The n contestant of EGMO are named C1,C2,⋯Cn. After the competition, they queue in front of the restaurant according to the following rules.[*]The Jury chooses the initial order of the contestants in the queue.
[*]Every minute, the Jury chooses an integer i with 1≤i≤n.[*]If contestant Ci has at least i other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly i positions.
[*]If contestant Ci has fewer than i other contestants in front of her, the restaurant opens and process ends.[*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices.
[*]Determine for every n the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
combinatoricsEGMOEGMO 2018monovariantProcesses