Olympiad problems

2015 Turkey Team Selection Test, 7

Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

2025 Bulgarian Spring Mathematical Competition, 9.4

Tags: prime , divisibility , sophie germain identity , functional equation , number theory , function

Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.

2018 Romania Team Selection Tests, 4

Tags: factorial , number theory , pell equation , sophie germain identity

Given an non-negative integer $k$, show that there are infinitely many positive integers $n$ such that the product of any $n$ consecutive integers is divisible by $(n+k)^2+1$.

1969 IMO Longlists, 30

Tags: composite number , sophie germain identity , number theory

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

1969 IMO, 1

Tags: sophie germain identity , composite number , factorization , number theory

Prove that there are infinitely many positive integers $m$, such that $n^4+m$ is not prime for any positive integer $n$.

1969 IMO Shortlist, 30

Tags: number theory , sophie germain identity , composite number

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$