$ denote the number of sequences of seven consecutive good integers that end with $n$. Calculate the smallest positive integer $N$ such that $S(N) \\geq 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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