| 1. Find all quadruples (a, b, c, d) of distinct positive integers satisfying a + b = cd and c + d = ab. | |
| 2. Does there exist an infinite set of points in space such that at least one, but only finitely many, points of the set belong to each plane? | |
| 3. A club has 3n+1 members. Every two members play just one of tennis, chess and table-tennis with each other. Each member has n tennis partners, n chess partners and n table-tennis partners. Show that there must be three members of the club, A, B and C such that A and B play chess together, B and C play tennis together and C and A play table-tennis together. |
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