The center of a sphere is on the surface of another sphere of equal radius. How does the common part of the two spheres relate to the volume of one of the spheres?
The common part of the two spheres is the sum of the volumes of two spherical caps, the height of which is $m=\frac{r}{2}$. The volume of a spherical cap, as is known,

$$
K_{s}=\frac{1}{3} \pi m^{2}(3 r-m),
$$

or, since $m=\frac{r}{2}$,

$$
K_{s}=\frac{5}{24} \pi r^{3}
$$

and

$$
2 K_{s}=\frac{5}{12} \pi r^{3}
$$

Thus, the sought ratio is

$$
\frac{2 K_{s}}{K}=\frac{\frac{5}{12} \pi r^{3}}{\frac{4}{3} \pi r^{3}}=\frac{5}{16}
$$

(Jánosy Gyula, Budapest.)

The problem was also solved by: Ádámffy E., Bánó L., Dömény I., Friedländer H., Fürchtgott M., Gunszt B., Haar A., Harsányi Z., Heimlich P., Hermann J., Jánosy Gy., Kertész G., Kiss J., Krampera Gy., Kürti I., Magyari F., Nagy Á., Pám M., Rássy P., Riesz M., Rosenberg J., Schöffer I., Schuster Gy., Söpkéz Gy., Strobl J.