Question 50
Determine all real numbers $x \gt 1$, $y \gt 1$, and $z \gt 1$, satisfying the equation
$$
x + y + z + \frac{3}{x-1} + \frac{3}{y-1} + \frac{3}{z-1} = 2(\sqrt{x+2} + \sqrt{y+2} + \sqrt{z+2})
$$
$$
x + y + z + \frac{3}{x-1} + \frac{3}{y-1} + \frac{3}{z-1} = 2(\sqrt{x+2} + \sqrt{y+2} + \sqrt{z+2})
$$