Given a $1000\times 1000$ board. At the beginning, all cells have $0$ written on it. In an operation, we are allowed to choose any $130\times 130$ subboard and increase every number in this subboard by $1$. We can perform this operation as many times as we like. What is the maximum number of cells with the same (non-zero) number?
We can choose non-overlapping $130\times 130$ subboards to cover a $910\times 910$ subboard, yielding $910^2$ squares with the same number, $1$. Is this optimal, or can we do better?
[Source: Based on Russian competition problem]