CMO 2010 Problem 5

Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_{n}=n !+n$. Show that if $P\left(a_{n}\right) / Q\left(a_{n}\right)$ is an integer for every $n$, then $P(n) / Q(n)$ is an integer for every integer $n$ such that $Q(n) \neq 0$.