| 1. Let α be a real number, not an odd multiple of π. Prove that tan α/2 is rational iff cos α and sin α are rational. |
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| 2. Show that the centers of the squares on the outside of the four sides of a rhombus form a square. |
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| 3. (a1, a2, ... , an) is a permutation of (1, 2, ... , n). Show that ∏ (ai - i) is even if n is odd. |
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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 30 Oct 03