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: 536

2016 Middle European Mathematical Olympiad, 4
Tags: number theory , number theory proposed , functional equation , function , Divisibility , algebra
Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$. Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.

2004 Unirea, 2
Tags: arithmetic sequence , Divisibility , algebra
Find the arithmetic sequences of $ 5 $ integers $ n_1,n_2,n_3,n_4,n_5 $ that verify $ 5|n_1,2|n_2,11|n_3,7|n_4,17|n_5. $

1990 IMO Shortlist, 23
Tags: modular arithmetic , number theory , Divisibility , IMO 1990 , Orders And LTE , P3
Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

1989 IMO Shortlist, 27
Tags: modular arithmetic , number theory , least common multiple , Divisibility , IMO Shortlist
Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n \minus{} 1.$

2013 Vietnam Team Selection Test, 2
Tags: number theory , Perfect Squares , Divisibility , function , infinitely many solutions
a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares. b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.

1975 IMO, 4
Tags: modular arithmetic , Divisibility , Digits , decimal representation , IMO , IMO 1975 , digit sum
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

1969 IMO Shortlist, 25
Tags: number theory , Divisibility , Frobenius , Additive Number Theory , IMO Shortlist , IMO Longlist
$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

2024 VJIMC, 4
Tags: number theory , Analytic Number Theory , Divisibility
Let $p>2$ be a prime and let \[\mathcal{A}=\{n \in \mathbb{N}: 2p \mid n \text{ and } p^2\nmid n \text{ and } n \mid 3^n-1\}.\] Prove that \[\limsup_{k \to \infty} \frac{\vert \mathcal{A} \cap [1,k]\vert}{k} \le \frac{2\log 3}{p\log p}.\]

1999 IMO Shortlist, 3
Tags: modular arithmetic , number theory , Integer sequence , Divisibility , Sequence , IMO Shortlist , Hi
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

2015 Turkey MO (2nd round), 1
Tags: number theory , Turkey , Divisibility , Perfect Square
$m$ and $n$ are positive integers. If the number \[ k=\dfrac{(m+n)^2}{4m(m-n)^2+4}\] is an integer, prove that $k$ is a perfect square.

1969 IMO Shortlist, 13
Tags: modular arithmetic , number theory , Divisibility , Additive Number Theory , IMO Shortlist , IMO Longlist
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

1988 IMO, 3
Tags: number theory , Divisibility , Diophantine equation , Perfect Square , Vieta Jumping , IMO , IMO 1988
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.

1967 IMO Longlists, 17
Tags: number theory , Divisibility , binomial coefficients , IMO , IMO 1967
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

2011 China National Olympiad, 3
Tags: modular arithmetic , algorithm , number theory , prime numbers , Divisibility
Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]

2023 239 Open Mathematical Olympiad, 5
Tags: number theory , Divisibility
Let $a{}$ and $b>1$ be natural numbers. Prove that there exists a natural number $n < b^2$ such that the number $a^n + n$ is divisible by $b{}$.

1990 IMO, 3
Tags: modular arithmetic , number theory , Divisibility , IMO 1990 , Orders And LTE , P3
Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

2022 Spain Mathematical Olympiad, 6
Tags: Spain , number theory , Divisibility
Find all triples $(x,y,z)$ of positive integers, with $z>1$, satisfying simultaneously that \[x\text{ divides }y+1,\quad y\text{ divides }z-1,\quad z\text{ divides }x^2+1.\]

1962 IMO Shortlist, 1
Tags: number theory , decimal representation , Divisibility , IMO , IMO 1962
Find the smallest natural number $n$ which has the following properties: a) Its decimal representation has a 6 as the last digit. b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

1969 Bulgaria National Olympiad, Problem 1
Tags: number theory , Divisibility
Prove that if the sum of $x^5,y^5$ and $z^5$, where $x,y$ and $z$ are integer numbers, is divisible by $25$ then the sum of some two of them is divisible by $25$.

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 $\phi (x)=x^2-x+1$

Russian TST 2016, P1
Tags: number theory , Divisibility
Let $a{}$ and $b{}$ be natural numbers greater than one. Let $n{}$ be a natural number for which $a\mid 2^n-1$ and $b\mid 2^n+1$. Prove that there is no natural $k{}$ such that $a\mid 2^k+1$ and $b\mid 2^k-1$.

2011 N.N. Mihăileanu Individual, 1
Tags: number theory , Divisibility
[b]a)[/b] Prove that $ 4040100 $ divides $ 2009\cdot 2011^{2011}+1. $ [i]Gabriel Iorgulescu[/i] [b]b)[/b] Let be three natural numbers $ x,y,z $ with the property that $ (1+\sqrt 2)^x=y^2+2z^2+2yz\sqrt 2. $ Show that $ x $ is even. [i]Marius Cavachi[/i]

2018 Thailand TSTST, 3
Tags: number theory , Divisibility , Bulgaria
Find all pairs of integers $m, n \geq 2$ such that $$n\mid 1+m^{3^n}+m^{2\cdot 3^n}.$$

2023 Abelkonkurransen Finale, 3b
Tags: number theory , Divisibility
Find all integers $a$ and $b$ satisfying \begin{align*} a^6 + 1 & \mid b^{11} - 2023b^3 + 40b, \qquad \text{and}\\ a^4 - 1 & \mid b^{10} - 2023b^2 - 41. \end{align*}

1999 Romania Team Selection Test, 10
Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.