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.  

For $\prod\limits_{cyc}(a-b)\neq0$ we obtain:
$$\sum_{cyc}\frac{a^2}{2a^2+bc}=\sum_{cyc}\frac{a^2}{a(a+b+c)+a^2-ab-ac+bc}=$$
$$=\sum_{cyc}\frac{a^2}{(a-b)(a-c)}=\frac{\sum\limits_{cyc}a^2(c-b)}{\prod\limits_{cyc}(a-b)}=1.$$
