49th Putnam 1988

The triangular numbers are 1, 3, 6, 10, ... , n(n+1)/2, ... . Prove that there are an infinite number of pairs a_i, b_i such that m is triangular iff a_im + b_i is triangular.

Solution

Let T_n = n(n + 1)/2. Let us try to express T_an+b in the form AT_n + B. Evidently this works if a = 2b + 1, with A = (2b + 1)² and B = b(b + 1)/2.

The only if part is not quite so obvious. Suppose that T = (2b + 1)²t + b(b+1)/2 is triangular. We must show that t is triangular. Put k = b(b+1). Then T = (4k+1)t + k/2. Since T is triangular, 8T+1 = 8(4k+1)t + (4k+1) is an odd square. But 4k+1 = (2b+1)² is an odd square, so 8t+1 is an odd square and hence t is triangular also.

49th Putnam 1988

© John Scholes
jscholes@kalva.demon.co.uk
1 Jan 2001