2021 AMC 12B Fall Problem 19

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

Answer: E

Concepts:regular polygon counting intersections

Difficulty rating: 2320

Small hint:

Consider each pair of polygons separately

Big hint:

Two inscribed polygons with

m

and

n

sides in general position cross at

2\min(m, n)

points

Solution:

For two convex polygons inscribed in the same circle with no shared vertices, each side of the smaller polygon crosses the larger polygon’s boundary exactly twice, so they meet at

2\min(m, n)

points.

Summing over all pairs:

(5,6), (5,7), (5,8)

give

10

each;

(6,7), (6,8)

give

12

each;

(7,8)

gives

14.

The total is

3 \cdot 10 + 2 \cdot 12 + 14 = 68.

Thus, the correct answer is E.