Consider a matrix of size
whose entries are real numbers of absolute value not exceeding
. The sum of all entries of the matrix is
. Let
be an even positive integer. Determine the least number
such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding
in absolute value.
%V0
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.