MathNet 2015-part-02 — 191 problems (EN–VI) Question 142

Some cities of a country are connected with roads. We say that a city $A$ belongs to a cycle of length $n$ if one can travel from $A$ through exactly $n-1$ other cities and return back to $A$. It is known that each city of the country belongs to a cycle of length 4 and also to a cycle of length 5.
Is it sure that
a) at least one city belongs to a cycle of length 3?
b) each city belongs to a cycle of length 3?