Question 26
Let $x, y, z$ be nonzero real numbers such that the equations
$$
\begin{aligned}
& x+\frac{1}{y}=y+\frac{1}{x} \\
& y+\frac{1}{z}=z+\frac{1}{y} \\
& z+\frac{1}{x}=x+\frac{1}{z}
\end{aligned}
$$
all hold. Show that two of the three variables must be equal.
$$
\begin{aligned}
& x+\frac{1}{y}=y+\frac{1}{x} \\
& y+\frac{1}{z}=z+\frac{1}{y} \\
& z+\frac{1}{x}=x+\frac{1}{z}
\end{aligned}
$$
all hold. Show that two of the three variables must be equal.