2014 IMO 第 2 题

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.

令 $n\geq 2$ 为整数。考虑一个由 $n^{2}$ 个单位方块组成的 $n\times n$ 棋盘。如果每一行和每一列都只包含一个车，则该板上的 $n$ 车的配置是和平的。找到最大的正整数 $k$，使得对于 $n$ 车的每个和平配置，有一个 $k\times k$ 方格，其任何 $k^{2}$ 单位方格上都不包含车。