2013 AMC 10A Problem 25

Question

All

20

diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?

49

65

70

96

128

Answer: A

Concepts:counting intersections regular polygon combinations

Difficulty rating: 2180

Small hint:

Start with

\binom84

intersections from four vertices

Big hint:

Correct for the center and the eight other triple-concurrence points

Solution:

If no three diagonals were concurrent, each choice of

4

vertices would determine one interior intersection, giving

\binom84=70

.

The

4

long diagonals through opposite vertices all meet at the center, so the center was counted

\binom42=6

times and should be counted once. Subtract

5

.

There are also

8

symmetric points where

3

diagonals meet. Each was counted

\binom32=3

times and should be counted once, so subtract

8(3-1)=16

.

The number of distinct interior intersection points is

70-5-16=49

.

Thus, A is the correct answer.