Japan Junior Mathematical Olympiad

By assumption there exists exactly $1$ person who won $j$ matches for $j=0,1,2,3,4,5$, so let $A_j$ be the person who won $j$ matches for each $j$ ($0\le j\le 5$). Then we can see that

$A_5$ won all matches, so, he won over $A_0,A_1,A_2,A_3,A_4$. $A_4$ lost to $A_5$, but won over $A_0,A_1,A_2,A_3$. $A_3$ lost to $A_4,A_5$, but won over $A_0,A_1,A_2$. $A_2$ lost to $A_3,A_4,A_5$, but won over $A_0,A_1$. $A_1$ lost to $A_2,A_3,A_4,A_5$, but won over $A_0$. $A_0$ lost to $A_1,A_2,A_3,A_4,A_5$.

In this way, we can see that if we can determine who won how many games, then we can tabulate all the win-loss combinations at the tournament. Hence the number of possible combinations is $6!=720$.