Question 60
A non-negative real number is written in each cell of a $100 \times 100$ chessboard. The board is called balanced if and only if the numbers in every column sum to 1 and the numbers in every row also sum to 1. Find the largest positive real number $x$ such that in every balanced board we can choose 100 cells, no two in the same column or the same row, such that the numbers in these cells are all at least $x$.