| 1. A pentagon inscribed in a circle has equal angles. Show that it has equal sides. |
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| 2. Let a be an integer, and let p(x1, x2, ... , xn) = ∑1n k xk. Show that the number of integral solutions (x1, x2, ... , xn) to p(x1, x2, ... , xn) = a, with all xi > 0 equals the number of integral solutions (x1, x2, ... , xn) to p(x1, x2, ... , xn) = a - n(n + 1)/2, with all xi ≥ 0. |
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| 3. R is a rectangle. Find the set of all points which are closer to the center of the rectangle than to any vertex. |
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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 1, 1894-1905, 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