Question 63
Frieren and a mimic use $113$ treasure chests to play a game. At the beginning, all chests are empty and unlocked. In each year, Frieren adds a gem to an unlocked chest, then the mimic acts according to the following rules:
- If there is only one unlocked chest, it unlocks all chests;
- If there is more than one unlocked chest, the mimic chooses one of the unlocked chests and locks it.
Prove that there exists a constant $C \gt 0$ such that, no matter how the mimic acts, Frieren can always guarantee that, at the end of any year, the numbers of gems in any two chests differ by at most $C$. (Remark: the mimic is not allowed to eat Frieren.)
- If there is only one unlocked chest, it unlocks all chests;
- If there is more than one unlocked chest, the mimic chooses one of the unlocked chests and locks it.
Prove that there exists a constant $C \gt 0$ such that, no matter how the mimic acts, Frieren can always guarantee that, at the end of any year, the numbers of gems in any two chests differ by at most $C$. (Remark: the mimic is not allowed to eat Frieren.)