Question

Regular polygons with $5,$ $6,$ $7,$ and $8{ }$ 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?

$52$

$56$

$60$

$64$

$68$

Solution:

For a regular $k$ -gon and a regular $n$ -gon inscribed in the same circle with $k\lt n$ and no shared vertices, their boundaries intersect in $2k$ points. Each side of the smaller polygon is crossed twice by the boundary of the larger polygon.

Therefore, sum over all pairs of polygons. The $5$ -gon contributes $2\cdot5$ intersections with each of the $6$ -, $7$ -, and $8$ -gons. The $6$ -gon contributes $2\cdot6$ intersections with each of the $7$ - and $8$ -gons. The $7$ -gon contributes $2\cdot7$ with the $8$ -gon.

The total is $3(10)+2(12)+14=68.$

Thus, the answer is E .

2021 AMC 10B Fall Problem 21

Solution:

Problem 21 in Other Years