| 1. a and b are positive integers. Show that there are at most a finite number of integers n such that an2 + b and a(n + 1)2 + b are both squares. | |
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2. The triangle ABC is not isosceles. The incircle touches BC, CA, AB at K, L, M respectively. N is the midpoint of LM. The line through B parallel to LM meets the line KL at D, and the line through C parallel to LM meets the line MK at E. Show that D, E and N are collinear.
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| 3. Find the minimum value of x2n + 2 x2n-1 + 3 x2n-2 + ... + 2n x + (2n+1) for real x. |
The original problems are in Hungarian. Many thanks to Carlos di Fiore for supplying an English translation for 1990-1993.
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(C) John Scholes
jscholes@kalva.demon.co.uk
10 Feb 2003
Last corrected/updated 19 Apr 2003