13. (YUG 4) Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to 1.
13. Let $S_{1}, S_{2}, S_{3}, S_{4}$ denote the areas of the faces of the tetrahedron, $V$ its volume, $h_{1}, h_{2}, h_{3}, h_{4}$ its altitudes, and $r$ the radius of its inscribed sphere. Since
$$
3 V=S_{1} h_{1}=S_{2} h_{2}=S_{3} h_{3}=S_{4} h_{4}=\left(S_{1}+S_{2}+S_{3}+S_{4}\right) r,
$$
it follows that
$$
\frac{1}{h_{1}}+\frac{1}{h_{2}}+\frac{1}{h_{3}}+\frac{1}{h_{4}}=\frac{1}{r} .
$$

In our case, $h_{1}, h_{2}, h_{3}, h_{4} \geq 1$, hence $r \geq 1 / 4$. On the other hand, it is clear that a sphere of radius greater than $1 / 4$ cannot be inscribed in a tetrahedron all of whose altitudes have length equal to 1 . Thus the answer is $1 / 4$.