IMO 1969 LL BUL11

Let Z be a set of points in the plane. Suppose that there exists

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IMO 1969 LL BUL11

Origin: BUL

Problem

Let Z 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. Let us call such a pair of points unjoinable. Prove that for each real r > 0 there exists an unjoinable pair of points separated by distance r.