| 1. a, b are integers. The solutions of y - 2x = a, y2 - xy + x2 = b are rational. Show that they must be integers. |
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| 2. A square has 10s digit 7. What is its units digit? |
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| 3. A, B are two points inside a given circle C. Show that there are infinitely many circles through A, B which lie entirely inside C. |
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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