VMO 2009 Problem 4

Let $a$ , $b$ , $c$ be three real numbers. For each positive integer number $n$ , $a^n + b^n + c^n$ is an integer number. Prove that there exist three integers $p$ , $q$ , $r$ such that $a$ , $b$ , $c$ are the roots of the equation $x^3 + px^2 + qx + r = 0$ .