CMO 2009 Problem 3

Define

$$f(x, y, z)=\frac{(x y+y z+z x)(x+y+z)}{(x+y)(x+z)(y+z)} .$$

Determine the set of real numbers $r$ for which there exists a triplet $(x, y, z)$ of positive real numbers satisfying $f(x, y, z)=r$.