| 1. Among any four people at a party there is one who has met the three others before the party. Show that among any four people at the party there must be one who has met everyone at the party before the party. | |
| 2. Let a1 = 1, a2, a3, ... be a sequence of positive integers such that ak < 1 + a1 + a2 + ... + ak-1 for all k > 1. Prove that every positive integer can be expressed as a sum of ais. | |
| 3. E is the midpoint of the side AB of the square ABCD, and F, G are any points on the sides BC, CD such that EF is parallel to AG. Show that FG touches the inscribed circle of the square. |
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 28 Apr 2003