2022 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 18 Easy

Let T_{k} be the transformation of the coordinate plane that first rotates the plane k degrees counterclockwise around the origin and then reflects the plane across the y-axis. What is the least positive integer n such that performing the sequence of transformations  T_{1},\ T_{2},\ T_{3}\ ,..,T_{n} returns the point (1,0) back to itself?

  • A.

    359

  • B.

    360

  • C.

    719

  • D.

    720

  • E.

    721

Answer:A

Let the angle between 'the point', 'the origin' and the positive x-axis be \alpha.

T_{x}: \alpha \rightarrow \alpha + x \rightarrow 180-(\alpha+x)

T_x+T_y: 180-(\alpha +x) \rightarrow 180-(\alpha +x)+y \rightarrow \alpha+x-y

Initially: 0^{\circ} after T_1, T_2 \dots T_n.

When n=2k, the angle:

(1-2)+(2-3)+\dots [(2k+1)-2k]=-k.

Back to the original position: k=360^{\circ}, n=720.

When n=2k+1, the angle:

(1-2)+(2-3)+\dots [(2k+1)-2k]+T_{2k+1}=-k+T_{2k+1}=180-(-k+2k+1)=179-k

When k=179, n=359.

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