6th Swedish 1966 Problem 4 Let f(x) = 1 + 2/x. Put f 1 (x) = f(x), f 2 (x) = f(f 1 (x)), f 3 (x) = f(f 2 (x)), ... . Find the solutions to x = f n (x) for n > 0.
x = -1 or 2. Solution f(x) = x iff 0 = x 2 - x - 2 = (x+1)(x-2). Now it is clear by a trivial induction that f n (x) = linear/linear. So f n (x) = x is a quadratic and so has at most two roots. But x = -1 and 2 are obviously roots, so they are the only roots. 6th Swedish 1966 © John Scholes jscholes@kalva.demon.co.uk 30 December 2003 Last corrected/updated 30 Dec 03