That is a rather neat solution

I assume you did not understand this:

$\displaystyle

\frac {a^2}{a^3 - abc + a} + \frac {b^2}{b^3 - abc + b} + \frac {c^2}{c^3 - abc + c}\ge \frac {(a + b + c)^2}{a^3 + b^3 + c^3 - 3abc + a + b + c}$

Try proving $\displaystyle \frac{x^2}{a} +\frac{y^2}{b} +\frac{z^2}{c} \geq \frac{(x+y+z)^2}{a+b+c}$

With a little bit of algebra, you will see this is equivalent to,

$\displaystyle \sum_{cyc}{x^2(\frac{b+c}{a})} \geq 2 \sum_{cyc}{xy}$

Which after a little bit of bunching shows clear AM-GM.

EDIT: Actually it directly follows from CS(in a slightly different form)

$\displaystyle \frac{x_{1}^{2}}{a_{1}}+\frac{x_{2}^{2}}{a_{2}}+.. .+\frac{x_{n}^{2}}{a_{n}}\geq\frac{(x_{1}+x_{2}+.. .+x_{n})^{2}}{a_{1}+a_{2}+...+a_{n}}$

where $\displaystyle x_{i}\in\mathbb{R}$ and $\displaystyle a_{i}\geq0$