<s>[INST] If $x^2=y+z$, $y^2=x+z$ and $z^2=x+y$, prove If $x^2=y+z$, $y^2=x+z$ and $z^2=x+y$,
Prove that 
$$\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=1$$.
My attempt:
$$L.H.S=\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}$$
$$=\frac{(y+1)(z+1)+(x+1)(z+1)+(x+1)(y+1)}{(x+1)(y+1)(z+1)}$$
$$=\frac{x^2+y^2+z^2+yz+xz+xy+3}{(x+1)(y+1)(z+1)}$$
What should I do next?
 [/INST] Note that
$$x^2 + x = y^2 + y = z^2 + z = x + y  + z$$
$$x(x + 1) = y(y + 1) = z(z + 1) = x + y + z$$
so
$$\begin{align}\frac{1}{x + 1} + \frac{1}{y + 1} + \frac{1}{z + 1} &= \frac{x}{x + y + z} + \frac{y}{x + y + z} + \frac{z}{x + y + z}\\
&= \frac{x + y + z}{x + y + z}\\&= 1\end{align}$$
</s>