TH-TMO 2009 Problem 7

Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$ . Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$ . Show that $S_n$ is an integer for all positive integers $n$ .