Question

Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1\leq n_2.$ If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is:

$2n_1$

$2n_2$

$n_1n_2$

$n_1+n_2$

none of these

Answer: A

Concepts:counting intersections extremal argument

Difficulty rating: 2130

Small Hint:

A line segment can enter and leave a convex polygon at most once

Big Hint:

Apply that bound to each side of the polygon with fewer sides

Solution:

Each side of $P_1$ lies on a line, and its intersection with the convex region $P_2$ is a single segment or empty. Thus that side crosses the boundary of $P_2$ at most twice. Across $n_1$ sides there are at most $2n_1,$ and a suitable thin convex $n_1$ -gon crossing a convex $n_2$ -gon attains this bound.

Therefore, the correct answer is A.

1968 AMC 12 Problem 30

Answer: A

Difficulty rating: 2130

Small Hint:

Big Hint:

Solution:

Problem 30 in Other Years