1968 AMC 12 Problem 20

Attempt Problem 20 of the 1968 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1968 AMC 12 solutions, or check the answer key.

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20.

The measures of the interior angles of a convex polygon of

n

sides are in arithmetic progression. If the common difference is

5^\circ

and the largest angle is

160^\circ,

then

n

equals:

9

10

12

16

32

Write the smallest angle as

160-5(n-1)

Equate the arithmetic-series sum to

180(n-2)

The smallest angle is

160-5(n-1).

The angle sum is

\frac n2

times

320-5(n-1),

and it equals

180(n-2).

This simplifies to

\begin{aligned} n^2+7n-144&=0,\\ (n-9)(n+16)&=0. \end{aligned}

Thus

n=9.

Therefore, the correct answer is A.

Problem 20 in other years