MathNet 2010 — Part 04 Question 23

Let $a$, $b$, $c$ be positive real numbers. Prove that
$$
\frac{a^2 b (b-c)}{a+b} + \frac{b^2 c (c-a)}{b+c} + \frac{c^2 a (a-b)}{c+a} \geq 0.
$$