We inscribe a regular hexahedron and a regular octahedron in a sphere; how do the radii of the spheres that can be inscribed in the hexahedron and octahedron compare?
If the radius of the given sphere is $r$, then the edge of the inscribed regular hexahedron is $\frac{2 r}{3} \sqrt{3}$, and that of the octahedron is $r \sqrt{2}$. The radius of the sphere inscribed in the hexahedron is equal to half the edge of the hexahedron, i.e., its size is $\frac{r}{3} \sqrt{3}$; the radius of the sphere inscribed in the octahedron is $\frac{r}{3} \sqrt{3}$. We can see, therefore, that the radii of the spheres are equal.

(Friedmann Bernát.)

The problem was also solved by: Feuer Mór, Freund Antal, Geist Emil, Goldstein Zsigmond, Hofbauer Ervin, Kántor Nándor, Klein Mór, Piovarcsy Jenő, Riesz Frigyes, Szabó István.