| 1. a and b are rationals. Show that if ax2 + by2 = 1 has a rational solution (in x and y), then it must have infinitely many. | |
| 2. The vertices of a convex n-gon are colored so that adjacent vertices have different colors. Prove that if n is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color. | |
| 3. A triangle has inradius r and circumradius R. Its longest altitude has length H. Show that if the triangle does not have an obtuse angle, then H ≥ r + R. When does equality hold? |
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