Question 112
A real number is written on each square of a $2024 \times 2024$ board such that sum of all real numbers on the board is equal to $2024$. The board is also entirely covered by $1 \times 2$ or $2 \times 1$ dominoes each consisting $2$ unit squares of the board such that no square is covered by two different dominoes. For each domino, Asli erases the two numbers it covers, writes $0$ on one of the squares and writes the sum of the two numbers on the other square. Find the maximal possible number $k$ such that regardless of how the real numbers were written and the dominos were placed initially Asli can guarantee that after her moves there exists a column or row such that the sum of all numbers on it is at least $k$.