China Western Mathematical Olympiad

The answer is $n=3$ or $n\ge 5$.

(1) For $n=3$, suppose $A$, $B$ and $C$ are three players, and the result of three matches are as follows: $A$ wins $B$, $B$ wins $C$, and $C$ wins $A$. These results obviously satisfy the condition.

(2) If $n=4$, suppose that the condition holds, i.e., in view of the results of all matches, every player is not out-performed by any other player. It is obvious that none of these four players wins in his three games, otherwise, the other three players will be not out-performed by this player. Similarly, none of these three players loses in his three games. It follows that each player wins one or two matches. For the player $A$, assume that $A$ wins $B$ and $D$, but loses to $C$, then both $B$ and $D$ win $C$; otherwise, they would not out-performed $A$. For the loser in the match $B$ vs. $D$, he only wins $C$, and so the loser is impossible to be not out-performed by the winner. Consequently, for $n=4$, the given condition could not happen.

(3) For $n=6$, one can construct the tournament results by means of the following directed graph, in which each black dot represents a player, and $\bullet \to \circ$ represents a match with the result that player $\bullet$ wins player $\circ$.



(4) If there exist tournament results such that each of $n$ players $A_i$ ($1\le i\le n$) is not out-performed by any other player, we will prove that the same holds for $n+2$ players as follows: Suppose that $M$ and $N$ are the additional players to the original $n$ players. Construct the game results of $M$ and $N$ as follows: $$A_i\to M,M\to N,N\to A_i$$ for all $i=1,2,\dots ,n$, and the game results among $A_i$'s are still the original ones. Now we want to check that these $n+2$ players satisfy the given condition. For any player $G\in \{A_1,A_2,\dots ,A_n\}$, then it suffices to consider the players $G,M,N$, and it reduces to the case $n=3$.



One can check that each of these three players $G,M,N$ is not out-performed by any one of the other two players. Hence, the tournament result of these $n+2$ players satisfies the required condition.

In particular, it follows from (1) and the induction step of (4) that the required condition holds for any odd $n$ with $n\ge 3$. Moreover, it follows from (3) and the induction step of (4) that the required condition holds for any even $n$ with $n\ge 6$, and it completes the proof. $\square$