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2014 Harvard-MIT Mathematics Tournament
32
32
Part of
2014 Harvard-MIT Mathematics Tournament
Problems
(1)
2014 Guts #32: A Classic System
Source:
8/26/2014
Find all ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of complex numbers with
a
2
+
b
2
≠
0
a^2+b^2\neq 0
a
2
+
b
2
=
0
,
a
+
10
b
a
2
+
b
2
=
5
a+\tfrac{10b}{a^2+b^2}=5
a
+
a
2
+
b
2
10
b
=
5
, and
b
+
10
a
a
2
+
b
2
=
4
b+\tfrac{10a}{a^2+b^2}=4
b
+
a
2
+
b
2
10
a
=
4
.
complex numbers