Problem
COM-B2-M01-P008 Tournament Wins
#8
★★★☆☆ Level 3 of 5
In a tournament, each of \(n\) players played each other exactly once, with no draws. Prove that some player won at least \(\frac{n-1}{2}\) games, and some player won at most \(\frac{n-1}{2}\) games.
Count the total number of wins.
There are \(\binom n2\) games, and each game gives exactly one win. The average number of wins per player is \(\frac{\binom n2}{n}=\frac{n-1}{2}\). Therefore someone has at least the average, and someone has at most the average.
A good place to discuss that the average may be fractional.