| 1. ABC is an acute-angled triangle. D is a variable point in space such that all faces of the tetrahedron ABCD are acute-angled. P is the foot of the perpendicular from D to the plane ABC. Find the locus of P as D varies. | |
| 2. A factory produces several types of mug, each with two colors, chosen from a set of six. Every color occurs in at least three different types of mug. Show that we can find three mugs which together contain all six colors. | |
| 3. What is the largest possible value of |a1 - 1| + |a2 - 2| + ... + |an - n|, where a1, a2, ... , an is a permutation of 1, 2, ... , n? |
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 1 Nov 2003