| 1. Show that for any real x, at least one of x, 2x, 3x, ... , (n-1)x differs from an integer by no more than 1/n. |
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| 2. The numbers 1, 2, ... , n are arranged around a circle so that the difference between any two adjacent numbers does not exceed 2. Show that this can be done in only one way (treating rotations and reflections of an arrangement as the same arrangement). |
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| 3. Given two points A, B and a line L in the plane, find the point P on the line for which max(AP, BP) is as short as possible. |
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The original problems are in Hungarian. They are available on the KöMaL archive on the web. They are also available in English (with solutions in): (Translated by Elvira Rapaport) József Kürschák, G Hajós, G Neukomm & J Surányi, Hungarian Problem Book 2, 1906-1928, MAA 1963. Out of print, but available in some university libraries.
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John Scholes
jscholes@kalva.demon.co.uk
20 Oct 1999
Last corrected/updated 1 Nov 03