Find the value of $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}$ 
Let $a$, $b$ and $c$ be real numbers such that $a + b + c = 0$ and define:
  $$P=\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}.$$
  What is the value of $P$?

This question came in the regional maths olympiad. I tried AM-GM and CS inequality but failed to get a result. Please give me some hint in how to solve this question.  

Write $c=-a-b$.
Then
$$\frac{a^2}{2a^2+bc}=\frac{a^2}{2a^2-ab-b^2}=\frac{a^2}{(2a+b)(a-b)}$$
and
$$\frac{c^2}{2c^2+ab}=\frac{(a+b)^2}{2a^2+5ab+2b^2}=\frac{(a+b)^2}{(2a+b)(a+2b)}.$$
The sum therefore equals
$$\frac{a^2(a+2b)-b^2(2a+b)+(a+b)^2(a-b)}{(2a+b)(a+2b)(a-b)}=\cdots$$
etc. (as long as the denominator is nonzero).
