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

2006 Germany Team Selection Test, 2
Tags: factorial , modular arithmetic , number theory , divisibility , exponential , chinese remainder theorem
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]

2021 Taiwan TST Round 1, N
Tags: number theory , sequence , divisibility
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

2022 Taiwan TST Round 1, 2
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 $$n=\frac{ab+3b+8}{a^2+b+3}.$$

1969 IMO Longlists, 24
Tags: algebra , polynomial , number theory , divisibility
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

1978 IMO Shortlist, 15
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 $f(B) = n.$

2017 JBMO Shortlist, NT2
Tags: number theory , divisibility , factorial
Determine all positive integers n such that $n^2/ (n - 1)!$

2014 India Regional Mathematical Olympiad, 3
Tags: number theory , divisibility , positive integer
Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.

1984 IMO, 2
Tags: algebra , polynomial , modular arithmetic , number theory , divisibility
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

2023 Thailand TST, 1
Tags: number theory , prime , divisibility
Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2024 Czech-Polish-Slovak Junior Match, 4
Tags: divisibility , floor function , number theory
How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?

2010 IMO Shortlist, 4
Tags: modular arithmetic , divisibility , number theory
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

2017 Thailand TSTST, 2
Tags: number theory , divisibility
$\text{(i)}$ Does there exist a positive integer $m > 2016^{2016}$ such that $\frac{2016^m-m^{2016}}{m+2016}$ is a positive integer? $\text{(ii)}$ Does there exist a positive integer $m > 2017^{2017}$ such that $\frac{2017^m-m^{2017}}{m+2017}$ is a positive integer? [i](Serbia MO 2016 P1)[/i]

2016 Rioplatense Mathematical Olympiad, Level 3, 6
Tags: number theory , sum of digits , sequence , divisibility
When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions: (i) $n$ divides $A_m$, (ii) $n$ divides $m$, (iii) $n$ divides the sum of the digits of $A_m$.

2010 Belarus Team Selection Test, 2.2
Tags: number theory , prime , divisibility
Let $p$ be a positive prime integer, $S(p)$ be the number of triples $(x,y,z)$ such that $x,y,z\in\{0,1,..., p-1\}$ and $x^2+y^2+z^2$ is divided by $p$. Prove that $S(p) \ge 2p- 1$. (I. Bliznets)

IMSC 2024, 6
Tags: polynomial , lifting the exponent , divisibility , number theory , algebra
Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that $$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$ is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial. [i]Proposed by Vlad Matei, Romania[/i]

2005 India IMO Training Camp, 2
Tags: function , number theory , algebra , divisibility , functional equation
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

2006 IMO Shortlist, 7
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]

2025 6th Memorial "Aleksandar Blazhevski-Cane", P5
Tags: number theory , divisibility , 2-adic valuation
Let $s < t$ be positive integers. Define a sequence by: $a_1 = s, a_2 = t$; $a_3$ is the smallest integer that's greater than $a_2$ and divisible by $a_1$; in general, $a_{n + 1}$ is the smallest integer greater than $a_n$ that's divisible by $a_1, a_2, ..., a_{n - 2}, a_{n - 1}$. [b]a)[/b] What is the maximum number of odd integers that can appear in such a sequence? (Justify your answer) [b]b)[/b] Prove that $a_{2025}$ is divisible by $2^{808}$, regardless of the choice of $s$ and $t$. Proposed by [i]Ilija Jovcevski[/i]

2014 India Regional Mathematical Olympiad, 6
Tags: combinatorics , number theory , divisibility
In the adjacent fi gure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3$?

2015 IFYM, Sozopol, 1
Tags: number theory , prime number , divisibility
Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.

1969 IMO Shortlist, 34
Tags: number theory , divisibility
$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$

2002 Federal Math Competition of S&M, Problem 3
Tags: number theory , divisibility
Let $m$ and $n$ be positive integers. Prove that the number $2n-1$ is divisible by $(2^m-1)^2$ if and only if $n$ is divisible by $m(2^m-1)$.

2015 Cono Sur Olympiad, 1
Tags: divisibility , number theory
Show that, for any integer $n$, the number $n^3 - 9n + 27$ is not divisible by $81$.

2021 China Team Selection Test, 2
Tags: number theory , number sequence , divisibility
Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.

2018 Bulgaria EGMO TST, 2
Tags: number theory , fermat's little theorem , divisibility
Let $m,n \geq 2$ be integers with gcd$(m,n-1) = $gcd$(m,n) = 1$. Prove that among $a_1, a_2, \ldots, a_{m-1}$, where $a_1 = mn+1, a_{k+1} = na_k + 1$, there is at least one composite number.