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: 545

2022 IMO, 5
Tags: number theory , divisibility , lte lemma , diophantine equation
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

2008 Germany Team Selection Test, 3
Tags: modular arithmetic , number theory , divisibility , extremal combinatorics , additive combinatorics
Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

2024 Baltic Way, 17
Tags: prime number , divisibility , bound , number theory , fermat's little theorem
Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

2010 Peru IMO TST, 9
Tags: modular arithmetic , number theory , sequence , divisibility
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

1988 IMO Longlists, 14
Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

2025 Bulgarian Winter Tournament, 12.3
Tags: large number , function , sequence , divisibility , polynomial , number theory
Determine all functions $f: \mathbb{Z}_{\geq 2025} \to \mathbb{Z}_{>0}$ such that $mn+1$ divides $f(m)f(n) + 1$ for any integers $m,n \geq 2025$ and there exists a polynomial $P$ with integer coefficients, such that $f(n) \leq P(n)$ for all $n\geq 2025$.

2008 Greece Team Selection Test, 1
Tags: algebra , polynomial , divisibility
Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$ is divisible by $\phi (x)=x^2-x+1$

2023 JBMO TST - Turkey, 4
Tags: number theory , divisibility
For a prime number $p$. Can the number of n positive integers that make the expression \[\dfrac{n^3+np+1}{n+p+1}\] an integer be $777$?

2020 AMC 10, 6
Tags: divisibility
How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$ $\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$

2020 Iran Team Selection Test, 5
Tags: fractional part , algebra , divisibility
For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$: $$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$ [i]Proposed by Mohammad Amin Sharifi[/i]

1979 IMO Longlists, 25
Tags: number theory , relatively prime , series summation , divisibility , prime
If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

2012 Bogdan Stan, 4
Tags: linear algebra , matrix , gcd , divisibility , induction
Prove that the elements of any natural power of a $ 2\times 2 $ special linear integer matrix are pairwise coprime, with the possible exception of the pairs that form the diagonals. [i]Vasile Pop[/i]

1978 IMO Longlists, 52
Tags: modular arithmetic , algebra , polynomial , number theory , divisibility
Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way: $(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$ $(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set. Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$

Kvant 2023, M2768
Tags: number theory , divisibility
Let $n{}$ be a natural number. The pairwise distinct nonzero integers $a_1,a_2,\ldots,a_n$ have the property that the number \[(k+a_1)(k+a_2)\cdots(k+a_n)\]is divisible by $a_1a_2\cdots a_n$ for any integer $k{}.$ Find the largest possible value of $a_n.$ [i]Proposed by F. Petrov and K. Sukhov[/i]

2023 Switzerland - Final Round, 6
Tags: number theory , prime , divisibility
Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

1992 IMO Longlists, 31
Tags: algebra , polynomial , number theory , divisibility , coefficient
Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$

1997 Slovenia National Olympiad, Problem 1
Tags: number theory , divisibility
Suppose that $m,n$ are integers greater than $1$ such that $m+n-1$ divides $m^2+n^2-1$. Prove that $m+n-1$ cannot be a prime number.

2005 Federal Math Competition of S&M, Problem 1
Tags: power mean , divisibility , algebra , number theory
Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.

2025 India National Olympiad, P1
Tags: recurrence , divisibility , induct
Consider the sequence defined by \(a_1 = 2\), \(a_2 = 3\), and \[ a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1}, \] for all integers \(k \geq 1\). Determine all positive integers \(n\) such that \[ \frac{a_n}{n} \] is an integer. Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.

2007 Germany Team Selection Test, 3
Tags: modular arithmetic , number theory , divisibility , exponential
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

2021 IMO Shortlist, N1
Tags: number theory , divisibility , cubefree
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2023 UMD Math Competition Part II, 3
Tags: number theory , combination , divisibility
Let $p$ be a prime, and $n > p$ be an integer. Prove that \[ \binom{n+p-1}{p} - \binom{n}{p} \] is divisible by $n$.

2007 IMO Shortlist, 2
Tags: modular arithmetic , number theory , divisibility
Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

2005 IberoAmerican, 3
Tags: modular arithmetic , algebra , number theory , divisibility
Let $p > 3$ be a prime. Prove that if \[ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m}, \] with $\gdc(n,m) = 1$, then $p^3$ divides $n$.

1997 IMO Shortlist, 14
Tags: number theory , prime number , divisibility , power of 2 , prime divisor
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.