Problem

Source: Israel National Olympiad 2018 Q1

Tags: logic, combinatorics



$n$ people sit in a circle. Each of them is either a liar (always lies) or a truthteller (always tells the truth). Every person knows exactly who speaks the truth and who lies. In their turn, each person says 'the person two seats to my left is a truthteller'. It is known that there's at least one liar and at least one truthteller in the circle. Is it possible that $n=2017?$ Is it possible that $n=5778?$