Question

Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive $x$ -axis. For any line $l,$ the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1,$ and the resulting line is reflected in $l_2.$ Let $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R(R^{(n-1)}(l)).$ Given that $l$ is the line $y=(\frac{19}{92})x,$ find the smallest positive integer $m$ for which $R^{(m)}(l)=l.$

Answer

Answer: 945

Concepts:reflection (geometry)transformation modular arithmetic

Difficulty rating: 2230

Small Hint:

Two reflections in intersecting lines compose to a rotation through twice the angle between the lines

Big Hint:

An unoriented line returns to itself when its accumulated rotation is a multiple of $\pi$

Solution:

The composition is rotation through $2\left(\frac{\pi}{54}-\frac{\pi}{70}\right)=\frac{8\pi}{945}.$ A line through the origin is unchanged by a rotation exactly when the rotation angle is a multiple of $\pi.$ Thus $m$ must satisfy $\frac{8m}{945}\in\mathbb Z.$ Since $\gcd(8,945)=1,$ the least such $m$ is $945.$

1992 AIME Problem 11

Answer: 945

Difficulty rating: 2230

Small Hint:

Big Hint:

Solution:

Problem 11 in Other Years