MathDB

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|

Problem Statement