| 1. Show that 5 divides 1n + 2n + 3n + 4n iff 4 does not divide n. |
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| 2. Let α = cot π/8, β = cosec π/8. Show that α satisfies a quadratic and β a quartic, both with integral coefficients and leading coefficient 1. |
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| 3. Let d be the greatest common divisor of a and b. Show that exactly d elements of {a, 2a, 3a, ... , ba} are divisible by b. |
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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
Last corrected/updated 30 Oct 2003