10,11

Find the length of the shortest path on the surface of a unit regular tetrahedron between the midpoints of its opposite edges.

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Let $M$ and $N$ be the midpoints of the opposite edges $A B$ and $C D$ of a regular tetrahedron $A B C D$ (Fig.1), and the path between $M$ and $N$ intersects the edge $A C$. Consider the net of the tetrahedron, consisting of four equilateral triangles: $A B C, D_1 A B, D_2 A C$, and $D_3 B C$ (Fig.2). Then the shortest path sought is the shortest path between the midpoint $M$ of segment $A B$ and the midpoint $N'$ of segment $D_2 C$. The length of this path is 1. Similarly, for the shortest paths intersecting the edges $A D, C D$ or $B D$. All other possible paths are obviously longer.

## Answer

1.00