| 1. In an infinite sequence of positive integers every element (starting with the second) is the harmonic mean of its neighbors. Show that all the numbers must be equal. | |
| 2. There are 4n segments of unit length inside a circle radius n. Show that given any line L there is a chord of the circle parallel or perpendicular to L which intersects at least two of the 4n segments. | |
| 3. For each arrangement X of n white and n black balls in a row, let f(X) be the number of times the color changes as one moves from one end of the row to the other. For each k such that 0 < k < n, show that the number of arrangements X with f(X) = n - k is the same as the number with f(X) = n + k. |
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