Question 1
Let $A,B$ be sets of positive integers with $|A|=|B|=n$, where $n$ is a positive integer, and with equal sums of their elements. Consider an $n\times n$ square grid. Prove that nonnegative integers can be placed in its cells so that (i) the row sums are the elements of $A$; (ii) the column sums are the elements of $B$; and (iii) at least $(n-1)^2+k$ entries are zero, where $k=|A\cap B|$.