| 1. a1, a2, ... , an is any finite sequence of positive integers. Show that a1! a2! ... an! < (S + 1)! where S = a1 + a2 + ... + an. |
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| 2. P, Q, R are three points in space. The circle CP passes through Q and R, the circle CQ passes through R and P, and the circle CR passes through P and Q. The tangents to CQ and CR at P coincide. Similarly, the tangents to CR and CP at Q coincide, and the tangents to CP and CQ at R coincide. Show that the circles are either coplanar or lie on the surface of the same sphere. |
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| 3. A1, A2, ... , An are points in the plane, no three collinear. The distinct points P and Q in the plane do not coincide with any of the Ai and are such that PA1 + ... + PAn = QA1 + ... + QAn. Show that there is a point R in the plane such that RA1 + ... + RAn < PA1 + ... + PAn. |
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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 3 Nov 03