| 1. ABC is an acute-angled non-isosceles triangle. H is the orthocenter, I is the incenter and O is the circumcenter. Show that if one of the vertices lies on the circle through H, I and O, then at least two vertices lie on it. | |
| 2. The Fibonacci numbers are defined by F1 = F2 = 1, Fn = Fn-1 + Fn-2. Suppose that a rational a/b belongs to the open interval (Fn/Fn-1, Fn+1/Fn). Prove that b ≥ Fn+1. | |
| 3. S is a convex 3n gon. Show that we can choose a set of triangles, such that the edges of each triangle are sides or diagonals of S, and every side or diagonal of S belongs to just one triangle. |
The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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© John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 28 Apr 2003