Question 163
Let $x, y, z$ be positive real numbers such that $x y + y z + z x = x + y + z$. Prove the inequality
$$
\frac{1}{x^{2} + y + 1} + \frac{1}{y^{2} + z + 1} + \frac{1}{z^{2} + x + 1} \leqslant 1.
$$
When is equality attained in the above inequality?
$$
\frac{1}{x^{2} + y + 1} + \frac{1}{y^{2} + z + 1} + \frac{1}{z^{2} + x + 1} \leqslant 1.
$$
When is equality attained in the above inequality?