Solve the following system of equations:

$$
\sqrt[\frac{x}{2}]{2 x-y}=2, \quad(2 x-y) \cdot 5^{\frac{x}{4}}=1000
$$
Substituting the expression for $2 x-y$ from the second equation into the first one,

$$
\begin{aligned}
& \sqrt[\frac{x}{4}]{1000 \cdot 5^{-\frac{x}{4}}}=\sqrt[\frac{x}{4}]{\sqrt{1000} \cdot 5^{-1}}=2 \\
& \frac{x}{\sqrt[4]{1000}}=10=\sqrt[3]{1000}, \quad \frac{x}{4}=3, \quad x=12
\end{aligned}
$$

Accordingly,

$$
2 x-y=1000 \cdot 5^{-3}=8, \quad y=2 x-8=16
$$

No other solution is possible, and the value pair $x=12, y=16$ satisfies both equations.

Kis István (Pécs, Zipernovszky K. Mechanical Engineering High School II. grade)

Note. Solutions that transition to logarithmic equations are not substantially different.