Problem

Source: VIII International Festival of Young Mathematicians Sozopol 2017, Theme for 10-12 grade

Tags: number theory



Find all pairs $(x,y)$, $x,y\in \mathbb{N}$ for which $gcd(n(x!-xy-x-y+2)+2,n(x!-xy-x-y+3)+3)>1$ for $\forall$ $n\in \mathbb{N}$.