1967 IMO Problems/Problem 5

Take |a1| >= |a2| >= ... >= |a8|. Suppose that |a1|, ... , |ar| are all equal and greater than |ar+1|. Then for sufficiently large n, we can ensure that |as|n < 1/8 |a1|n for s > r, and hence the sum of |as|n for all s > r is less than |a1|n. Hence r must be even with half of a1, ... , ar positive and half negative.

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

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