MathDB

Consider a white 100×100 square. Several cells (not necessarily neighbouring) were painted black. In each row or column that contains some black cells their number is odd. Hence we may consider the middle black cell for this row or column (with equal numbers of black cells in both opposite directions). It so happened that all the middle black cells of such rows lie in different columns and all the middle black cells of the columns lie in different rows.
a) Prove that there exists a cell that is both the middle black cell of its row and the middle black cell of its column.
b) Is it true that any middle black cell of a row is also a middle black cell of its column?

Problem Statement