| 1. P is a point inside a convex quadrilateral ABCD such that the areas of the triangles PAB, PBC, PCD and PDA are all equal. Show that one of its diagonals must bisect the area of the quadrilateral. | |
| 2. What is the largest possible number of triples a < b < c that can be chosen from 1, 2, 3, ... , n such that for any two triples a < b < c and a' < b' < c' at most one of the equations a = a', b = b', c = c' holds? | |
| 3. PQRS is a convex quadrilateral whose vertices are lattice points. The diagonals of the quadrilateral intersect at E. Prove that if the sum of the angles at P and Q is less than 180o then the triangle PQE contains a lattice point apart from P and Q either on its boundary or in its interior. |
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