1966 AMC 12 Problem 21

Question

An “

n

-pointed star” is formed as follows: the sides of a convex polygon are numbered consecutively

1,

2,

\ldots,

k,

\ldots,

n,

n\geq5;

for all

n

values of

k,

sides

k

and

k+2

are non-parallel, sides

n+1

and

n+2

being respectively identical with sides

1

and

2;

prolong the

n

pairs of sides numbered

k

and

k+2

until they meet. (A figure is shown for the case

n=5.

)

Let

S

be the degree-sum of the interior angles at the

n

points of the star; then

S

equals:

180

360

180(n+2)

180(n-2)

180(n-4)

Relate each star-tip angle to two adjacent interior angles of the original polygon

The original polygon’s interior-angle sum is

180(n-2)

degrees

Let the original polygon’s interior angles be

a_1,\ldots,a_n.

The star angle made from sides

k

and

k+2

equals

a_k+a_{k+1}-180^\circ.

Summing cyclically counts each

a_k

twice. Hence

\begin{aligned} S&=2\sum_{k=1}^n a_k-180n\\ &=2\cdot180(n-2)-180n\\ &=180(n-4). \end{aligned}

Therefore, the correct answer is E.

Problem 21 in other years