A sphere of unit radius can be inscribed in a tetrahedron. The tangent planes of the sphere, parallel to the faces of the tetrahedron, cut off a smaller tetrahedron from the tetrahedron. What is the sum of the radii of the spheres inscribed in these smaller tetrahedra?
Let the vertices of the tetrahedron be $A, B, C, D$, its volume $V$, its surface area $A$, and the radius of the inscribed sphere $r=1$. The plane tangent to the tetrahedron and parallel to the face $BCD$ cuts off the tetrahedron $AB_1C_1D_1$. (See the figure.) Let the radius of the sphere inscribed in this tetrahedron be denoted by $r_A$, and the height from vertex $A$ of the tetrahedron $ABCD$ be denoted by $m_A$. The tetrahedra $AB_1C_1D_1$ and $ABCD$ are similar, so $\frac{r_A}{1} = \frac{m_A - 2}{m_A} = 1 - \frac{2}{m_A}$. Calculating similarly for the radii of the spheres inscribed in the tetrahedra cut off by the planes parallel to the other faces:

\[
r_A + r_B + r_C + r_D = 4 - 2\left(\frac{1}{m_A} + \frac{1}{m_B} + \frac{1}{m_C} + \frac{1}{m_D}\right)
\]

If the area of the triangle $BCD$ is denoted by $t_A$, the volume of the tetrahedron is: $V = \frac{t_A \cdot m_A}{3}$, from which $\frac{1}{m_A} = \frac{t_A}{3V}$. Calculating similarly for the other heights:

\[
\frac{1}{m_A} + \frac{1}{m_B} + \frac{1}{m_C} + \frac{1}{m_D} = \frac{t_A + t_B + t_C + t_D}{3V} = \frac{A}{3V}
\]

The volume of the tetrahedron can also be expressed using the radius of the inscribed sphere, $V = \frac{A \cdot r}{3}$, from which (considering that $r=1$) $A = 3V$. Therefore, the right-hand side of (2) is: $\frac{A}{3V} = \frac{3V}{3V} = 1$, and thus from (1) $r_A + r_B + r_C + r_D = 4 - 2 = 2$.

Remark. The planar counterpart of the problem is as follows: In a triangle, the radii of the circles inscribed in the triangles cut off by the tangents to the sides parallel to the sides of the triangle are $r_a, r_b, r_c$. Then $r_a + r_b + r_c = r$.

![](https://cdn.mathpix.com/cropped/2024_05_02_54176265ef274cbed37eg-1.jpg?height=656&width=849&top_left_y=1081&top_left_x=627)