Problem 3.
Alice and Bob alternatively play the following game on a 100x100 grid, with Alice going first. Initially, the grid is empty. They alternatively pick one number from 1 to 100², that is not already written in one of the cells, and choose an empty cell and put this number in this cell. When no empty cell has remained, Alice computes the sum of numbers in each row, and her score is the greatest of these 100 sums. Bob computes the sum of the numbers in each column, and his score is the greatest of these 100 sums. Alice wins if her score is greater than Bob's score, and Bob wins if his score is greater than Alice's score. In any other case, no one wins.
Find if any of the two players has a winning strategy, and if so, find which one of the two.