| 1. Several people visited a library yesterday. Each one visited the library just once (in the course of yesterday). Amongst any three of them, there were two who met in the library. Prove that there were two moments T and T' yesterday such that everyone who visited the library yesterday was in the library at T or T' (or both). |
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| 2. Three circles C1, C2, C3 in the plane touch each other (in three different points). Connect the common point of C1 and C2 with the other two common points by straight lines. Show that these lines meet C3 in diametrically opposite points. |
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| 3. (x1, y1, z1) and (x2, y2, z2) are triples of real numbers such that for every pair of integers (m, n) at least one of x1m + y1n + z1, x2m + y2n + z2 is an even integer. Prove that one of the triples consists of three integers. |
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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
24 Mar 2003
Last corrected/updated 24 Mar 2003