Problem:
Let and be complex numbers such that and . Let . The maximum possible value of can be written as , where and are relatively prime positive integers. Find . (Note that , for , denotes the measure of the angle that the ray from to makes with the positive real axis in the complex plane.)
Solution:
Consider a geometric interpretation. Point is on the unit circle centered at the origin . Point is on the circle with radius centered at the origin. Because dividing two complex numbers subtracts their complex arguments, is the angle between and the vector from to , that is, . The tangent of this angle is clearly maximized when is maximized which occurs when is tangent to the unit circle at . When this happens, is a right triangle with right angle . Then the Pythagorean Theorem shows . It follows that . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions