2012
Problem - 1419
In a round-robin tournament with $6$ teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end on the tournament?
There will be totally $C_6^2=15$ games therefore the total points all the teams will obtain is $15$. Therefore, a six-way tie is impossible because this will imply each team gets $2.5$ points. However the given condition indicates all the teams must get a whole point score. Therefore, the maximum number of tied winner is $5$ which means each team wins $3$ games. This output is in fact possible:
- Team $6$ losses all its$5$ games.
- Team $1$ wins $2$, $4$, and $6$.
- Team $2$ wins $3$, $5$, and $6$.
- Team $3$ wins $4$, $1$, and $6$.
- Team $4$ wins $5$, $2$, and $6$.
- Team $5$ wins $1$, $3$, and $6$