Selection tests for the Gulf Mathematical Olympiad 2013

We have $$\begin{array}{rl}& \frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\\ & =\frac{a^4}{a^3+a^{2}b+a{b}^2}+\frac{b^4}{b^3+b^{2}c+b{c}^2}+\frac{c^4}{c^3+c^{2}a+c{a}^2}\\ & \ge \frac{{\left(a^2+b^2+c^2\right)}^2}{a^3+a{b}^2+a{c}^2+b{a}^2+b^3+b{c}^2+c{a}^2+c{b}^2+c^3}\end{array}$$ by applying Cauchy-Schwarz inequality.

On the other hand $$\begin{array}{rl}\frac{{\left(a^2+b^2+c^2\right)}^2}{a^3+a{b}^2+a{c}^2+b{a}^2+b^3+b{c}^2+c{a}^2+c{b}^2+c^3}& =\frac{a^2+b^2+c^2}{a+b+c}\\ & \ge \frac{a+b+c}{3}\end{array}$$ by applying again Cauchy-Schwarz inequality.

The equality holds when $a=b=c$.