| 1. A circle C touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles. |
|
| 2. Show that if we choose any n+2 distinct numbers from the set {1, 2, 3, ... , 3n} there will be two whose difference is greater than n and smaller than 2n. |
|
| 3. ABC is a triangle. The point A' lies on the side opposite to A and BA'/BC = k, where 1/2 < k < 1. Similarly, B' lies on the side opposite to B with CB'/CA = k, and C' lies on the side opposite to C with AC'/AB = k. Show that the perimeter of A'B'C' is less than k times the perimeter of ABC. |
|
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 30 Oct 2003