| 1. Show that a positive integer n is the sum of two squares iff 2n is the sum of two squares. |
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| 2. Show that 1/n + 1/(n+1) + 1/(n+2) + ... + 1/n2 > 1 for integers n > 1. |
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| 3. Show that for every acute-angled triangle ABC there is a point in space P such that (1) if Q is any point on the line BC, then AQ subtends an angle 90o at P, (2) if Q is any point on the line CA, then BQ subtends an angle 90o at P, and (3) if Q is any point on the line AB, then CQ subtends an angle 90o at P. |
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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