Problem

Source: Turkey JBMO TST 2024 P2

Tags: combinatorics



A real number is written on each square of a $2024 \times 2024$ chessboard. It is given that the sum of all real numbers on the board is $2024$. Then, the board is covered by $1 \times 2$ or $2\times 1$ dominos such that there isn't any square that is covered by two different dominoes. For each domino, Aslı deletes $2$ numbers covered by it and writes $0$ on one of the squares and the sum of the two numbers on the other square. Find the maximum number $k$ such that after Aslı finishes her moves, there exists a column or row where the sum of all the numbers on it is at least $k$ regardless of how dominos were replaced and the real numbers were written initially.