2012 APMO 第 5 题

Into each box of a $2012 \times 2012$ square grid, a real number greater than or equal to $0$ and less than or equal to $1$ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to $1$ , no matter how the grid is split into $2$ such rectangles. Determine the maximum possible value for the sum of all the $2012 \times 2012$ numbers inserted into the boxes.

在 $2012 \times 2012$ 方格的每个框中，插入一个大于或等于 $0$ 且小于或等于 $1$ 的实数。考虑通过绘制一条与网格的水平边或垂直边平行的线，将网格分割为由网格框组成的 $2$ 非空矩形。假设对于至少一个生成的矩形，该矩形内的框中的数字总和小于或等于 $1$ ，无论网格如何分割为 $2$ 这样的矩形。确定插入框中的所有 $2012 \times 2012$ 数字总和的最大可能值。