At what distance from the center should a plane intersect a sphere of radius $R$ so that the volume of the smaller spherical cap is equal to the volume of a cone whose base is the base of the cap and whose apex is at the center of the sphere.
If the distance of the intersecting plane from the center is $x$, then the height of the spherical cap is $R-x$. The spherical cap and the cone form a spherical segment, whose volume is

$$
K=\frac{2}{3} R^{2} \pi(R-x)
$$

The volume of the cone is

$$
k=\frac{1}{3}\left(R^{2}-x^{2}\right) \pi x=\frac{1}{3}(R-x)(R+x) \pi x
$$

According to the problem,

$$
K=2 k
$$

thus

$$
R^{2}=(R+x) x
$$

or

$$
x^{2}+R x-R^{2}=0
$$

from which

$$
x=\frac{R}{2}(\sqrt{5}-1)
$$

(1) can also be written as

$$
x^{2}=R(R-x)
$$

Thus, the radius must be divided into two parts according to the golden ratio, because according to (2), the part falling towards the center $(x)$ is the geometric mean between the entire radius and the other part of the radius.

(Lajos Kertész.)

Number of solutions: 32.