HMMT November, 2023

Solution:
Taking $\log_{2}$ both sides of the first equation gives
$$\begin{aligned} & \log_{2} x \log_{y} z = 1 \\ & \frac{\log_{2} x \log_{2} z}{\log_{2} y} = 1 \end{aligned}$$
Performing similar manipulations on the other two equations, we get
$$\begin{aligned} & \frac{\log_{2} x \log_{2} z}{\log_{2} y} = 1 \\ & \frac{\log_{2} y \log_{2} x}{\log_{2} z} = 2 \\ & \frac{\log_{2} z \log_{2} y}{\log_{2} x} = 3 \end{aligned}$$
Multiplying the first and second equation gives $\left(\log_{2} x\right)^{2} = 2$ or $\log_{2} x = \pm \sqrt{2}$. Multiplying the second and third equation gives $\left(\log_{2} y\right)^{2} = 6$ or $\log_{2} y = \pm \sqrt{6}$. Thus, we have
$$\log_{x} y = \frac{\log_{2} y}{\log_{2} x} = \pm \frac{\sqrt{6}}{\sqrt{2}} = \pm \sqrt{3}$$