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 4 of 5: Repeat by columns
Detailed analysis
Define column interval sums C in the same way. The identical argument gives a column b with sums above and below b at most 1. The cell at row a and column b is bounded by 1.