The surface area of a spherical segment is $n$ times the lateral surface area of the cone that can be inscribed in the corresponding slice. What is the ratio of the height of the segment to the diameter of the sphere?
Let the height of the segment be $m$, the radius of the sphere be $r$, and the radius of the base of the cone be $\rho$. According to the problem:

$$
2 r \pi m = m \rho \pi \sqrt{\rho^{2} + m^{2}}
$$

but

$$
\rho = \sqrt{2 r m - m^{2}}
$$

thus

$$
2 r m = n \sqrt{4 r^{2} m^{2} - 2 r m^{3}}
$$

squaring both sides of this equation:

$$
4 r^{2} m^{2} = n^{2} \left(4 r^{2} m^{2} - 2 r m^{3}\right)
$$

or

$$
2 r = n^{2} (2 r - m) \\
m n^{2} = 2 r (n^{2} - 1)
$$

from which

$$
\frac{m}{2 r} = \frac{n^{2} - 1}{n^{2}}
$$

(Smodics Kázmér, Veszprém.)

The problem was also solved by: Bayer B., Benedek Zs., Czank K., Demeter J., Fekete N., Filkorn J., Frank A., Gellért J., Grosz K., Keesz J., Kerekes T., Krausz B., Krisztián Gy., Kürth A., Lukhaub Gy., Lupsa Gy., Messik G., Póka Gy., Sasvári G., Sasvári J., Spitzer H., Weisz A.