1980 AMC 12 Problem 13

Question

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)

Represent a

90^\circ

counterclockwise turn by multiplication by

i

The displacement vectors form an infinite geometric series with ratio

\frac{i}{2}

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.

Problem 13 in other years