9.5. A football team coach loves to experiment with the lineup. During training sessions, he divides 20 available field players into two teams of 10 players each, adds goalkeepers, and arranges a game between the teams. He wants any two field players to end up on different teams at some training session. What is the minimum number of training sessions that must take place for this to happen?
# 9.5. 5 Training Sessions.

Note that four training sessions are insufficient. Let's take 10 football players who played on the same team during the first training session. During the second training session, at least five of them will be on the same team again. During the third training session, at least three of these five will be on the same team again. Finally, two of these three will be on the same team during the fourth training session.

However, five training sessions are sufficient. Here is an example of team distribution.

|  | Team 1 | Team 2 |
| :--- | :---: | :---: |
| Training 1 | $1-10$ | $11-20$ |
| Training 2 | $1-5,11-15$ | $6-10,16-20$ |
| Training 3 | $1-3,6-7,11-13,16-17$ | $4-5,8-10,14-15,18-20$ |
| Training 4 | $1-2,4,6,8,11-12,14,16,18$ | $3,5,7,9-10,13,15,17,19-20$ |
| Training 5 | $2,4,6,8,10,12,14,16,18,20$ | $1,3,5,7,9,11,13,15,17,19$, |