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?