MathDB

11

Part of 1992 AIME Problems

Problems(1)

Lines and Transformations

Source:

2/17/2006
Lines l1l_1 and l2l_2 both pass through the origin and make first-quadrant angles of π70\frac{\pi}{70} and π54\frac{\pi}{54} radians, respectively, with the positive x-axis. For any line ll, the transformation R(l)R(l) produces another line as follows: ll is reflected in l1l_1, and the resulting line is reflected in l2l_2. Let R(1)(l)=R(l)R^{(1)}(l)=R(l) and R(n)(l)=R(R(n−1)(l))R^{(n)}(l)=R\left(R^{(n-1)}(l)\right). Given that ll is the line y=1992xy=\frac{19}{92}x, find the smallest positive integer mm for which R(m)(l)=lR^{(m)}(l)=l.
geometrygeometric transformationreflectionrotationmodular arithmetic