6.5. 2023 Minecraft players gathered and divided into two servers. Every minute, someone was upset that there were more players on their server than on the other, and switched to the other server. Over 2023 minutes, each player changed servers only once. How many people could have initially gathered on the first server? Find all possible options and prove that there are no others.
Solution. Let's call the servers A and B. If any of the servers is empty at the beginning, then after 1012 transitions, it will become the largest, and all those who have not changed servers will be on the other one. That is, both servers are not empty at the beginning. Note that at the moment when a player who started playing on server A switches to server B, no more than 1011 players have gathered on this server. From this moment on, server B cannot gather more than 1012 players, because when it does, more people start playing on it than on server A, and players cannot leave from there at this moment. Therefore, at the end, no more than 1012 people are playing on server B, and no fewer than 1011 on server A. Similarly, we get that at the end of the game, no more than 1012 people are playing on server A, and no fewer than 1011 on server B. This means that the only possible case is when at the end of the game, 1011 people gather on one server, and 1012 on the other. But each participant changed servers exactly once, which means that the initial situation coincides with the final situation. That is, at the beginning, 1011 people are on one server, and 1012 on the other.

Answer. 1011 people or 1012 people.