| 1. How many n-digit decimal integers have all digits 1, 2 or 3. How many also contain each of 1, 2, 3 at least once? |
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| 2. Prove that 5n + 2 3n-1 + 1 = 0 (mod 8). |
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| 3. ABCD is a quadrilateral with vertices in that order. Prove that AC is perpendicular to BD iff AB2 + CD2 = BC2 + DA2. |
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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