We inscribe a sphere in an equilateral cone, to which we lay an tangent plane parallel to the base of the cone. We then inscribe another sphere in the resulting cone, and so on. Determine the sum of the volumes of the spheres if this procedure is continued to infinity. The slant height of the cone is $l=2$.
The radius of a sphere inscribed in an equilateral cone is equal to one-third of the cone's height. Therefore, if the height of the cone is \( m \), then the radii of the spheres are: \( \frac{m}{3}, \frac{m}{9}, \frac{m}{27} \), etc. Thus, the sum of the volumes of these spheres is:

\[
S=\frac{4}{3} \pi\left(\frac{m}{3}\right)^{3} \cdot\left(1+\frac{1}{27}+\frac{1}{27^{2}}+\ldots\right)=\frac{2 \pi m^{3}}{39}
\]

Since in this case \( m=\sqrt{3} \), we have

\[
S=\frac{2 \pi \sqrt{3}}{13}
\]

(Dávid Engel, Déva.)

The problem was also solved by: Baumann J., Bayer B., Beck I., Burján K., Czank K., Faith F., Filkorn J., Hein I., Holzmann I. M., Keesz J., Kerekes T., Krausz B., Gy. Krisztián, A. Kürth, M. Lindtner, Gy. Lukhaub, Gy. Lupsa, Gy. Perl, G. Sasvári, J. Scharff, R. Scheuer, K. Smodics, O. Tézner, T. Ulmer, A. Weisz.