7th Australian 1986 Problem A1 Given a positive integer n and real k > 0, what is the largest possible value for (x 1 x 2 + x 2 x 3 + x 3 x 4 + ... + x n-1 x n ), where x i are non-negative real numbers with sum k?
k 2 /4 Solution Let x h = max(x i ). Then x 1 x 2 + x 2 x 3 + x 3 x 4 + ... + x h-1 x h + x h x h+1 + x h+1 x h+2 + ... + x n-1 x n ≤ x 1 x h + x 2 x h + ... + x h-1 x h + x h x h+1 + x h x h+2 + ... + x h x n = x h (k - x h ) ≤ k 2 /4 (last step AM/GM). But k 2 /4 can be realized by x 1 = x 2 = k/2 and others 0. Solutions are also available in Australian Mathematical Olympiads 1979-1995 by H Lausch and P J Taylor, ISBN 0858896451, published by Australian Mathematics Trust. It can be ordered by email. 7th Aus 1986 © John Scholes jscholes@kalva.demon.co.uk 5 Jan 2004 Last corrected/updated 5 Jan 04