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31 votes

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?

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

31 votes

Best answer

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 $\Rightarrow$

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 $ABC,ABD,BCD,ACD,\ldots$

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

Correct Answer: $D$

0

35 votes

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.

6 votes

suppose there are $7$ teams.

Team |
A | B | C | D | E | F | G |

Points |
0 | 0 | 0 | 0 | 0 | 0 | 0 |

$A$ wins all the matches but loses to $B$

Team |
${\color{Green} A}$ | B | C | D | E | F | G |

Points |
${\color{Green}{10}}$ | $2$ | 0 | 0 | 0 | 0 | 0 |

now since $A$ has played with all the teams next $B$ plays with remaining teams(i.e. $C,D,E,F,G$) and wins all the matches but loses to $C$

Team |
${\color{Green} A}$ | ${\color{Green} B}$ | C | D | E | F | G |

Points |
${\color{Green}{10}}$ | ${\color{Green}{10}}$ | $2$ | 0 | 0 | 0 | 0 |

now since $A,B$ have played with all the teams next $C$ plays with remaining teams(i.e. $D,E,F,G$) and wins all the matches.

Team |
${\color{Green} A}$ | ${\color{Green} B}$ | ${\color{Green}{C}}$ | D | E | F | G |

Points |
${\color{Green}{10}}$ | ${\color{Green}{10}}$ | ${\color{Green}{10}}$ | 0 | 0 | 0 | 0 |

Now all the remaining teams cannot make a score $\geq 10$ so only $A,B,C$ can enter the next round.

And the score of each of $A,B,C =10$

$\therefore$ Option $D.$ $10$ is the correct answer.

3 votes

$\text{A B C | D E F G}$

Suppose $\text{E, F, G}$ have disqualified. So we have to consider the minimum points these teams can score. It will not be $0$ as $\text{E,F,G}$ play matches among themselves.

Minimum points $\text{E,F,G}$ can get is $2,2,2$ resp.($\text{E-F F-G E-G}$ ties)

The total number of points is $42$ {$21$ games- $\binom{7}{2}$}

So $42-6 =36$ points left.

This can be distributed among teams $\text{A,B,C,D}$ in:

$\text{9 9 9 9}$

$\text{10 10 8 8}$ and on...

$8$ and $9$ cannot guarantee a team to qualify to second round as more than one team scores the same. So the minimum points to qualify to second round is $10$.

Suppose $\text{E, F, G}$ have disqualified. So we have to consider the minimum points these teams can score. It will not be $0$ as $\text{E,F,G}$ play matches among themselves.

Minimum points $\text{E,F,G}$ can get is $2,2,2$ resp.($\text{E-F F-G E-G}$ ties)

The total number of points is $42$ {$21$ games- $\binom{7}{2}$}

So $42-6 =36$ points left.

This can be distributed among teams $\text{A,B,C,D}$ in:

$\text{9 9 9 9}$

$\text{10 10 8 8}$ and on...

$8$ and $9$ cannot guarantee a team to qualify to second round as more than one team scores the same. So the minimum points to qualify to second round is $10$.