IMO 1967 Problem 5

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let $$c_n = \sum^8_{k=1} a^n_k$$ for $n=1,2,3,\ldots$ . Given that among the numbers of the sequence $(c_n)$ , there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$