SCHWEITZER 2005 Problem 8

Determine all continuous, strictly monotone functions $\phi : \mathbb{R}^+\to\mathbb{R}$ such that $$F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right)$$ is homogeneous of degree 1, ie $F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+$ NoteF(x,y)=F(y,x) and F(x,x)=2x