KZ-MO 2000 Problem 6

For positive numbers $a$ , $b$ and $c$ satisfying the equality $a + b + c = 1$ , prove the inequality $$\frac {a ^ 7 + b ^ 7} {a ^ 5 + b ^ 5} + \frac {b ^ 7 + c ^ 7} {b ^ 5 + c ^ 5} + \frac {c ^ 7 + a ^ 7} {c ^ 5 + a ^ 5} \geq \frac {1} {3}.$$