MathDB

Given positive integers

m

and

n

. Let

P

and

Q

be two collections of

m \times n

numbers of

0

and

1

, arranged in

m

rows and

n

columns. An example of such collections for

m=3

and

n=4

\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right].

Let those two collections satisfy the following properties: (i) On each row of

P

, from left to right, the numbers are non-increasing, (ii) On each column of

Q

, from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in

P

equals to the same row in

Q

, (iv) The sum of numbers on the column in

P

equals to the same column in

Q

. Show that the number on row

i

and column

j

P

equals to the number on row

i

and column

j

Q

for

i=1,2,\dots,m

and

j=1,2,\dots,n

.
Proposer: Stefanus Lie

Problem Statement