| 1. Show that a graph has an even number of points of odd degree. |
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| 2. P is any point inside an acute-angled triangle. D is the maximum and d is the minimum distance PX for X on the perimeter. Show that D ≥ 2d, and find when D = 2d. |
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| 3. x1 < x2 < x3 < x4 are real. y1, y2, y3, y4 is any permutation of x1, x2, x3, x4. What are the smallest and largest possible values of (y1 - y2)2 + (y2 - y3)2 + (y3 - y4)2 + (y4 - y1)2. |
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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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© John Scholes
jscholes@kalva.demon.co.uk
6 January 2003
Last corrected/updated 6 Jan 03