CZ-SK-MO 2011 Problem 6

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any $x,y\in\mathbb{R}^+$ , we have $$f(x)f(y)=f(y)f\Big(xf(y)\Big)+\frac{1}{xy}.$$