Question 1
(GBR 3) IMO5. Find, with proof, all functions $f$ defined on the nonnegative real numbers and taking nonnegative real values such that
(i) $f[xf(y)]f(y)=f(x+y)$,
(ii) $f(2)=0$ but $f(x)\ne0$ for $0\le x\lt 2$.
(i) $f[xf(y)]f(y)=f(x+y)$,
(ii) $f(2)=0$ but $f(x)\ne0$ for $0\le x\lt 2$.