Question from the Russian Olympiad. (Translated with Google Translate, with a translation fix.):
In a circle there are $181$ people, each of whom is either a knight or a liar (liars always lie, and knights always tell the truth). Each of those standing said: “Two places away from me there is at least one liar.” Find the smallest possible number of liars among these $181$ people.
I first estimated that there were $5$ people then the minimum number of liars is $2$. because $1$ said $3$ liars $2$ said $4$ liars $3$ said $5$ liars and $4$ said $1$ liars and $5$ said $2$ liars. then if $1$ liar then $4$ and $3$ knights. Whence it follows that $4$ is also a liar and $2$ knight. Well, then the minimum number of liars in our case is $90$.
Question:is it correct answer?
EDIT: I’ll try to explain it. Let's say if there are people in a circle with numbers $1,2,3,4,5,6$ ..., $181$ then this means that #$1$ says that either #$3$ or #$180$ is a liar, #$2$ says that either #$4$ or #$181$ is a liar and so forth.