20th Balkan 2003 Problem 3 Find all real-valued functions f(x) on the rationals such that: (1) f(x + y) - y f(x) - x f(y) = f(x) f(y) - x - y + xy, for all x, y (2) f(x) = 2 f(x+1) + 2 + x, for all x and (3) f(1) + 1 > 0.
f(m/n) = -m/n + ½ m/n . Solution Put g(x) = f(x) + x. Then the conditions become: g(x+y) = g(x) g(y), g(x) = 2 g(x+1), and g(1) > 0. Hence g(1) = 1/2 and, by a trivial induction, g(n) = 1/2 n for integers n. By another easy induction g(nx) = g(x) n , so g(m/n) n = g(m) = 1/2 m and hence g(m/n) = 1/2 m/n . 20th Balkan 2003 © John Scholes jscholes@kalva.demon.co.uk 6 July 2003 Last corrected/updated 16 Jan 04