1991 AMC 12 第 12 题

Question

一个凸六边形的内角度数构成一个由正整数组成的等差数列。设其最大内角的度数为

m^\circ

。

m^\circ

的最大可能值为

The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers. Let

m^\circ

be the measure of the largest interior angle of the hexagon. The largest possible value of

m^\circ

is

165^\circ

167^\circ

170^\circ

175^\circ

179^\circ

把各角写成

a,a+d,\ldots,a+5d

，并利用它们的和

Write the angles as

a,a+d,\ldots,a+5d

and use their sum

所得方程迫使

d

为偶数，而凸性要求最大角小于

180^\circ

The equation forces

d

to be even, and convexity bounds the largest angle below

180^\circ

内角和为

720^\circ

，所以

\begin{aligned} 6a+15d&=720,\\ 2a+5d&=240 \end{aligned}\text{。}

因此

d

为偶数。最大角为

a+5d=120+\frac52d

，且必须小于

180

。所以

d\lt24

，允许的最大偶数为

d=22

。此时

a=65

，且

m=65+5(22)=175

。

因此正确答案为 D。

The angle sum is

720^\circ,

so

\begin{aligned} 6a+15d&=720,\\ 2a+5d&=240. \end{aligned}

Hence

d

is even. The largest angle is

a+5d=120+\frac52d,

which must be less than

180.

Thus

d\lt24,

and the largest allowable even value is

d=22.

It gives

a=65

and

m=65+5(22)=175.

Thus the correct answer is D.

其他年份的第 12 题