| 1. AC is the long diagonal of a parallelogram ABCD. The perpendiculars from C meet the lines AB and AD at P and Q respectively. Show that AC2 = AB·AP + AD·AQ. |
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| 2. Find three distinct positive integers a, b, c such that 1/a + 1/b + 1/c is an integer. |
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| 3. The real quadratics ax2 + 2bx + c and Ax2 + 2Bx + C are non-negative for all real x. Show that aAx2 + 2bBx + cC is also non-negative for all real x. |
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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
Last corrected/revised 1 Nov 03