MathDB

A square with side

2008

is broken into regions that are all squares with side

1

. In every region, either

0

1

is written, and the number of

1

's and

0

's is the same. The border between two of the regions is removed, and the numbers in each of them are also removed, while in the new region, their arithmetic mean is recorded. After several of those operations, there is only one square left, which is the big square itself. Prove that it is possible to perform these operations in such a way, that the final number in the big square is less than

\frac{1}{2^{10^6}}

Problem Statement