1968 AMC 12 第 20 题

Question

一个

n

边凸多边形的内角度数成等差数列。若公差为

5^\circ

，最大角为

160^\circ

，则

n

等于：

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

将最小角写成

160-5(n-1)

Write the smallest angle as

160-5(n-1)

令等差数列的和等于

180(n-2)

Equate the arithmetic-series sum to

180(n-2)

最小角为

160-5(n-1)

。内角和为

\frac n2

乘以

320-5(n-1)

，并且等于

180(n-2)

。化简得

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

因此

n=9

。

因此，正确答案是 A。

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.

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