MathDB

2014 Team #5: Magnitude of Difference of Magnitudes Constant

Source:

March 2, 2014

absolute value

Problem Statement

Prove that there exists a nonzero complex number

c

and a real number

d

such that

\left|\left|\dfrac1{1+z+z^2}\right|-\left|\dfrac1{1+z+z^2}-c\right|\right|=d

for all

z

with

|z|=1

and

1+z+z^2\neq 0

. (Here,

|z|

denotes the absolute value of the complex number

z

, so that

|a+bi|=\sqrt{a^2+b^2}

for real numbers

a,b