| 1. a >= 1, b >= 1 and c > 0 are reals and n is a positive integer. Show that ( (ab + c)n - c) <= an ( (b + c)n - c). | |
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2. ABC is a face of a convex irregular triangular prism (the triangular faces are not necessarily congruent or parallel). The diagonals of the quadrilateral face opposite A meet at A'. The points B' and C' are defined similarly. Show that the lines AA', BB' and CC' are concurrent.
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| 3. There are 998 red points in the plane, no three collinear. What is the smallest k for which we can always choose k blue points such that each triangle with red vertices has a blue point inside? |
The original problems are in Hungarian. Many thanks to Carlos di Fiore for supplying an English translation for 1990-1993.
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(C) John Scholes
jscholes@kalva.demon.co.uk
10 Feb 2003
Last corrected/updated 19 Apr 2003