| 1. Given any four distinct points in the plane, show that the ratio of the largest to the smallest distance between two of them is at least √2. | |
| 2. x, y, z are positive reals less than 1. Show that at least one of (1 - x)y, (1 - y)z and (1 - z)x does not exceed 1/4. | |
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3. Two circles centers O and O' are disjoint. PP' is an outer tangent (with P on the circle center O, and P' on the circle center O'). Similarly, QQ' is an inner tangent (with Q on the circle center O, and Q' on the circle center O'). Show that the lines PQ and P'Q' meet on the line OO'.
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The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003