OPM 2022 Problem 3

The Proenc has a new $8\times 8$ chess board and requires composing it into rectangles that do not overlap, so that: (i) each rectangle has as many white squares as black ones; (ii) there are no two rectangles with the same number of squares. Determines the maximum value of $n$ for which such a decomposition is possible. For this value of $n$ , determine all possible sets ${A_1,... ,A_n}$ , where $A_i$ is the number of rectangle $i$ in squares, for which a decomposition of the board under the conditions intended actions is possible.