APMO 2012/02

[Nadim Ul Abrar]

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.