2009 IMO Shortlist C4

NLD For an integer $m \geq 1$, we consider partitions of a $2^{m} \times 2^{m}$ chessboard into rectangles consisting of cells of the chessboard, in which each of the $2^{m}$ cells along one diagonal forms a separate rectangle of side length 1 . Determine the smallest possible sum of rectangle perimeters in such a partition.

NLD 对于整数 $m \geq 1$，我们考虑将 $2^{m} \times 2^{m}$ 棋盘划分为由棋盘单元组成的矩形，其中沿一条对角线的每个 $2^{m}$ 单元形成边长为 1 的单独矩形。确定该分区中矩形周长的最小可能总和。