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permutation of numbers on a matrix so that \sum b_{ij} < 2

Source: Austrian Polish 1984 APMC

April 30, 2020

linear algebramatrix

Problem Statement

m\times n

matrix

(a_{ij})

of real numbers satisfies

|a_{ij}| <1

and

\sum_{i=1}^m a_{ij}= 0

for all

j

. Show that one can permute the entries in each column in such a way that the obtained matrix

(b_{ij})

satisfies

\sum_{j=1}^n b_{ij} < 2

for all

i