| 1. A library has one exit and one entrance and a blackboard at each. Only one person enters or leaves at a time. As he does so he records the number of people found/remaining in the library on the blackboard. Prove that at the end of the day exactly the same numbers will be found on the two blackboards (possibly in a different order). | |
| 2. Sn is a square side 1/n. Find the smallest k such that the squares S1, S2, S3, ... can be put into a square side k without overlapping. | |
| 3. Let pk(x) = 1 - x + x2/2! - x3/3! + ... + (-x)2k/(2k)! Show that it is non-negative for all real x and all positive integers k. |
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