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?

$49$

$65$

$70$

$96$

$128$

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.

2013 AMC 10A Problem 25

Solution:

Problem 25 in Other Years