Let \(n \geq 2\) be an integer. Consider an \(n \times n\) chessboard consisting of \(n^2\) unit squares. A configuration of \(n\) rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer \(k\) such that, for each peaceful configuration of \(n\) rooks, there is a \(k \times k\) square which does not contain a rook on any of its \(k^2\) unit squares.
This problem is from the IMO.This problem is part of this set.