MathDB

Matrices with defined property

Source: Iran TST 2013:TST 2,Day 1,Problem 1

April 23, 2013

functionlinear algebramatrixcombinatorics proposedcombinatorics

Problem Statement

Nonnegative real numbers

p_{1},\ldots,p_{n}

and

q_{1},\ldots,q_{n}

are such that

p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}

Among all the matrices with nonnegative entries having

p_i

as sum of the

i

-th row's entries and

q_j

as sum of the

j

-th column's entries, find the maximum sum of the entries on the main diagonal.