Question 87
Find all triples $(x, y, z)$ of real numbers larger than $1$, so that
$$
\left(\frac{x}{2} + \frac{1}{x} - 1\right) \left(\frac{y}{2} + \frac{1}{y} - 1\right) \left(\frac{z}{2} + \frac{1}{z} - 1\right) = \left(1 - \frac{x}{yz}\right) \left(1 - \frac{y}{xz}\right) \left(1 - \frac{z}{xy}\right).
$$
$$
\left(\frac{x}{2} + \frac{1}{x} - 1\right) \left(\frac{y}{2} + \frac{1}{y} - 1\right) \left(\frac{z}{2} + \frac{1}{z} - 1\right) = \left(1 - \frac{x}{yz}\right) \left(1 - \frac{y}{xz}\right) \left(1 - \frac{z}{xy}\right).
$$