IMC 2008 · Problem 3

Let $p$ be a polynomial with integer coefficients and let $a_1 < a_2 < \dots < a_k$ be integers.

a) Prove that there exists $a \in \mathbb{Z}$ such that $p(a_i)$ divides $p(a)$ for all $i = 1, 2, \dots, k$.

b) Does there exist an $a \in \mathbb{Z}$ such that the product $p(a_1) \cdot p(a_2) \cdot \dots \cdot p(a_k)$ divides $p(a)$?