1992 AIME Problems/Problem 11
Lines and both pass through the origin and make first-quadrant angles of and radians, respectively, with the positive x-axis. For any line , the transformation produces another line as follows: is reflected in , and the resulting line is reflected in . Let and . Given that is the line , find the smallest positive integer for which .
Let be a line that makes an angle of with the positive -axis. Let be the reflection of in , and let be the reflection of in .
The angle between and is , so the angle between and must also be . Thus, makes an angle of with the positive -axis.
Similarly, since the angle between and is , the angle between and the positive -axis is .
Thus, makes an angle with the positive -axis. So makes an angle with the positive -axis.
Therefore, iff is an integral multiple of . Thus, . Since , , so the smallest positive integer is .
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