4. In a football tournament, 12 teams participated. By September, they had played several games, and no two teams had played each other more than once. It is known that the first team played exactly 11 games. Three teams played 9 games each. One team played 5 games. Four teams played 4 games each. Two teams played only one game each. And

## MATHEMATICS

9th GRADE

the information about the twelfth team was lost. How many games did the 12th team play?
Answer: 5 games.

Solution. Let the first team K1 play 11 games - i.e., once with everyone. Teams K2 and K3 played 1 game each - these are games with team K1.

There are 9 teams left (K4-K12). Three of these teams (K4, K5, K6) played 9 games each. One of these games was with K1. And 8 with all teams K4-K12 (except themselves).

Teams K7, K8, K9, K10 played 4 games each. It is clear that these games were with teams K1, K4, K5, K6. (And there were no other games for these teams)

Team K11 played 5 games. 4 of them were with K1, K4, K5, K6. And it is certain that she did not play with teams K2, K3, K7, K8, K9, K10. Thus, she played the fifth game with team K12.

In total, team K12 played 5 games (with K1, K4-6, K11) (and did not play with K2, K3, K7-K10).

Criteria. Correct solution - 7 points.

Fully correctly described who played with whom, but there is no answer to the question of the problem (or it is somehow incorrect) - 6 points.

Noted that the team that played 11 games played with everyone (without further progress) - 0 points. If it is also noted that two teams played only with it - **1** point.

A school student can use graphs for their solution without explaining the terms used.