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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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system

Consider an integer \( n \geq 3 \).

In a tournament of \( n \) nations, each nation plays against the other \( n-1 \) nations.

The ranking is based on the number of matches won (a match can only be won or lost). In the case of a tie, the ranking is determined by looking at the number of points scored.

A clean sweep is achieved by winning all matches. Receiving the wooden spoon means losing all matches.

a) Do there exist tournaments that can result in these scores:

Tournament of 6 nations:
\[
\begin{array}{ccc}
\text{Team} & \text{Wins} & \text{Losses} \\
A & 5 & 0 \\
B & 4 & 1 \\
C & 4 & 1 \\
D & 1 & 4 \\
E & 1 & 4 \\
F & 0 & 5 \\
\end{array}
\]

Tournament of 5 nations:
\[
\begin{array}{ccc}
\text{Teams} & \text{Wins} & \text{Losses} \\
A & 3 & 1 \\
B & 3 & 1 \\
C & 2 & 2 \\
D & 1 & 3 \\
E & 1 & 3 \\
\end{array}
\]

b) Demonstrate that the integers \( n \) for which there exists a tournament where the winner has as many wins as losses are odd integers.

c) For which values of \( n \) does a tournament exist where the runner-up has more losses than wins?

d) For which values of \( n \) does a tournament exist where the top three can be ranked without considering the number of points scored?

e) For what minimal value of \( n \) does a tournament exist where the top three can be ranked without considering the number of points scored, given that there was no clean sweep?

f) Let \( k \geq 0 \) be a fixed integer. For what minimal value of \( n \) does a tournament exist where the top three can be ranked without considering the number of points scored, given that the first player has at least \( k \) losses?

Present the answer in LaTex format: \boxed{Your answer}
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