Question

蚂蚁 Andy 生活在坐标平面上，目前位于 $(-20, 20)$ ，面朝东方，也就是 $x$ 轴正方向。Andy 先移动 $1$ 个单位，然后向左转 $90^{\circ}$ 。接着他向北移动 $2$ 个单位，再向左转 $90^{\circ}$ 。然后他向西移动 $3$ 个单位，再向左转 $90^{\circ}$ 。Andy 继续这样移动，每次移动距离增加 $1$ 个单位，并且总是左转。他第 $2020\text{次}$ 左转所在点的坐标是什么？

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?

$(-1030,-994)$

$(-1030,-990)$

$(-1026,-994)$

$(-1026,-990)$

$(-1022,-994)$

答案：B

知识点：坐标几何找规律

难度评级：1480

小提示：

追踪每四次移动的净位移。

Track the net displacement over four moves

大提示：

$2020$ 次移动正好是 $505$ 个完整的四步循环。

There are exactly $505$ complete four-move cycles in $2020$ moves

解答：

前四步依次为向东 $1$ 、向北 $2$ 、向西 $3$ 、向南 $4$ 。净位移为 $(1-3,2-4)=(-2,-2)\text{，}$ 而且他重新面向东方。

因为 $2020=4\cdot 505$ ，Andy 共完成 $505$ 个这样的循环。从 $(-20,20)$ 出发，最终位置为 $\begin{aligned} &(-20,20)+505(-2,-2) \\ &\quad =(-1030,-990) \end{aligned}\text{。}$

所以正确答案是 B。

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.

2020 AMC 10B 第 13 题

答案：B

难度评级：1480

小提示：

大提示：

解答：

其他年份的第 13 题