UA-UMO 2005 Problem 112

The sum of positive real numbers $a,b,c$ equals $1$ . Prove that:

$\sqrt{\frac{1}{a}-1}\sqrt{\frac{1}{b}-1}+\sqrt{\frac{1}{b}-1}\sqrt{\frac{1}{c}-1}+\sqrt{\frac{1}{c}-1}\sqrt{\frac{1}{a}-1} \ge 6.$