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Problem 1. A $2026\times 2026$ board is said to be bordeaux if at least one of its $2026^2$ unit cells is coloured red. A rectangular region made of cells is oddly-rectangular if it contains an odd number of red cells. Determine the largest positive integer $M$ such that, in every possible $2026 \times 2026$ bordeaux board, there exists an oddly-rectangular region of at least $M$ cells.
Note: A rectangular region has sides that are parallel to the sides of the board and contains all of its interior.
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