Solutions to $x^3+y^3+z^3 = x^2 + y^2 + z^2=x+y+z=0$ I need to prove that $xyz=1$. 
dealing trew this problem I get that $x+y= \frac{2}{3}$ and that results that $z = -\frac{2}{3}$.
after all i got $xyz= - \frac{10}{27}$

When I was before dealing with this, I over and over get that $xyz=0$.
You can't prove this for rational numbers x,y,z . There is at least one counter example: 
$x = y = z = 0$ is a solution which contradicts $xyz = 1$.
hint: 

Assume $xyz=1$ 
then $1|xyz$ 
so $xyz=1a$