KURSCHAK 2021 Problem 2

In neverland, there are $n$ cities and $n$ airlines. Each airline serves an odd number of cities in a circular way, that is, if it serves cities $c_1,c_2,\dots,c_{2k+1}$ , then they fly planes connecting $c_1c_2,c_2c_3,\dots,c_1c_{2k+1}$ . Show that we can select an odd number of cities $d_1,d_2,\dots,d_{2m+1}$ such that we can fly $d_1\rightarrow d_2\rightarrow\dots\rightarrow d_{2m+1}\rightarrow d_1$ while using each airline at most once.