Fall 2021 AMC 10 B

Answer (E): Consider a regular $m$-gon and a regular $n$-gon, with $m\le n$, inscribed in the same circle with no shared vertices. If $A$ and $B$ are adjacent vertices of the $m$-gon, then minor arc $AB$ contains at least one vertex of the $n$-gon. Thus side $AB$ intersects exactly two sides of the $n$-gon inside the circle, and the two polygons intersect in exactly $2m$ points. It follows that the pentagon intersects the hexagon, the heptagon (the regular polygon with $7$ sides), and the octagon in $10$ points each; the hexagon intersects the heptagon and the octagon in $12$ points each; and the heptagon intersects the octagon in $14$ points. The total number of points of intersection is $3\cdot 10+2\cdot 12+14=68$.