MathDB

In a sports tournament involving

N

teams, each team plays every other team exactly one. At the end of every match, the winning team gets

1

point and losing team gets

0

points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows:

x_1 \ge x_2 \ge \cdots \ge x_N .

Prove that for any

1\le k \le N

\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}

Problem Statement