34th CanMO 2002 Problem 1 What is the largest possible number of elements in a subset of {1, 2, 3, ... , 9} such that the sum of every pair (of distinct elements) in the subset is different?
{1, 2, 3, 5, 8} has five elements and all pairs with a different sum. If there is a subset with 6 elements, then it has 15 pairs, each with sum at least 1 + 2 = 3 and at most 8 + 9 = 17. There are only 15 numbers at least 3 and at most 17, so each of them must be realised. But the only pair with sum 3 is 1,2 and the only pair with sum 17 is 8, 9 and then 1 + 9 = 2 + 8. So six elements is impossible. 34th CanMO 2002 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002