AMC10 2020 B

AMC10 2020 B · Q13

AMC10 2020 B · Q13. It mainly tests Sequences & recursion (algebra), Coordinate geometry.

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 2020th left turn?

蚂蚁 Andy 生活在一个坐标平面上，目前位于 $(-20, 20)$，面朝东（即正 $x$ 方向）。Andy 移动 1 个单位，然后左转 $90^\circ$。然后，他移动 2 个单位（向北）并左转 $90^\circ$。接着移动 3 个单位（向西）并再次左转 $90^\circ$。Andy 继续这个过程，每次移动距离增加 1 个单位，并且总是左转。Andy 进行第 2020 次左转时的位置是哪里？

(A) (-1030, -994) (-1030, -994)

(B) (-1030, -990) (-1030, -990)

(D) (-1026, -990) (-1026, -990)

(E) (-1022, -994) (-1022, -994)

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Note that Andy’s moves east and west cause changes in his $x$-coordinate, and moves north and south cause changes in his $y$-coordinate. Summing all the changes gives the $x$- and $y$-coordinates of Andy’s final location: \[ x=-20+(1-3)+(5-7)+\cdots+(2017-2019) \] \[ =-20+505(-2) \] \[ =-1030 \] and \[ y=20+(2-4)+(6-8)+\cdots+(2018-2020) \] \[ =20+505(-2) \] \[ =-990. \] The final location is $(-1030,-990)$.

答案（B）：注意，Andy 向东和向西的移动会改变他的 $x$ 坐标，向北和向南的移动会改变他的 $y$ 坐标。把所有变化量相加，就得到 Andy 最终位置的 $x$ 与 $y$ 坐标： \[ x=-20+(1-3)+(5-7)+\cdots+(2017-2019) \] \[ =-20+505(-2) \] \[ =-1030 \] 以及 \[ y=20+(2-4)+(6-8)+\cdots+(2018-2020) \] \[ =20+505(-2) \] \[ =-990. \] 最终位置是 $(-1030,-990)$。

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