Consider four, mutually touching, equal-radius spheres. Imagine a fifth sphere that touches the four spheres and calculate the ratio of the radius of the fifth sphere to the radius of a given sphere.
The lines connecting the centers of the spheres form a regular tetrahedron. The given spheres are concentric with the sphere circumscribed around the tetrahedron, either internally or externally tangent. Therefore, the radii of the tangent spheres can be obtained by adding or subtracting the radius of a given sphere from the radius of the sphere circumscribed around the tetrahedron. If the radius of the given spheres is \( r \), and the radius of the sought spheres is \( R \), then:

\[
R = \frac{2 r}{4} \sqrt{6} \pm r = \frac{r}{2} \sqrt{6} \pm r
\]

and thus

\[
R: r = \left(\frac{1}{2} \sqrt{6} \pm 1\right): 1
\]

(Károly Oltay.)

The problem was also solved by: Andráschek F., Freibauer E., Juvancz I., Kárf J., Krausz B., Gy. Krisztián, Gy. Lukhaub, L. Miliczer, N. Pollák, G. Sasvári, Á. Schieb, J. Weisz.