| 1. Let r > 1 denote the ratio of two adjacent sides of a parallelogram. Determine how the largest possible value of the acute angle included by the diagonals depends on r. | |
| 2. Prove that after removing any n-3 diagonals of a convex n-gon, it is always possible to choose n-3 non-intersecting diagonals amongst those remaining, but that n-2 diagonals can be removed so that it is not possible to find n-3 non-intersecting diagonals amongst those remaining. | |
| 3. For k = 1, 2, ... , n, Hk is a disjoint union of k intervals of the real line. Show that one can find [(n + 1)/2] disjoint intervals which belong to different Hk. |
The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003