Let
be a set of
points in space. From the family of all segments with endpoints in
,
segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of
yellow segments, where
, arranged in order of increasing length.
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Let $A$ be a set of $n$ points in space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$ , arranged in order of increasing length.