IMO 1967 Problem B2 a 1 , ... , a 8 are reals, not all zero. Let c n = a 1 n + a 2 n + ... + a 8 n for n = 1, 2, 3, ... . Given that an infinite number of c n are zero, find all n for which c n is zero.
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  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 c n = 0 for all odd n. s 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. 9th IMO 1967 © John Scholes jscholes@kalva.demon.co.uk 29 Sep 1998 Last corrected/updated 29 Sep 1998