Question 143
All subscripts in this problem are to be considered modulo $6$, that means for example that $\omega_{7}$ is the same as $\omega_{1}$. Let $\omega_{1}, \ldots, \omega_{6}$ be circles of radius $r$, whose centers lie on a regular hexagon of side length $1$. Let $P_{i}$ be the intersection of $\omega_{i}$ and $\omega_{i+1}$ that lies further from the center of the hexagon, for $i=1, \ldots, 6$. Let $Q_{i}$, $i=1 \ldots 6$, lie on $\omega_{i}$ such that $Q_{i}, P_{i}, Q_{i+1}$ are colinear. Find the number of possible values of $r$.