Question 4
A4 (MON) 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 $f:\mathbb R^+\to\mathbb R$ such that
$$f(x)f(y)=y^\alpha f\left(\frac x2\right)+x^\beta f\left(\frac y2\right)\qquad(x,y\in\mathbb R^+).$$
$$f(x)f(y)=y^\alpha f\left(\frac x2\right)+x^\beta f\left(\frac y2\right)\qquad(x,y\in\mathbb R^+).$$