Question 90
Let $n \ge 3$ be a positive integer. There are $n$ Jujutsushis in total from $k$ different schools, labeled from 1 to $n$. It is known that when two Jujutsushis fight, the one with the smaller label will win. It is also known that if $\{x_1, x_2, \dots, x_n\}$ is a permutation of $\{1, 2, \dots, n\}$, then among the $n-1$ fights between Jujutsushi number $x_1$ and $x_2$, $x_2$ and $x_3$, $\dots$, $x_{n-1}$ and $x_n$, the set of winners will contains at least one Jujutsushi from each of the $k$ schools. Prove that $n \ge 2^k$.