2020 AMC 10B Problem 13

Problem 13 of 25IntermediateGeometry

Andy the Ant lives on a coordinate plane and is currently at

(-20, 20)

facing east (that is, in the positive

x

-direction). Andy moves

1

unit and then turns

90^{\circ}

left. From there, Andy moves

2

units (north) and then turns

90^{\circ}

left. He then moves

3

units (west) and again turns

90^{\circ}

left. Andy continues this process, increasing his distance each time by

1

unit and always turning left. What is the location of the point at which Andy makes the

2020\text{th}

left turn?

Show solution

Solution

In the first four moves, Andy goes

1

east,

2

north,

3

west, and

4

south. The net change is

(1-3,2-4)=(-2,-2),

and he is again facing east. Since

2020=4\cdot 505,

Andy completes

505

such cycles. Starting from

(-20,20),

his final position is

\begin{aligned} &(-20,20)+505(-2,-2) \\ &\quad =(-1030,-990). \end{aligned}

Thus, B is the correct answer.