2021 Fall AMC 10B Problems/Problem 21
Problem 21
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?
Solution 1
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.
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
-sided polygon has the least number of sides
times, and the
-sided polygon has the least number of sides
time.
Therefore the number of intersections is .
~kingofpineapplz