1980 AMC 12 第 13 题

Question

一只可忽略其大小的小虫从坐标平面的原点出发。它先向右移动

1

个单位到达

(1,0)

。然后逆时针转

90^\circ

，再移动

\frac12

个单位到达

(1,\frac12)

。此后它继续以同样方式移动，每次逆时针转

90^\circ

，并且移动距离是前一次的一半。它最接近下列哪个点？

A bug (of negligible size) starts at the origin on the coordinate plane. First it moves

1

unit right to

(1,0).

Then it makes a

90^\circ

turn counterclockwise and travels

\frac12

a unit to

(1,\frac12).

If it continues in this fashion, each time making a

90^\circ

turn counterclockwise and traveling half as far as in the previous move, to which of the following points will it come closest?

(\frac23,\frac23)

(\frac45,\frac25)

(\frac23,\frac45)

(\frac23,\frac13)

(\frac25,\frac45)

用乘以

i

表示逆时针旋转

90^\circ

Represent a

90^\circ

counterclockwise turn by multiplication by

i

各段位移向量构成公比为

\frac{i}{2}

的无穷等比级数

The displacement vectors form an infinite geometric series with ratio

\frac{i}{2}

把位移向量表示为复数，依次为

1,\ \frac i2,\ \left(\frac i2\right)^2,\ldots\text{。}

它们的和是

\frac1{1-\frac{i}{2}} =\frac{1+\frac{i}{2}}{1+\frac{1}{4}} =\frac45+\frac25i\text{。}

因此小虫趋近于

(\frac{4}{5},\frac{2}{5})

。

因此，正确答案是 B。

As complex numbers, the successive displacement vectors are

1,\ \frac i2,\ \left(\frac i2\right)^2,\ldots.

Their sum is

\frac1{1-\frac{i}{2}} =\frac{1+\frac{i}{2}}{1+\frac{1}{4}} =\frac45+\frac25i.

Thus the bug approaches

(\frac{4}{5},\frac{2}{5}).

Therefore, the correct answer is B.

其他年份的第 13 题