Let's determine the value of the following expression:

$$
X=\sqrt{x \sqrt{y \sqrt{z \sqrt{x \sqrt{y \sqrt{z \cdots}}}}}}
$$
I. Solution.

$$
\begin{gathered}
X=x^{\frac{1}{2}} y^{\frac{1}{4}} z^{\frac{1}{8}} x^{\frac{1}{16}} y^{\frac{1}{32}} z^{\frac{1}{64}} x^{\frac{1}{128}} y^{\frac{1}{256}} z^{\frac{1}{512}} \ldots \\
=x^{\frac{1}{2}+\frac{1}{16}+\frac{1}{128}+\ldots} y^{\frac{1}{4}+\frac{1}{32}+\frac{1}{256}+\ldots} z^{\frac{1}{8}+\frac{1}{64}+\frac{1}{512}+\ldots} \\
=x^{\frac{4}{7}} y^{\frac{2}{7}} z^{\frac{1}{7}}=\sqrt[7]{x^{4} y^{2} z}
\end{gathered}
$$

(Nándor Deutsch, Losoncz.)

II. Solution.

$$
\begin{gathered}
X^{8}=\sqrt{x \sqrt{y \sqrt{z \sqrt{x \sqrt{y \sqrt{z \ldots}}}}}} \\
X^{8}=x^{4} y^{2} z X \\
X^{7}=x^{4} y^{2} z \\
X=\sqrt[7]{x^{4} y^{2} z}
\end{gathered}
$$

The problem was also solved by: Csete A., Czank K., Engel R., Filkorn J., Freibauer E., Glass M., Hendel J., Kohn B., Kornis Ö., Krausz B., Krisztián Gy., Kürth A., Lindtner M., Lupta Gy., Neumann J., Pálfy F., Pollák N., Porkoláb J., Rozlosnik P., Sasvári G., Sasvári J., Szibelth S., Weisz J.