Northeastern WOOTers Mock AIME I Problems/Problem 10
If are complex numbers such that then find the value of .
Our strategy is to take advantage of degrees of freedom. The given condition appears extremely weak (that is, it offers little information), yet apparently it uniquely determines . Counterintuitively, this very fact offers lots of information.
Degree of Freedom 1: Translation
Observe that replacing , , with , , , respectively, has no effect on the condition. Then, by setting , we can set without loss of generality. Substituting this into the condition and clearing denominators yields Then , with ; this implies .
Degree of Freedom 2: Dilation
Observe that replacing , , , with , , , respectively, has no effect on the condition. Then, an appropriate can be chosen such that ; that is, without loss of generality, .
Degree of Freedom 3: Rotation
Let's take a closer look at the given condition. We have already changed it into , . Let and . By methods such as De Moivre's Theorem, we determine the condition is true if and only if Since this relationship is supposedly enough to fix , we can set without loss of generality.
From here, we determine and . Then we can compute