| 1. The square 0 ≤ x ≤ n, 0 ≤ y ≤ n has (n+1)2 lattice points. How many ways can each of these points be colored red or blue, so that each unit square has exactly two red vertices? | |
| 2. ABC is any non-equilateral triangle. P is any point in the plane different from the vertices. Let the line PA meet the circumcircle again at A'. Define B' and C' similarly. Show that there are exactly two points P for which the triangle A'B'C' is equilateral and that the line joining them passes through the circumcenter. | |
| 3. k is a non-negative integer and the integers a1, a2, ... , an give at least 2k different remainders on division by n+k. Prove that among the ai there are some whose sum is divisible by n+k. |
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 19 Apr 2003