2021 AMC 10 B Fall
Complete problem set with solutions and individual problem pages
Regular polygons with , and sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?(2021 AMC Fall 10B, Question #21)
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Imagine we have regular polygons with and sides and inscribed in a circle without sharing a vertex. We see that each side of the polygon with sides (the polygon with fewer sides) will be intersected twice. (We can see this because to have a vertex of the mgon on an arc subtended by a side of the n-gon, there will be one intersection to "enter" the arc and one to "exit" the arc. KingRavi)
This means that we will end up with times the number of sides in the polygon with fewer sides.
If we have polygons with , and sides, we need to consider each possible pair of polygons and count their intersections. Throughout 6 of these pairs, the -sided polygon has the least number of sides times, the 6 sided polygon has the least number of sides times, and the -sided polygon has the least number of sides 1 time. Therefore the number of intersections is .
