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 1 of 5: Exhibit a construction
Detailed analysis
Put 1 in one cell and in its four orthogonally adjacent cells, and put 0 everywhere else. Every straight split leaves one side with sum at most 1, so the total 5 is attainable.