1967 IMO Problems/Problem 5

Take $|a_1| >= |a_2| >= ... >= |a_8|$ . Suppose that $|a_1|, ... , |a_r|$ are all equal and greater than $|a_{r+1}|$ . Then for sufficiently large $n$ , we can ensure that $|a_s|n < \frac{1}{8} |a_1|n$ for $s > r$ , and hence the sum of $|a_s|n$ for all $s > r$ is less than $|a_1|n$ . Hence $r$ must be even with half of $a_1, ... , a_r$ positive and half negative.

If that does not exhaust the $a_i$ , then in a similar way there must be an even number of $a_i$ with the next largest value of $|a_i|$ , with half positive and half negative, and so on. Thus we find that $cn = 0$ for all odd $n$ .

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1967 IMO Problems/Problem 5