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.

2021 AMC 12B Fall Problem 19

Answer: E

Difficulty rating: 2320

Small Hint:

Big Hint:

Solution:

Problem 19 in Other Years