Question 92
Prove that for all non-negative real numbers $x, y, z$, not all equal to $0$, the following inequality holds
$$
\frac{2x^{2}-x+y+z}{x+y^{2}+z^{2}}+\frac{2y^{2}+x-y+z}{x^{2}+y+z^{2}}+\frac{2z^{2}+x+y-z}{x^{2}+y^{2}+z} \geqslant 3
$$
Determine all the triples $(x, y, z)$ for which the equality holds.
$$
\frac{2x^{2}-x+y+z}{x+y^{2}+z^{2}}+\frac{2y^{2}+x-y+z}{x^{2}+y+z^{2}}+\frac{2z^{2}+x+y-z}{x^{2}+y^{2}+z} \geqslant 3
$$
Determine all the triples $(x, y, z)$ for which the equality holds.