[ Parallelism of lines and planes $]$  [ Systems of segments, lines, and circles ]

In space, there are $n$ segments, no three of which are parallel to the same plane. For any two segments, the line connecting their midpoints is perpendicular to both segments. What is the largest $n$ for which this is possible?
Answer: $n=2$.

Suppose that $n \geq 3$. Take three segments $a, b$, and $c$ and draw three lines connecting the midpoints of each pair. Each of the segments $a, b, c$ is perpendicular to the two lines emanating from its midpoint, and therefore it is perpendicular to the plane passing through all three drawn lines. Thus, the three segments $a, b, c$ are parallel to one plane, which contradicts the condition. (Article XXX "Problems XL MMO - answers, hints, solutions", Kvant 1978 issue 2 p. 64, problem 10)