MathDB

Denote by

Re(z)

and

Im(z)

the real part and imaginary part, respectively, of a complex number

z

; that is, if

z = a + bi

, then

Re(z) = a

and

Im(z) = b

. Suppose that there exists some real number

k

such that

Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right)

for some complex number

w

with

||w||=\frac{\sqrt3}{2}

Re(w) > 0

, and

Im(w) \ne 0

. If

k

can be expressed as

\frac{\sqrt{a}-b}{c}

for integers

a

b

c

with

a

squarefree, find

a + b + c

Problem Statement