Quiz

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Let's take, for example, a polyhedron formed by joining two regular tetrahedra along one of their faces. Its center of gravity will be in the middle. This point will be the midpoint of the faces by which the tetrahedra are joined. Let's cut off almost at the very edge of the common face after joining the tetrahedra, removing a pyramid from both ends – a regular tetrahedron. And let's hollow out three faces symmetrically inward on each side. The position of the center of gravity of the polyhedron formed in this way will not change, but if we place it on any face after this, the projection of the center of mass onto the plane of this face will be outside it – which means the polyhedron will tip over onto its edges.

Interesting picture. I must admit, my spatial imagination is not quite enough to understand how the center of mass is projected onto its "outer" faces. But in any case, Yashchur is right, if it is not possible to "place" it on at least one face out of all of them, then such a polyhedron does not meet the conditions of the problem. In this case, "cutting off the pyramids" is allowed, but "hollowing out" three faces inward is not. It simply won't be able to lie on these three faces.

Such a face will always be found (the polyhedron will "fall" until it "finds" such a face), so the answer is no.

In principle, everything is correct, but why will such a face always be found? It's not entirely obvious to me, heh.

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