Baltic Way 2022 · Problem 6

Mattis is hosting a badminton tournament for 40 players on 20 courts numbered from 1 to 20. The players are distributed with 2 players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court 1, and the player who won on court 20 stay in place. For the remaining 38 players, the winner on court $i$ moves to court $i+1$ and the loser moves to court $i-1$. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?