| 1. The diagonals of a trapezium are perpendicular. Prove that the product of the two lateral sides is not less than the product of the two parallel sides. | |
| 2. Two delegations A and B, with the same number of delegates, arrived at a conference. Some of the delegates knew each other already. Prove that there is a non-empty subset A' of A such that either each member in B knew an odd number of members from A', or each member of B knew an even number of members from A'. | |
| 3. 2kn+1 diagonals are drawn in a convex n-gon. Prove that among them there is a broken line having 2k+1 segments which does not go through any point more than once. Moreover, this is not necessarily true if kn diagonals are drawn. |
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