OBM 1997 Problem 3

a) Show that there are no functions $f, g: \mathbb R \to \mathbb R$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x \in \mathbb R$ .

b) Let $S$ be the set of all real numbers greater than 1. Show that there are functions $f, g : S \to S$ satsfying the condition above.