| 1. Show that given four non-coplanar points A, B, P, Q there is a plane with A, B on one side and P, Q on the other, and with all the points the same distance from the plane. |
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| 2. Show that we cannot factorise x4 + 2x2 + 2x + 2 as the product of two quadratics with integer coefficients. |
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| 3. Let S be any finite set of distinct positive integers which are not divisible by any prime greater than 3. Prove that the sum of their reciprocals is less than 3. |
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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