Problem

Source: Bundeswettbewerb Mathematik 2020, Round 2 - Problem 4

Tags: combinatorics, inequalities, nonnegative, Sum



In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number. Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.