OMCC 2012 Problem 3

Let $a,b,c$ be real numbers that satisfy $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c} =1$ and $ab+bc+ac >0$ .

Show that $$a+b+c - \frac{abc}{ab+bc+ac} \ge 4$$