| 1. ABCD is a convex quadrilateral with AB + BD ≤ AC + CD. Prove that AB < AC. |
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| 2. Every planar section of a three-dimensional body B is a disk. Show that B must be a ball. |
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| 3. A tournament is arranged amongst a finite number of people. Every person plays every other person just once and each game results in a win to one of the players (there are no draws). Show that there must a person X such that, given any other person Y in the tournament, either X beat Y, or X beat Z and Z beat Y for some Z. |
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The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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© John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 28 Apr 2003