BG-MO 2016 Problem 6

Let $n$ be positive integer.A square $A$ of side length $n$ is divided by $n^2$ unit squares. All unit squares are painted in $n$ distinct colors such that each color appears exactly $n$ times. Prove that there exists a positive integer $N$ , such that for any $n>N$ the following is true: There exists a square $B$ of side length $\sqrt{n}$ and side parallel to the sides of $A$ such that $B$ contains completely cells of $4$ distinct colors.