2006 IMO Shortlist C4

A cake has the form of an $n \times n$ square composed of $n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$. Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $\mathcal{A}$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle. (Taiwan)

蛋糕的形状是由 $n^{2}$ 个单位正方形组成的 $n \times n$ 正方形。草莓位于某些单位方块上，以便每一行或每一列恰好包含一个草莓；称这种排列为$\mathcal{A}$。让 $\mathcal{B}$ 是另一个这样的安排。假设蛋糕左上角有一个顶点的每个网格矩形包含的排列 $\mathcal{B}$ 的草莓不少于排列 $\mathcal{A}$ 的草莓。证明$\mathcal{B}$可以通过执行多个切换从$\mathcal{A}$获得，定义如下：切换包括选择一个只有两个草莓的网格矩形，位于其右上角和左下角，并将这两个草莓移动到该矩形的另外两个角。 (台湾)