BG-MO 2006 Problem 2

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$ $$f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}$$ a) Prove that $f(2x)=4f(x)$ for all $x>0$ ; b) Find all such functions.

Nikolai Nikolov, Oleg Mushkarov