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1. A straight line cuts the side AB of the triangle ABC at C1, the side AC at B1 and the line BC at A1. C2 is the reflection of C1 in the midpoint of AB, and B2 is the reflection of B1 in the midpoint of AC. The lines B2C2 and BC intersect at A2. Prove that sin B1A1C/sin C2A2B = B2C2/B1C1.
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| 2. Given any 22 points in the plane, no three collinear. Show that the points can be divided into 11 pairs, so that the 11 line segments defined by the pairs have at least five different intersections. | |
| 3. There are 30 boxes each with a unique key. The keys are randomly arranged in the boxes, so that each box contains just one key and the boxes are locked. Two boxes are broken open, thus releasing two keys. What is the probability that the remaining boxes can be opened without forcing them? |
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