IMO Shortlist 1985 problem 8
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Avg: 0,0 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.
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![m \geq \frac{2q}{n}](/media/m/6/2/9/629db9208fe5c8e7f86eca273f6debf8.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1985