13th Australian 1992 Problem B2 R is the real numbers and S is R excluding the point 2/3. Find all functions f : S → R such that 2 f(x) + f(2x/(3x - 2) ) = 996x for all x.
1992x(x-1)/(3x-2) Solution Note that if we substitute x = 2y/(3y-2) into 2x/(3x-2) we get y. Thus we have 2 f(2x/(3x-2)) + f(x) = 996·2x/(3x-2), so f(2x/(3x-2)) + f(x)/2 = 996x/(3x-2). Subtracting from the relation given, we get (3/2) f(x) = 996(3x-3)/(3x-2) and hence f(x) = 1992x(x-1)/(3x-2). Thanks to Suat Namli 13th Aus 1992 © John Scholes jscholes@kalva.demon.co.uk 5 February 2004 Last updated/corrected 5 Feb 04