IMO Shortlist 1997 problem 22
Dodao/la:
arhiva2. travnja 2012. Does there exist functions
![f,g: \mathbb{R}\to\mathbb{R}](/media/m/f/0/6/f065aff539b1c5eba615378de4b433f7.png)
such that
![f(g(x)) = x^2](/media/m/6/d/9/6d9396ba499ffbeb778bae42b369d3f3.png)
and
![g(f(x)) = x^k](/media/m/0/5/2/05245168916acb177a3804145cfae4bf.png)
for all real numbers
a) if
![k = 3](/media/m/1/1/a/11aef2d3221cf7416a8fb1e60cbe6ccf.png)
?
b) if
![k = 4](/media/m/3/2/9/329bda79c08558723f84f38b89d8d3e9.png)
?
%V0
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 $k = 4$?
Izvor: Međunarodna matematička olimpijada, shortlist 1997