Olympiad problems

2017 JBMO Shortlist, NT2

Determine all positive integers n such that $n^2/ (n - 1)!$

2012 APMO, 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.

1960 IMO, 1

Tags: decimal representation , divisibility , number theory , algebra

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

2014 APMO, 3

Tags: divisibility , hensel , surjective , quadratic residue

Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$. [i]Warut Suksompong, Thailand[/i]

2010 Postal Coaching, 1

Tags: combinatorics , permutation , divisibility

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

1998 IMO Shortlist, 1

Tags: divisibility , number theory , polynomial

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

2022 Germany Team Selection Test, 2

Tags: divisibility , cubefree , number theory

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, 43

Tags: modular arithmetic , divisibility , number theory

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$

2008 Germany Team Selection Test, 1

Tags: number theory , divisibility

Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

2009 Brazil Team Selection Test, 1

Tags: combinatorics , permutation , divisibility

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

2017 Macedonia JBMO TST, 1

Tags: prime number , divisibility , number theory

Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.

2023 Indonesia TST, 1

Tags: prime , divisibility , number theory

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2023 Pan-American Girls’ Mathematical Olympiad, 5

Tags: divisibility

Find all pairs of primes $(p,q)$ such that $6pq$ divides $$p^3+q^2+38$$

2017 Azerbaijan BMO TST, 2

Tags: divisibility , rmn

Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.

2020 Moldova Team Selection Test, 6

Tags: algebra , divisibility

Let $n$, $(n \geq3)$ be a positive integer and the polynomial $f(x)=(1+x) \cdot (1+2x) \cdot (1+3x) \cdot ... \cdot (1+nx)$ $= a_0+a_1 \cdot x+a_2 \cdot x^2+a_3 \cdot x^3+...+a_n \cdot x^n$. Show that the number $a_3$ divides the number $k=C^2_{n+1} \cdot (2 \cdot C^2_n \cdot C^2_{n+1}-3 \cdot a_2).$

2012 IMO Shortlist, N8

Tags: prime , gauss , divisibility , modular arithmetic , number theory

Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

1996 Estonia Team Selection Test, 1

Tags: divisibility , perfect cube , number theory

Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.

1974 IMO, 3

Tags: binomial coefficient , summation , divisibility , modular arithmetic , number theory

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

2022 JBMO TST - Turkey, 1

Tags: integer , divisibility , number theory

For positive integers $a$ and $b$, if the expression $\frac{a^2+b^2}{(a-b)^2}$ is an integer, prove that the expression $\frac{a^3+b^3}{(a-b)^3}$ is an integer as well.

2018 China Team Selection Test, 5

Tags: divisibility , binomial coefficient , number theory

Given a positive integer $k$, call $n$ [i]good[/i] if among $$\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n}$$ at least $0.99n$ of them are divisible by $k$. Show that exists some positive integer $N$ such that among $1,2,...,N$, there are at least $0.99N$ good numbers.

2018 OMMock - Mexico National Olympiad Mock Exam, 4

Tags: number theory , divisibility , sum of digits

For each positive integer $n$ let $s(n)$ denote the sum of the decimal digits of $n$. Find all pairs of positive integers $(a, b)$ with $a > b$ which simultaneously satisfy the following two conditions $$a \mid b + s(a)$$ $$b \mid a + s(b)$$ [i]Proposed by Victor Domínguez[/i]

1969 IMO Shortlist, 49

Tags: number theory , combinatorics , divisibility , invariant

$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$