a1, ... , a8 are reals, not all zero. Let cn = a1n + a2n + ... + a8n for n = 1, 2, 3, ... . Given that an infinite number of cn are zero, find all n for which cn is zero.
Solution
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.
Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
29 Sep 1998
Last corrected/updated 29 Sep 1998