23rd VMO 1985 Problem A2 Find all real-valued functions f(n) on the integers such that f(1) = 5/2, f(0) is not 0, and f(m) f(n) = f(m+n) + f(m-n) for all m, n.
f(n) = 2 n + 1/2 n Solution Putting m = n = 0, we get f(0) 2 = 2f(0), so f(0) = 2. Putting n = 1, we get f(m+1) = 5/2 f(m) + f(m-1). That is a standard linear recurrence relation. The associated quadratic has roots 2, 1/2, so the general solution is f(n) = A 2 n + B/2 n . f(0) = 1 gives A + B = 2, f(1) = 5/2 gives 2A + B/2 = 5/2, so A = B = 1. It is now easy to check that this solutions satisfies the conditions. Thanks to Suat Namli 23rd VMO 1985 © John Scholes jscholes@kalva.demon.co.uk 12 February 2004 Last corrected/updated 12 Feb 04