| 1. If x2 + y2 = u2 + v2 = 1 and xu + yv = 0 for real x, y, u, v, find xy + uv. |
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| 2. S is a set of 16 squares on an 8 x 8 chessboard such that there are just 2 squares of S in each row and column. Show that 8 black pawns and 8 white pawns can be placed on these squares so that there is just one white pawn and one black pawn in each row and column. |
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3. A and B are points on the circle C, which touches a second circle at a third point P. The lines AP and BP meet the second circle again at A' and B' respectively. Show that triangles ABP and A'B'P are similar.
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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