| 1. ABCDE is a regular pentagon (with vertices in that order) inscribed in a circle of radius 1. Show that AB·AC = √5. |
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| 2. The roots of the quadratic x2 - (a + d) x + ad - bc = 0 are α and β. Show that α3 and β3 are the roots of x2 - (a3 + d3 + 3abc + 3bcd) x + (ad - bc)3 = 0. |
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| 3. Show that 2903n - 803n - 464n + 261n is divisible by 1897. |
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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