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.  

If $a=3$ and $b=-1$ and $c=-2$ we get a value $1$.
But
$$\sum_{cyc}\frac{a^2}{2a^2+bc}-1=-\frac{abc(a+b+c)\sum\limits_{cyc}(a^2-ab)}{\prod\limits_{cyc}(2a^2+bc)}=0.$$
Thus, $P=1$ for all $a+b+c=0$ and $\prod\limits_{cyc}(2a^2+bc)\neq0$
and we are done!
