| 1. Given any 5 points A, B, P, Q, R (in the plane) show that AB + PQ + QR + RP <= AP + AQ + AR + BP + BQ + BR. | |
| 2. n > 2 is even. The squares of an n x n chessboard are painted with n2/2 colors so that there are exactly two squares of each color. Prove that one can always place n rooks on squares of different colors so that no two are in the same row or column. | |
| 3. Divide the positive integer n by the numbers 1, 2, 3, ... , n and denote the sum of the remainders by r(n). Prove that for infinitely many n we have r(n) = r(n+1). |
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