IMO Shortlist 1969 problem 11
Dodao/la:
arhiva2. travnja 2012. ![(BUL 5)](/media/m/b/d/a/bda5346c4a3e74aa4cb20742beb374b1.png)
Let
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
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.](/media/m/e/3/9/e39b9e0e2c861e98e4cb5db918594886.png)
Let us call such a pair of points unjoinable. Prove that for each real
![r > 0](/media/m/d/c/2/dc2437e41160bf7216a2f7d96a71ac81.png)
there exists an unjoinable pair of points separated by distance
%V0
$(BUL 5)$ 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 $r.$
Izvor: Međunarodna matematička olimpijada, shortlist 1969