I think possible with 9 only.

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In a tournament with $7$ teams, each team plays one match with every other team. For each match, the team earns two points if it wins, one point if it ties, and no points if it loses. At the end of all matches, the teams are ordered in the descending order of their total points (the order among the teams with the same total are determined by a whimsical tournament referee). The first three teams in this ordering are then chosen to play in the next round. What is the minimum total number of points a team must earn in order to be guaranteed a place in the next round?

- $13$
- $12$
- $11$
- $10$
- $9$

1 vote

lets first try with n=9 ( our tarhet is to get only top three teams get 9 points . If more than three teams will get 9 points then choosing top three will also depend upon referee )

A wins over D,E,F,G and ties with B so total 9 points

B wins over D,E,F,G and ties with A so total 9 points ( if A ties with B then B also ties with A)

now try for C to getting 9 points so first three(A,B,C) will qualify for next round

C ties with D (1 point) , wins over E,F,G( 6 points ) , no other possibilities for C

hence with 9 points we cant choose top three teams guarantely.

Now n=10

A wins over D,E,F,G and ties with B,C so total 10 points

B wins over D,E,F,G and ties with A,C so total 10 points

C wins over D,E,F,G and ties with B,A so total 10 points

for others ( D,E,F,G) we cant get 10 points

Hence 10 is the minimum total number of points a team must earn in order to be guaranteed a place in the next round.

ANS is D

A wins over D,E,F,G and ties with B so total 9 points

B wins over D,E,F,G and ties with A so total 9 points ( if A ties with B then B also ties with A)

now try for C to getting 9 points so first three(A,B,C) will qualify for next round

C ties with D (1 point) , wins over E,F,G( 6 points ) , no other possibilities for C

hence with 9 points we cant choose top three teams guarantely.

Now n=10

A wins over D,E,F,G and ties with B,C so total 10 points

B wins over D,E,F,G and ties with A,C so total 10 points

C wins over D,E,F,G and ties with B,A so total 10 points

for others ( D,E,F,G) we cant get 10 points

Hence 10 is the minimum total number of points a team must earn in order to be guaranteed a place in the next round.

ANS is D