Vmo 2006 a3
Source:
February 28, 2006
geometryrectanglemodular arithmeticcombinatorics proposedcombinatorics
Problem Statement
Let , be two positive integers greater than 3. Consider the table of size ( rows and columns) formed with unit squares. We are putting marbles into unit squares of the table following the instructions:
each time put 4 marbles into 4 unit squares (1 marble per square) such that the 4 unit squares formes one of the followings 4 pictures (click [url=http://www.mathlinks.ro/Forum/download.php?id=4425]here to view the pictures).
In each of the following cases, answer with justification to the following question: Is it possible that after a finite number of steps we can set the marbles into all of the unit squares such that the numbers of marbles in each unit square is the same?
a) , ;
b) , .