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2024 IMC
1
1
Part of
2024 IMC
Problems
(1)
Equation with complex numbers on the unit circle
Source: IMC 2024, Problem 1
8/7/2024
Determine all pairs
(
a
,
b
)
∈
C
×
C
(a,b) \in \mathbb{C} \times \mathbb{C}
(
a
,
b
)
∈
C
×
C
of complex numbers satisfying
∣
a
∣
=
∣
b
∣
=
1
|a|=|b|=1
∣
a
∣
=
∣
b
∣
=
1
and
a
+
b
+
a
b
‾
∈
R
a+b+a\overline{b} \in \mathbb{R}
a
+
b
+
a
b
∈
R
.
complex numbers