Question

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 his progress, 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?

$(-1030,-994)$

$(-1030,-990)$

$(-1026,-994)$

$(-1026,-990)$

$(-1022,-994)$

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 $(-20,20)+505(-2,-2)=(-1030,-990).$

Thus, B is the correct answer.

2020 AMC 10B Problem 13

Solution:

Problem 13 in Other Years