| 1. A rectangle has its vertices at lattice points and its sides parallel to the axes. Its smaller side has length k. It is divided into triangles whose vertices are all lattice points, such that each triangle has area ½. Prove that the number of the triangles which are right-angled is at least 2k. | |
| 2. A polynomial in n variables has the property that if each variable is given one of the values 1 and -1, then the result is positive whenever the number of variables set to -1 is even and negative when it is odd. Prove that the degree of the polynomial is at least n. | |
| 3. A, B, C, D are points in the plane, no three collinear. The lines AB and CD meet at E, and the lines BC and DA meet at F. Prove that the three circles with diameters AC, BD and EF either have a common point or are pairwise disjoint. |
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