| 1. Prove that 2p-1(2p - 1) is perfect when 2p - 1 is prime. [A perfect number equals the sum of its (positive) divisors, excluding the number itself.] |
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| 2. α and β are real and a = sin α, b = sin β, c = sin(α+β). Find a polynomial p(x, y, z) with integer coefficients, such that p(a, b, c) = 0. Find all values of (a, b) for which there are less than four distinct values of c. |
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| 3. ABCD is a rhombus. CA is the circle through B, C, D; CB is the circle through A, C, D; CC is the circle through A, B, D; and CD is the circle through A, B, C. Show that the angle between the tangents to CA and CC at B equals the angle between the tangents to CB and CD at A. |
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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 03