RMM 2020 Problem 3

Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$ , and $m$ distinct airports $c_1, \dots, c_m$ , where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$ ; $c_2$ and $c_3$ ; $\dots$ ; $c_{m-1}$ and $c_m$ ; $c_m$ and $c_1$ .

Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.