International Mathematical Olympiad 1986 Problem 5

Find all functions $f$, defined on the non-negative real numbers and taking non-negative real values, such that:

(i) $f(xf(y))f(y) = f(x + y)$ for all $x, y \geq 0$,

(ii) $f(2) = 0$,

(iii) $f(x) \neq 0$ for $0 \leq x < 2$.