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Purple Comet Problems
2018 Purple Comet Problems
14
Purple Comet 2018 HS problem 14
Purple Comet 2018 HS problem 14
Source:
March 19, 2020
complex numbers
Problem Statement
A complex number
z
z
z
whose real and imaginary parts are integers satis fies
(
R
e
(
z
)
)
4
+
(
R
e
(
z
2
)
)
2
+
∣
z
∣
4
=
(
2018
)
(
81
)
\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)
(
R
e
(
z
)
)
4
+
(
R
e
(
z
2
)
)
2
+
∣
z
∣
4
=
(
2018
)
(
81
)
, where
R
e
(
w
)
Re(w)
R
e
(
w
)
and
I
m
(
w
)
Im(w)
I
m
(
w
)
are the real and imaginary parts of
w
w
w
, respectively. Find
(
I
m
(
z
)
)
2
\left(Im(z) \right)^2
(
I
m
(
z
)
)
2
.
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