| 1. Show that the only rational solution to x3 + 3y3 + 9z3 - 9xyz = 0 is x = y = z = 0. | |
| 2. The polynomial xn + a1xn-1 + ... + an-1x + 1 has non-negative coefficients and n real roots. Show that its value at 2 is at least 3n. | |
| 3. The n+1 points P1, P2, ... , Pn, Q lie in the plane and no 3 are collinear. Given any two distinct points Pi and Pj, there is a third point Pk such that Q lies inside the triangle PiPjPk. Prove that n must be odd. |
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