2021 Fall AMC 12B Problem 19

Problem 19 of 25HarderGeometry Counting & Probability

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?

Show solution

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.

Answer choices

Solution

More practice