VN-TST 2007 Problem 1

Given two sets $A, B$ of positive real numbers such that: $|A| = |B| =n$ ; $A \neq B$ and $S(A)=S(B)$ , where $|X|$ is the number of elements and $S(X)$ is the sum of all elements in set $X$ . Prove that we can fill in each unit square of a $n\times n$ square with positive numbers and some zeros such that:

a) the set of the sum of all numbers in each row equals $A$ ;

b) the set of the sum of all numbers in each column equals $A$ .

c) there are at least $(n-1)^{2}+k$ zero numbers in the $n\times n$ array with $k=|A \cap B|$ .