2022 AMC 12A Problem 18

Problem 18 of 25IntermediateGeometry

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,

\ldots,

T_n

returns the point

(1,0)

back to itself?

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Solution

Rotating a point at angle

\theta

k^\circ

gives

\theta+k,

and reflecting across the

y

-axis sends angle

\phi

180-\phi.

T_k

sends

\theta

(180-k)-\theta.

Starting from

(1,0)

at angle

0,

applying

T_1,T_2,\ldots

gives angles

179,-1,178,-2,177,\ldots.

After an even number

2m

of steps the angle is

-m,

and after an odd number

2m+1

it is

179-m.

For the point to return, the angle must be a multiple of

360^\circ.

The even case needs

m=360,

i.e.

n=720.

The odd case needs

179-m=0,

i.e.

m=179

and

n=359,

where the net reflection fixes

(1,0).

The least such

n

359.

Thus, the correct answer is A.