| 1. What integers a, b, c satisfy a2 + b2 + c2 + 3 < ab + 3b + 2c ? | |
| 2. D is a closed disk radius R. Show that among any 8 points of D one can always find two whose distance apart is less than R. | |
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3. A pyramid has square base and equal sides. It is cut into two parts by a plane parallel to the base. The lower part (which has square top and square base) is such that the circumcircle of the base is smaller than the circumcircles of the lateral faces. Show that the shortest path on the surface joining the two endpoints of a spatial diagonal lies entirely on the lateral faces.
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The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003