| 1. Show that (a + a')(c + c') ≥ (b + b')2 for all real numbers a, a', b, b', c, c' such that aa' > 0, ac ≥ b2, a'c' ≥ b'2. |
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| 2. Find the highest power of 2 dividing 2n! |
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3. ABC is acute-angled. A' is a point on the semicircle diameter BC (lying on the opposite side of BC to A). B' and C' are similar. Show how to construct such points so that AB' = AC', BC' = BA' and CA' = CB'.
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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: Andy Liu, Hungarian Problem Book III, 1929-1943, MAA 2001. ISBN 0883856441.
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(C) John Scholes
jscholes@kalva.demon.co.uk
6 January 2003
Last corrected/updated 3 Nov 03