$\underline{\text { Frankin B.R. }}$.

A traveler visited a village where every person either always tells the truth or always lies.

The villagers stood in a circle, and each one told the traveler whether the person to their right was truthful. Based on these statements, the traveler was able to uniquely determine what fraction of the village's population consisted of liars. Determine what this fraction is.
Let $x$ be the fraction of liars. Imagine that all truthful residents became liars, and all liars "reformed." Then the traveler would hear the same thing! Indeed, the truthfulness of any resident has changed, but the truthfulness of the neighbor they are talking about has also changed. But the fraction of truthful people in this circle is $1-x$. Thus, the traveler cannot distinguish a circle with a fraction of liars $x$ from a circle with a fraction of liars $1-x$. Therefore, he could determine the fraction of liars only when $x=1-x$. But this means that $x=1 / 2$.

## Answer

$50 \%$.