| 1. Show that no triangle has two sides each shorter than its corresponding altitude (from the opposite vertex). |
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| 2. a, b, c, d are integers. For all integers m, n we can find integers h, k such that ah + bk = m and ch + dk = n. Show that |ad - bc| = 1. |
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3. ABC is an equilateral triangle with area 1. A' is the point on the side BC such that BA' = 2·A'C. Define B' and C' similarly. Show that the lines AA', BB' and CC' enclose a triangle with area 1/7.
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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 6 Jan 03