Problem 2

Into each box of a 2012 by 2012 square grid, insert a real number between 0 and 1 inclusive. Split the grid into two non-empty rectangles of boxes by a line parallel to a side of the grid. Suppose that for every such split at least one resulting rectangle has sum at most 1. Determine the maximum possible sum of all inserted numbers.

Step 2 of 5: Define row interval sums

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R(x,y)=\sum_{i=x}^{y}\sum_j r_{i,j},\quad R(x,y)=0\text{ if }x>y.

Detailed analysis

Let R(x,y) be the sum in rows x through y. Choose a as large as possible with R(1,a-1) at most 1, then choose c as small as possible with R(c+1,n) at most 1, where n=2012.