# Task 5. Maximum 20 points

In Moscow, a tennis tournament is being held. Each team consists of 3 players. Each team plays against every other team, with each participant of one team playing against each participant of the other exactly one match. Due to time constraints, a maximum of 200 matches can be played in the tournament. How many teams can participate in the tournament so that everyone gets to play?

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# Solution:

Assume there are two teams - then each of the three members of one team plays against each of the other - i.e., $3 * 3=9$ games. The number of team pairs can be $200: 9=22.2 \ldots$ a maximum of 22. Two teams form only one pair; three teams form three pairs. If there is a fourth team, add 3 more pairs, so in total 6. With each subsequent team, one more is added to the number of pairs from the previous number: (2 teams - 1 pair, 3 teams $-1+2$ pairs, 4 teams $-1+2+3$ pairs, etc.). Thus, on the seventh day, it is possible to form $1+2+3+4+5+6=21$ pairs. On the eighth day, it is already $21+7=28$, which exceeds the maximum of 22. Therefore, the answer is: 7 teams.

## Criteria:

Correctly derived the correct answer - 20 points The chain of reasoning is correct, but due to an arithmetic error, an incorrect answer is obtained - 5 points.