Problem:
Lines ℓ1​ and ℓ2​ both pass through the origin and make first-quadrant angles of 70π​ and 54π​ radians, respectively, with the positive x-axis. For any line ℓ, the transformation R(ℓ) produces another line as follows: ℓ is reflected in ℓ1​, and the resulting line is then reflected in ℓ2​. Let R(1)(ℓ)=R(ℓ), and for integer n≥2 define R(n)(ℓ)= R(R(n−1)(ℓ)). Given that ℓ is the line y=9219​x, find the smallest positive integer m for which R(m)(ℓ)=ℓ.
Solution:
Let λ0​ and λ be lines through the origin making angles of θ0​ and θ, respectively, with the positive x-axis. When λ is reflected in λ0​, the resulting line λ′ makes an angle of
θ0​+(θ0​−θ)=2θ0​−θ
with the positive x-axis. Thus, if λ is reflected in ℓ1​, then the result is a line λ1​ that passes through the origin and makes an angle of 270π​−θ with the positive x-axis. Reflecting λ1​ in the line ℓ2​ gives a line λ2​ through the origin that makes an angle of
254π​−(270π​−θ)=−9458π​+θ
with the positive x-axis. Thus R(λ) is obtained by rotating λ through −9458π​ radians and R(m)(λ) is obtained by rotating λ through −8mπ/945 radians. For R(m)(λ)=λ to hold, 9458m​ must be an integer. The smallest positive integer value of m for which this is true is 945​.
The problems on this page are the property of the MAA's American Mathematics Competitions
