1. That is a rather neat solution

Originally Posted by JaneBennet
By the way, some people think this problem is easy.

Art of Problem Solving Forum

Unfortunately I don’t understand the steps in their solution.
I assume you did not understand this:
$
\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 $\frac{x^2}{a} +\frac{y^2}{b} +\frac{z^2}{c} \geq \frac{(x+y+z)^2}{a+b+c}$
With a little bit of algebra, you will see this is equivalent to,

$\sum_{cyc}{x^2(\frac{b+c}{a})} \geq 2 \sum_{cyc}{xy}$
Which after a little bit of bunching shows clear AM-GM.

EDIT: Actually it directly follows from CS(in a slightly different form)
$\frac{x_{1}^{2}}{a_{1}}+\frac{x_{2}^{2}}{a_{2}}+.. .+\frac{x_{n}^{2}}{a_{n}}\geq\frac{(x_{1}+x_{2}+.. .+x_{n})^{2}}{a_{1}+a_{2}+...+a_{n}}$
where $x_{i}\in\mathbb{R}$ and $a_{i}\geq0$

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