| 1. Prove that n! n! > nn for n > 2. |
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| 2. Let A and B be diagonally opposite vertices of a cube. Prove that the midpoints of the 6 edges not containing A or B form a regular (planar) hexagon. |
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| 3. If d is the greatest common divisor of a and b, and D is the greatest common divisor of A and B, show that dD is the greatest common divisor of aA, aB, bA and bB. |
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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