VMO 2006 Problem 3

Let $m$ , $n$ be two positive integers greater than 3. Consider the table of size $m\times n$ ( $m$ rows and $n$ 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 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) $m=2004$ , $n=2006$ ;

b) $m=2005$ , $n=2006$ .

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