AT-OEMO-REG 2006 Problem 2

Let $n>1$ be a positive integer an $a$ a real number. Determine all real solutions $(x_1,x_2,\dots,x_n)$ to following system of equations: $x_1+ax_2=0$ $x_2+a^2x_3=0$ … $x_k+a^kx_{k+1}=0$ … $x_n+a^nx_1=0$