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5
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2022 PUMaC Team #5
Source:
September 9, 2023
algebra
Problem Statement
You’re given the complex number
ω
=
e
2
i
π
/
13
+
e
10
i
π
/
13
+
e
16
i
π
/
13
+
e
24
i
π
/
13
\omega = e^{2i\pi/13} + e^{10i\pi/13} + e^{16i\pi/13} + e^{24i\pi/13}
ω
=
e
2
iπ
/13
+
e
10
iπ
/13
+
e
16
iπ
/13
+
e
24
iπ
/13
, and told it’s a root of a unique monic cubic
x
3
+
a
x
2
+
b
x
+
c
x^3 +ax^2 +bx+c
x
3
+
a
x
2
+
b
x
+
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are integers. Determine the value of
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
.
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