FKMO 2015 Problem 3

There are at least $3$ subway stations in a city. In this city, there exists a route that passes through more than $L$ subway stations, without revisiting. Subways run both ways, which means that if you can go from subway station A to B, you can also go from B to A. Prove that at least one of the two holds. $\text{(i)}$ . There exists three subway stations $A$ , $B$ , $C$ such that there does not exist a route from $A$ to $B$ which doesn't pass through $C$ . $\text{(ii)}$ . There is a cycle passing through at least $\lfloor \sqrt{2L} \rfloor$ stations, without revisiting a same station more than once.