Consider a matrix of size
![n\times n](/media/m/1/c/a/1caee5824fd124b98d47c32a5a96cad3.png)
whose entries are real numbers of absolute value not exceeding
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
. The sum of all entries of the matrix is
![0](/media/m/7/b/8/7b8b0b058cf5852d38ded7a42d6292f5.png)
. Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be an even positive integer. Determine the least number
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
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.