MK-MO 2022 Problem 4

Sofia and Viktor are playing the following game on a $2022 \times 2022$ board:

• Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table;
• Then Viktor without seeing the table, chooses a positive integer $n$ ;
• After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them;
• Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before. If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes.

Proposed by Viktor Simjanoski