SCHWEITZER 2013 Problem 8

Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which $$f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right)$$ for all ${x,y \in \Bbb{R}} ({f^{-1}}$ denotes the inverse of ${f})$ . Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}.$ Proposed by Zoltán Daróczy