Question 139
Prove that
$$
\left( \frac{a+b}{c^2} + \frac{b+c}{a^2} + \frac{c+a}{b^2} \right) \cdot \left( \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} \right) \ge 3 + \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}
$$
for any positive real numbers $a$, $b$ and $c$.
$$
\left( \frac{a+b}{c^2} + \frac{b+c}{a^2} + \frac{c+a}{b^2} \right) \cdot \left( \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} \right) \ge 3 + \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}
$$
for any positive real numbers $a$, $b$ and $c$.