RO-NMO 2004 Problem 122

Let $f \in \mathbb Z[X]$ . For an $n \in \mathbb N$ , $n \geq 2$ , we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$ , for all $x \in \mathbb Z$ .

(a) Prove that $f_n$ is well defined.

(b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$ , $n \geq 2$ , the function $f_n$ is surjective.

Bogdan Enescu