CN-TST 2021 Problem 2

Given positive integers $n$ and $k$ , $n > k^2 >4.$ In a $n \times n$ grid, a $k$ -group is a set of $k$ unit squares lying in different rows and different columns. Determine the maximal possible $N$ , such that one can choose $N$ unit squares in the grid and color them, with the following condition holds: in any $k$ -group from the colored $N$ unit squares, there are two squares with the same color, and there are also two squares with different colors.