3. Each of the four spheres of radius $\mathrm{R}$ touches the other three. Calculate the radius $r$ of the sphere that can be inscribed between them.
3. The centers of the given spheres are the vertices of a regular tetrahedron with edge length \( a = 2R \). The center of the sought sphere is its centroid, which we denote as \( T \). Now, it holds that \( R + r = |A T| \).

![](https://cdn.mathpix.com/cropped/2024_05_30_db59002be042233d39ccg-11.jpg?height=545&width=491&top_left_y=370&top_left_x=839)

The centroid divides the median of the tetrahedron in the ratio \( |A T| : |T O| = 3 : 1 \), i.e., \( |A T| = \frac{3}{4} |A O| = \frac{3}{4} v \), where \( v \) is the height of the tetrahedron.

From triangle \( A O K \), we get:

\[
\begin{aligned}
|A O|^{2} & = |A K|^{2} - |O K|^{2} \\
v^{2} & = \left( \frac{a \sqrt{3}}{2} \right)^{2} - \left( \frac{a \sqrt{3}}{6} \right)^{2} \text{, i.e.,} \\
v & = a \sqrt{\frac{2}{3}} = 2R \sqrt{\frac{2}{3}}
\end{aligned}
\]

Now, \( |A T| = \frac{R \sqrt{6}}{2} \) and finally,

\[
r = \frac{R(\sqrt{6} - 2)}{2}
\]

5 points