| 1. A cube has all 4 vertices of one face at lattice points and integral side-length. Prove that the other vertices are also lattice points. | |
| 2. Show that for any integer k > 2, there are infinitely many positive integers n such that the lowest common multiple of n, n+1, ... , n+k-1 is greater than the lowest common multiple of n+1, n+2, ... , n+k. | |
| 3. The integers are colored with 100 colors, so that all the colors are used and given any integers a < b and A < B such that b - a = B - A, with a and A the same color and b and B the same color, we have that the whole intervals [a, b] and [A, B] are identically colored. Prove that -1982 and 1982 are different colors. |
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 28 Apr 2003