This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 14

2006 Germany Team Selection Test, 2
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]
factorial modular arithmetic chinese remainder theorem exponential number theory divisibility Chinese remainder theorem Euler's theorem bijection factorization p-adic valuation prime factorization quadratic residues

2013 ELMO Problems, 3
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]
modular arithmetic chinese remainder theorem relatively prime number theory construction greedy algorithm mathematical induction pigeonhole principle recurrence relations

2024 Brazil Cono Sur TST, 3
Find all positive integers $m$ that have some multiple of the form $x^2+5y^2+2024$, with $x$ and $y$ integers.
quadratic residue chinese remainder theorem number theory hensel s lemma divisibility bijection pigeonhole principle quadratic residues

2024 Thailand Mathematical Olympiad, 9
Prove that for all positive integers $n$, there exists a sequence of positive integers $a_1,a_2,\dots,a_n$ and $d_1,d_2,\dots,d_n$ satisfying all of the following three conditions. [list] [*] $\binom{2a_i}{a_i}$ is divisible by $d_i$ for all $i=1,2,\dots,n$ [*] $d_{i+1}=d_i+1$ for all $i=1,2,\dots, n-1$ [*] $d_i\neq m^k$ for all $i=1,2,\dots, n$ and positive integers $m$ and $k$ such that $k\geq 2$ [/list]
chinese remainder theorem number theory Chinese remainder theorem modular arithmetic sequences

2006 France Team Selection Test, 3
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]
modular arithmetic chinese remainder theorem number theory divisibility Chinese remainder theorem

2013 ELMO Shortlist, 5
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]
modular arithmetic chinese remainder theorem relatively prime number theory construction greedy algorithm mathematical induction pigeonhole principle recurrence relations

2005 IMO Shortlist, 4
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]
factorial modular arithmetic chinese remainder theorem exponential number theory divisibility Chinese remainder theorem Euler's theorem bijection factorization p-adic valuation prime factorization quadratic residues

2006 Germany Team Selection Test, 2
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]
factorial modular arithmetic chinese remainder theorem exponential number theory divisibility Chinese remainder theorem Euler's theorem bijection factorization p-adic valuation prime factorization quadratic residues

2017 Ukraine Team Selection Test, 12
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]
modular arithmetic chinese remainder theorem relatively prime number theory construction greedy algorithm mathematical induction pigeonhole principle recurrence relations

2022 Korea -Final Round, P5
Find all positive integers $m$ such that there exists integers $x$ and $y$ that satisfies $$m \mid x^2+11y^2+2022.$$
quadratic residue chinese remainder theorem number theory hensel s lemma divisibility Cauchy-Schwarz inequality Chinese remainder theorem mathematical induction modular arithmetic quadratic residues

2013 ELMO Shortlist, 5
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]
modular arithmetic chinese remainder theorem relatively prime number theory construction greedy algorithm mathematical induction pigeonhole principle recurrence relations

2006 France Team Selection Test, 3
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]
modular arithmetic chinese remainder theorem number theory divisibility Chinese remainder theorem

2005 IMO Shortlist, 6
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]
modular arithmetic chinese remainder theorem number theory divisibility Chinese remainder theorem

2006 Taiwan TST Round 1, 3
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]
modular arithmetic chinese remainder theorem number theory divisibility Chinese remainder theorem