# 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 150 matches can be played in the tournament. How many teams can participate in the tournament so that everyone gets to play?
# 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 $150: 9 = 16.6 \ldots$ a maximum of 16. 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, the number of pairs increases by one more: (2 teams - 1 pair, 3 teams $-1+2$ pairs, 4 teams $-1+2+3$ pairs, etc.) Thus, if there are 6 teams, $1+2+3+4+5=15$ pairs. On the seventh day, it is already $15+6=21$, which exceeds the maximum of 16. Therefore, the answer is: 6 teams.

## Criteria:

Correct answer obtained with sound reasoning - 20 points Correct chain of reasoning, but an arithmetic error leads to an incorrect answer - 5 points.

## Federal State Budget Educational Institution of Higher Education Russian Presidential Academy of National Economy and Public Administration

## RANEPA School Olympiad in Economics 2022 - 2023 Academic Year 8-9 grade Final Stage Variant 2.