Question 153
Find all positive real solutions of the system of equations
$$x + \frac{1}{x} - w = 2,$$
$$y + \frac{1}{y} - w = 2,$$
$$z + \frac{1}{x} + w = 2,$$
$$y + \frac{1}{z} + w = 2.$$
$$x + \frac{1}{x} - w = 2,$$
$$y + \frac{1}{y} - w = 2,$$
$$z + \frac{1}{x} + w = 2,$$
$$y + \frac{1}{z} + w = 2.$$