JBMO 2023 | Problem 3 - Solution - 26 June 2023

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.

Solution
The second player have the wining strategy.He place horizondal domino in the row $1,2,3,...,99$ but not in $100$ and make the couples:
$(100^2,1),(100^2-1,2),.....,(100^2/2+1,100^2/2)$
Τhen it follows if $A$ plays a number on someone domino then $B$ plays the number of his pair on the same domino.If $A$ playw a number at last row then $B$ place the other number of the pair in the last row mixed.
So $B$ can make all sums of $A$ equal and for him to be at leasy two diferrent so he win.