A sphere with radius $r$ is inscribed in a hemisphere, a regular hexahedron is inscribed in this sphere, a sphere is inscribed in this hexahedron, and a regular tetrahedron is inscribed in this sphere. How do the volumes of these bodies relate to each other?
(Second time corrected. 11 A regular hexahedron is inscribed in a hemisphere of radius $r$, a sphere is inscribed in this hexahedron, and a regular tetrahedron is inscribed in this sphere. Show that the volumes of these bodies are in the ratio:

$$
27 \pi \sqrt{2}: 18 \sqrt{3}: 3 \pi \sqrt{3}: 2
$$[^0]


[^0]:    ${ }^{1}$ We received only two correct solutions.