Let's calculate the sum of the following infinite geometric progression:

$$
\frac{\sqrt{2}+1}{\sqrt{2}-1}, \frac{1}{2-\sqrt{2}}, \frac{1}{2}, \ldots
$$
The geometric progression ratio of Minthogy is

$$
\frac{\sqrt{2}-1}{\sqrt{2}}<1
$$

we can apply the known sum formula:

$$
S=\frac{a}{1-q}=\frac{\sqrt{2}+1}{\sqrt{2}-1}:\left(1-\frac{\sqrt{2}-1}{\sqrt{2}}\right)=\frac{\sqrt{2}+1}{\sqrt{2}-1} \cdot \sqrt{2}=4+3 \sqrt{2} .
$$

(László Szántó, Pécs.)

Number of solutions: 42.