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

2011 All-Russian Olympiad, 1
Tags: floor function , combinatorics , Divisibility
For some 2011 natural numbers, all the $\frac{2010\cdot 2011}{2}$ possible sums were written out on a board. Could it have happened that exactly one third of the written numbers were divisible by three and also exactly one third of them give a remainder of one when divided by three?

1993 IMO Shortlist, 2
Tags: modular arithmetic , number theory , Divisibility , prime numbers , composite numbers , IMO Shortlist
A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

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

Dumbest FE I ever created, 1.
Tags: function , number theory , functional equation , Divisibility , Euler s Phi Function
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$, $$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$

2012 Brazil Team Selection Test, 3
Tags: modular arithmetic , number theory , prime , Divisibility
Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

2024 Romania Team Selection Tests, P3
Tags: number theory , Divisibility
Let $n{}$ be a positive integer and let $a{}$ and $b{}$ be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer $k{}$ such that at least one of the numbers $a^k-b$ and $b^k-a$ is divisible by $2^n.$ [i]Cătălin Liviu Gherghe[/i]

2015 Israel National Olympiad, 7
Tags: Fibonacci , Fibonacci sequence , prime numbers , Divisibility , number theory , combinatorics
The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number. [list=a] [*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$. [*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$. [/list]

1998 IMO Shortlist, 3
Tags: modular arithmetic , pigeonhole principle , number theory , Divisibility , IMO Shortlist , combinatorics
Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.

2011 IMO Shortlist, 1
Tags: algorithm , number theory , prime numbers , Divisibility , IMO Shortlist , number of divisors
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

1993 IMO Shortlist, 4
Tags: induction , number theory , Subsets , Additive Number Theory , IMO Shortlist , Divisibility
Show that for any finite set $S$ of distinct positive integers, we can find a set $T \supseteq S$ such that every member of $T$ divides the sum of all the members of $T$. [b]Original Statement:[/b] A finite set of (distinct) positive integers is called a [b]DS-set[/b] if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some [b]DS-set[/b].

2022 Greece Team Selection Test, 1
Tags: number theory , IMO Shortlist , Divisibility , cubefree , Hi
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 $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2007 Nicolae Coculescu, 4
Tags: Totient , Euler , number theory , Divisibility
Prove that $ p $ divides $ \varphi (1+a^p) , $ where $ a\ge 2 $ is a natural number, $ p $ is a prime, and $ \varphi $ is Euler's totient. [i]Cristinel Mortici[/i]

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]

2021 South Africa National Olympiad, 1
Tags: number theory , Divisibility
Find the smallest and largest integers with decimal representation of the form $ababa$ ($a \neq 0$) that are divisible by $11$.

1980 Bundeswettbewerb Mathematik, 1
Tags: game , number theory , Divisibility , Digits
Six free cells are given in a row. Players $A$ and $B$ alternately write digits from $0$ to $9$ in empty cells, with $A$ starting. When all the cells are filled, one considers the obtained six-digit number $z$. Player $B$ wins if $z$ is divisible by a given natural number $n$, and loses otherwise. For which values of $n$ not exceeding $20$ can $B$ win independently of his opponent’s moves?

2009 Germany Team Selection Test, 2
Tags: number theory , modular arithmetic , Sequence , Divisibility , IMO Shortlist
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2022 Korea Winter Program Practice Test, 6
Tags: number theory , Divisibility
Determine all positive integers $(x_1,x_2,x_3,y_1,y_2,y_3)$ such that $y_1+ny_2^n+n^2y_3^{2n}$ divides $x_1+nx_2^n+n^2x_3^{2n}$ for all positive integer $n$.

2009 Belarus Team Selection Test, 2
Tags: number theory , modular arithmetic , Sequence , Divisibility , IMO Shortlist
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

1976 Bundeswettbewerb Mathematik, 1
Tags: number theory , Divisibility
Prove that if $n$ is an odd natural number, then $1^n +2^n +\cdots +n^n$ is divisible by $n^2$.

1988 IMO Shortlist, 9
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.

STEMS 2021 Math Cat B, Q2
Tags: polynomial , power of 2 , prime numbers , Divisibility , kobayashi
Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$

2010 Belarus Team Selection Test, 6.1
Tags: function , algebra , modular arithmetic , number theory , Divisibility , IMO Shortlist
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

1990 IMO Longlists, 75
Tags: number theory , Divisibility , binary representation , minimization , IMO Shortlist
Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that \[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\] is an integer.

1985 Traian Lălescu, 1.1
Tags: Divisibility , algebra , Polynomials , polynomial
Prove that for all $ n\ge 2 $ natural numbers there exist $ a_n\in\mathbb{Q} $ such that $$ X^{2n}+a_nX^n+1\Huge\vdots X^2+\frac{1}{2}X+1, $$ and that there isn´t any $ a_n\in\mathbb{R}\setminus\mathbb{Q} $ with this property.

2015 Romania Team Selection Tests, 1
Tags: Sum , Divisibility , number theory , Romanian TST , GCD
Let $a$ be an integer and $n$ a positive integer . Show that the sum : $$\sum_{k=1}^{n} a^{(k,n)}$$ is divisible by $n$ , where $(x,y)$ is the greatest common divisor of the numbers $x$ and $y$ .