| 1. x1, x2, ... , xn is any sequence of positive reals and yi is any permutation of xi. Show that ∑ xi/yi ≥ n. |
|
| 2. S is a finite set of points in the plane. Show that there is at most one point P in the plane such that if A is any point of S, then there is a point A' in S with P the midpoint of AA'. |
|
|
3. Each vertex of a triangular prism is labeled with a real number. If each number is the arithmetic mean of the three numbers on the adjacent vertices, show that the numbers are all equal.
|
|
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.
Eötvös home
(C) John Scholes
jscholes@kalva.demon.co.uk
6 January 2003
Last corrected/updated 6 Jan 03