Northeastern WOOTers Mock AIME I Problems/Problem 10
Problem 10
If are complex numbers such that
then find the value of
.
Solution
The 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
cis
and
cis
. By methods such as De Moivre's Theorem, we determine the condition is true if and only if
Since this relationship supposedly fixes
, we can set
without loss of generality.
From here, we determine and
. Then we can compute