Hanoi Open Mathematics Competition 2012 Senior Question 11

The equation \(x^3+px^2+qx+r=0\) has three real roots \(x_1,x_2,x_3\), where \(p,q,r\) are integers. Put \(S_n=x_1^n+x_2^n+x_3^n\) for \(n=1,2,\ldots\). Prove that \(S_{2012}\) is an integer.