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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?

1. 13
2. 12
3. 11
4. 10
5. 9
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I think possible with 9 only.

Let the $7$ Teams be $A,B,C,D,E,F,G$ and so each team plays total 6 matches.

Suppose, Team $A$ wins over $E,F,G$ and draws with $B,C,D$ hence total points scored by Team A = $9$ points

Now, Team $B$ wins over $E,F,G$ and draws with $A,C,D$ hence total points scored by Team B = $9$ points

Similarly, happens for next two teams $C$ and $D$ .

Hence, Finalized scores are =>

A = 9
B = 9
C = 9
D = 9
E = ? (Less than or equal to 4)
F = ? ("...")
G = ? ("...")

Given that the order among the teams with the same total are determined by a whimsical tournament referee.

So, He/She can order the top $3$ teams like $ABC$,$ABD$,$BCD$,$ACD$ .......

But, Question says " team must earn in order to be guaranteed a place in the next round "

Hence, Not to depend on that whimsical referee, the minimum total number of points a team must earn in order to be guaranteed a place in the next round = $9+1$ = $10$ points

why extra 1 is added with 9?
Beacause that will guarantee the top 3 positions and there is no need to depend on referee.

If someone wants to get into the next round he/she must secure at least 3rd position. That means that a team must earn as many points as the $3$rd team does to keep alive the hope of going into the next round. (sometimes +1 , we will get to that later.)

We assume that these winners are in the order $A\rightarrow B\rightarrow C$.

Now we will try to increase the points of team $C$ such that $\text{points}(A,B) \geq C$ and $C$ also maintain $3$rd position.

Consider teams $A,B,C$

There can be a situation when winners $A,B,C$ all three team beat $D,E,F,G$ and play draw among them. Then $A,B,C$ will get $10$ points each.

• $\Rightarrow$ So, $3$rd team $C$ can get maximum $10$ points.

$C$ can not get $11$ points. Because in that case it has to beat one of the winners , and it will move to higher position but we need $C$ at 3rd posotion only.

What happens to $D$ ? the $4$th position holder ? He can get maximum $6$ after $\text{three}$ consecutive loss to $A,B \text{ and }C$ by beating $E,F,G$.

• $\Rightarrow$ So, If a team gets $10$ points , that team definitely get into the next round.
• Points of $C$ and $D$ are not equal in this case and we need not worry about referee.
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