Question 89
Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 1$. Prove that
$$
2\left(\frac{a b}{a + b} + \frac{b c}{b + c} + \frac{c a}{c + a}\right) + 1 \geq 6(a b + b c + c a).
$$
$$
2\left(\frac{a b}{a + b} + \frac{b c}{b + c} + \frac{c a}{c + a}\right) + 1 \geq 6(a b + b c + c a).
$$