| 1. Given n positive integers ai, define Sk = S aik, A = S2/S1, and C = ( S3/n)1/3. For each of n = 2, 3 which of the following is true: (1) A >= C; (2) A <= C; or (3) A may be > C or < C, depending on the choice of ai? | |
| 2. Let f1(k) be the sum of the (base 10) digits of k. Define fn(k) = f1(fn-1(k) ). Find f1992(21991). | |
| 3. A finite number of points are given in the plane, no three collinear. Show that it is possible to color the points with two colors so that it is impossible to draw a line in the plane with exactly three points of the same color on one side of the line. |
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