Problem 1049

80.39. Consider the infinite unfolding of a regular tetrahedron with edge length 1 on a plane - see Fig. 55, the trace of each vertex is marked with the same letter. Let $M$ and $N$ be two points on the surface of the tetrahedron. Consider all their images on the unfolding. The points $M_{1}, M_{2}, \ldots$, corresponding to point $M$, lie at the nodes of a lattice of equilateral triangles with side length 2. Consider that image of vertex $N$ which lies inside one of these triangles $M_{1} M_{2} M_{3}$ - point $N_{1}$. It remains to prove that one of the distances from $N_{1}$ to the vertices of the triangle $M_{1} M_{2} M_{3}$ is no more than $2 \sqrt{3}$. This follows from the fact that the triangle is divided into three quadrilaterals $O P_{1} P_{2} M_{3}$, $O P_{1} P_{3} M_{2}$, and $O P_{2} P_{3} M_{1}$, where $O$ is the center of the triangle $M_{1} M_{2} M_{3}$,

$$15^{*}$$

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Fig. 55

$P_{1}, P_{2}, P_{3}$ are the feet of the perpendiculars dropped from $O$ to the sides, since point $N_{1}$ lies in one of them.