IMO Shortlist 1994 problem A4
Dodao/la:
arhiva2. travnja 2012. Let
![\mathbb{R}](/media/m/1/4/0/140a3cd0f5aa77f0f229f3ae2e64c0a6.png)
denote the set of all real numbers and
![\mathbb{R}^+](/media/m/4/d/d/4dd6182efc1bb170a565248a692ee278.png)
the subset of all positive ones. Let
![\alpha](/media/m/f/c/3/fc35d340e96ae7906bf381cae06e4d59.png)
and
![\beta](/media/m/c/e/f/cef1e3bcf491ef3475085d09fd7d291e.png)
be given elements in
![\mathbb{R},](/media/m/d/f/a/dfa767ece3040d7b7692a39c224c9c1f.png)
not necessarily distinct. Find all functions
![f: \mathbb{R}^+ \mapsto \mathbb{R}](/media/m/e/0/a/e0aee0c0efcb928a310fec0b41c87a64.png)
such that
%V0
Let $\mathbb{R}$ denote the set of all real numbers and $\mathbb{R}^+$ the subset of all positive ones. Let $\alpha$ and $\beta$ be given elements in $\mathbb{R},$ not necessarily distinct. Find all functions $f: \mathbb{R}^+ \mapsto \mathbb{R}$ such that
$$f(x)f(y) = y^{\alpha} f \left( \frac{x}{2} \right) + x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^+.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1994