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19
Purple Comet 2009 HS Problem 19
Purple Comet 2009 HS Problem 19
Source:
September 3, 2011
complex numbers
Problem Statement
If
a
a
a
and
b
b
b
are complex numbers such that
a
2
+
b
2
=
5
a^2 + b^2 = 5
a
2
+
b
2
=
5
and
a
3
+
b
3
=
7
a^3 + b^3 = 7
a
3
+
b
3
=
7
, then their sum,
a
+
b
a + b
a
+
b
, is real. The greatest possible value for the sum
a
+
b
a + b
a
+
b
is
m
+
n
2
\tfrac{m+\sqrt{n}}{2}
2
m
+
n
where
m
m
m
and
n
n
n
are integers. Find
n
.
n.
n
.
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