KöMaL 2024 A884

We fill in an $n\times n$ table with real numbers such that the sum of the numbers in each row and each coloumn equals 1. For which values of $K$ is the following statement true: if the sum of the absolute values of the negative entries in the table is at most $K$, then it's always possible to choose $n$ positive entries of the table such that each row and each coloumn contains exactly one of the chosen entries.