EGMO '18 P3
Posted: Sun Jun 28, 2020 5:28 pm
The $n$ contestants of EGMO are named $C_1,C_2,...,C_n$. After the competition, they queue in front of the restaurant according to the following rules.
1. The Jury chooses the initial order of the contestants in the queue.
2. Every minute the Jury chooses an integer $i$ with $1\leq i\leq n$.
i) If contestant $C_i$ has at least $i$ other contestant in front of her , then she pays one euro to the Jury and moves forward in the queue by exactly
$i$ positions.
ii) If contestant $C_i$ has fewer than $i$ contestant in front of her, the restaurant opens and the process ends.
a) Prove that the process cannot continue indefinitely,regarsless of the Jury's choices.
b) Determine for every n, the maximum number of euros that the Jury can collect by cunningly choosig the initial order and the sequence of moves.
1. The Jury chooses the initial order of the contestants in the queue.
2. Every minute the Jury chooses an integer $i$ with $1\leq i\leq n$.
i) If contestant $C_i$ has at least $i$ other contestant in front of her , then she pays one euro to the Jury and moves forward in the queue by exactly
$i$ positions.
ii) If contestant $C_i$ has fewer than $i$ contestant in front of her, the restaurant opens and the process ends.
a) Prove that the process cannot continue indefinitely,regarsless of the Jury's choices.
b) Determine for every n, the maximum number of euros that the Jury can collect by cunningly choosig the initial order and the sequence of moves.