| 1. A is a set of integers which is closed under addition and contains both positive and negative numbers. Show that the difference of any two elements of A also belongs to A. | |
| 2. A convex n-gon is divided into triangles by diagonals which do not intersect except at vertices of the n-gon. Each vertex belongs to an odd number of triangles. Show that n must be a multiple of 3. | |
| 3. For a vertex X of a quadrilateral, let h(X) be the sum of the distances from X to the two sides not containing X. Show that if a convex quadrilateral ABCD satisfies h(A) = h(B) = h(C) = h(D), then it must be a parallelogram. |
The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
Kürschák home
(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003