Question 168
Find all functions $f : \mathbb{N} \to \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0 + a_1x + \dots + a_nx^n$ has at least one integer root, so does $f(a_0) + f(a_1)x + \dots + f(a_n)x^n$.