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Part of 1969 IMO Shortlist

Problems(1)

Existence of unjoinable pair of points at given distance.

Source:

9/29/2010
(BUL5)(BUL 5) Let ZZ be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of Z.Z. Let us call such a pair of points unjoinable. Prove that for each real r>0r > 0 there exists an unjoinable pair of points separated by distance r.r.
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