IMO 1983 LL COL20

Origin: COL

Problem

Let f and g be functions from the set A to the same set A. We define f to be a functional nth root of g (n is a positive integer) if f n(x) = g(x), where f n(x) = f n−1(f(x)). (a) Prove that the function g : R \toR, g(x) = 1/x has an infinite number of nth functional roots for each positive integer n. (b) Prove that there is a bijection from R onto R that has no nth func- tional root for each positive integer n.