Question 110
Let $x$, $y$, $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$.
a) Prove the inequality
$$
x + y + z \geq \sqrt{\frac{x y + 1}{2}} + \sqrt{\frac{y z + 1}{2}} + \sqrt{\frac{z x + 1}{2}}
$$
b) When does the equality hold?
a) Prove the inequality
$$
x + y + z \geq \sqrt{\frac{x y + 1}{2}} + \sqrt{\frac{y z + 1}{2}} + \sqrt{\frac{z x + 1}{2}}
$$
b) When does the equality hold?