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1. Prove that three half-lines from a given point contain three face diagonals of a cuboid iff the half-lines make with each other three acute angles whose sum is 180o.
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| 2. Given n > 2, find the largest h and the smallest H such that h < x1/(x1 + x2) + x2/(x2 + x3) + ... + xn/(xn + x1) < H for all positive real x1, x2, ... , xn. | |
| 3. k numbers are chosen at random from the set {1, 2, ... , 100}. For what values of k is the probability ½ that the sum of the chosen numbers is even? |
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