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 $c_n = 0.$

Solution

$c_n$ must be zero for all odd $n$.

Proof: WLOG suppose that $a_1 \geq a_2 \geq ... \geq a_8$. If $a_1+a_8 > 0$ then for sufficiently high odd $n$, $c_n$ will be dominated by $a_1$ alone i.e. it will always be positive. Similarly if $a_1+a_8 < 0$; hence $a_1=-a_8$. Now for odd $n$ these terms cancel, so we can repeat for the remaining values. Now all the terms cancel for all odd $n$. Since some $a_i$ is nonzero $c_n > 0$ for even $n$.

The above solution was written by Fiachra and 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.

 1967 IMO (Problems) • Resources Preceded byProblem 4 1 • 2 • 3 • 4 • 5 • 6 Followed byProblem 6 All IMO Problems and Solutions