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.  

plugging $c=-a-b$ in your term, we get $${\frac {{a}^{2}}{2\,{a}^{2}+b \left( -a-b \right) }}+{\frac {{b}^{2}}{
 \left( -a-b \right) a+2\,{b}^{2}}}+{\frac { \left( -a-b \right) ^{2}
}{ab+2\, \left( -a-b \right) ^{2}}}
$$
simplifying this we get $1$
