Track / Stage 6 / 33 of 400 #1033 of 1964

Problem 1033

National olympiad, first round

Geometry Difficulty 6.0 Prove it

Petya made a pyramid out of glass rods. The pyramid has 373 lateral edges and the same number of edges in the base. Petya wondered: "Is it possible to parallelly translate each of the 746 edges of the pyramid so that they form a closed broken line in space?" Is Petya's idea feasible?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Answer: No, it is not possible.

Solution. Suppose that the implementation of Petya's idea is possible, and consider the closed broken line formed by 746 edges. Introduce a coordinate system such that the plane $Oxy$ is parallel to the base of the pyramid, the axis $Oz$ is perpendicular to the base of the pyramid, and the height of the pyramid is 1, with the origin $O$ coinciding with one of the vertices of the closed broken line.

Now, let's move along our broken line, starting from point $O$ . Each time we cross an edge that lies in the base, we move in a plane parallel to $Oxy$ , i.e., the $z$ -coordinate of the vertex of the broken line does not change. If, however, we pass along an edge that was a lateral edge, we change the $z$ -coordinate by exactly 1.

Thus, when we have traversed all 746 edges and returned to point $O$ , the $z$ -coordinate of the vertex, on the one hand, should become 0, and on the other hand, it should be odd, since we changed its parity 373 times. Contradiction.

## Variant IV

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.