Let's inscribe an equilateral cone into a sphere. At what distance from the center of the sphere should we cut these two bodies with a plane parallel to the base of the cone so that the difference in the areas of the sections is equal to the area of the base of the cone?
Let the sought distance be $x$, the radius of the sphere be $r$, the radius of the spherical cap be $y$, and the radius of the conical section be $z$. According to the problem:

$$
y^{2} \pi - z^{2} \pi = \frac{3}{4} r^{2} \pi
$$

or

$$
y^{2} - z^{2} = \frac{3}{4} r^{2}
$$

But

$$
y^{2} = r^{2} - x^{2}
$$

and

$$
z^{2} = (r - x)^{2} \operatorname{tg}^{2} 30^{\circ} = \frac{1}{3}(r - x)^{2}
$$

Substituting (2) and (3) into (1), we get:

$$
x^{2} - \frac{r}{2} x + \frac{r^{2}}{16} = 0
$$

from which

$$
x = \frac{r}{4}
$$

(Tivadar Kerekes, Déva.)

The problem was also solved by: Appel S., Bayer B., Boros J., Burján K., Czank K., Freibauer E., Filkorn J., Krausz B., Gy. Krisztián, Gy. Lukhaub, Gy. Lupsa, R. Obláth, G. Petrogalli, Gy. Perl, G. Sasvári, J. Schuller, J. Weisz.