DE-MO 2008 Problem 3

Find all functions $f$ defined on non-negative real numbers having the following properties: (i) For all non-negative $x$ it is $f(x) \geq 0$ . (ii) It is $f\left(1\right)=\frac 12$ . (iii) For all non-negative numbers $x,y$ it is $f\left( y \cdot f(x) \right) \cdot f(x) = f(x+y)$ .