1999 AMC 12 第 7 题

Question

一个凸六边形最多可以有多少个锐角？

What is the largest number of acute angles that a convex hexagon can have?

2

3

4

5

6

凸多边形的外角和为

360^\circ

。

The exterior angles of a convex polygon sum to

360^\circ

一个锐内角对应一个大于

90^\circ

的外角。

An acute interior angle gives an exterior angle greater than

90^\circ

每个锐内角对应一个大于

90^\circ

的外角。因为凸多边形的外角和为

360^\circ

，其中最多有三个外角能大于

90^\circ

。这个上界可以达到：取一个等边六边形，其外角交替为

100^\circ

和

20^\circ

。它的边方向分成两组三条，每组内相差

120^\circ

，所以图形闭合；内角交替为

80^\circ

与

160^\circ

。因此最大可能数为三。

所以正确答案是 B。

Each acute interior angle corresponds to an exterior angle greater than

90^\circ.

Since the exterior angles of a convex polygon sum to

360^\circ,

at most three of them can exceed

90^\circ.

This bound is attainable: take a hexagon with equal side lengths and exterior angles alternating

100^\circ

and

20^\circ.

Its edge directions split into two triples

120^\circ

apart, so it closes, and its interior angles alternate between

80^\circ

and

160^\circ.

Hence the largest possible number is three.

Thus, the correct answer is B.

其他年份的第 7 题