| 1. a, b are positive reals. Show that 1/x + 1/(x-a) + 1/(x+b) = 0 has two real roots one in [a/3, 2a/3] and the other in [-2b/3, -b/3]. |
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| 2. ABC is a triangle. The bisector of ∠C meets AB at D. Show that CD2 < CA·CB. |
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| 3. The set {1, 2, 3, 4, 5} is divided into two parts. Show that one part must contain two numbers and their difference. |
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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