Question 109
Determine all functions $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ with the following properties:
a) $f(1)=0$,
b) $f(x)\gt 0$ for all $x\gt 1$,
c) Applies to all $x, y \geq 0$ with $x+y\gt 0$
$$
f(x f(y)) f(y)=f\left(\frac{x y}{x+y}\right) .
$$
a) $f(1)=0$,
b) $f(x)\gt 0$ for all $x\gt 1$,
c) Applies to all $x, y \geq 0$ with $x+y\gt 0$
$$
f(x f(y)) f(y)=f\left(\frac{x y}{x+y}\right) .
$$