IMO 2023 Shortlist C5
Elisa has 2023 treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of t...
Category: Combinatorics
Problem
Elisa has 2023 treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules: • if more than one chests are unlocked, it locks one of them, or • if there is only one unlocked chest, it unlocks all the chests. Given that this process goes on forever, prove that there is a constant C with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds C, regardless of how the fairy chooses the chests to lock. (Israel)