ISL 1997 Problem 22

Does there exist functions $f,g: \mathbb{R}\to\mathbb{R}$ such that $f(g(x)) = x^2$ and $g(f(x)) = x^k$ for all real numbers $x$

a) if $k = 3$ ?

b) if $k = 4$ ?

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.