1967 IMO Problems/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 $c_n = 0.$


It can be found here [1]

$\textbf{Note:}\hspace{4000pt}$ Problem 5 on this (https://artofproblemsolving.com/wiki/index.php/1967_IMO_Problems) page is equivalent to this since the only difference is that they are phrased differently.

Invalid username
Login to AoPS