| 1. Show that for p an odd prime and n a positive integer, there is at most one divisor d of n2p such that d + n2 is a square. | |
| 2. I is the incenter of the triangle ABC and A' is the center of the excircle opposite A. The bisector of angle BIC meets the side BC at A". The points B', C', B", C" are defined similarly. Show that the lines A'A", B'B", C'C" are concurrent. | |
| 3. A coin has probability p of heads and probability 1-p of tails. The outcome of each toss is independent of the others. Show that it is possible to choose p and k, so that if we toss the coin k times we can assign the 2k possible outcomes amongst 100 children, so that each has the same 1/100 chance of winning. [A child wins if one of its outcomes occurs.] |
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