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1983 IMO Longlists
53
Prove that a ∈ {0, 1, . . ., n} and z_k ∈ {1, i}
Prove that a ∈ {0, 1, . . ., n} and z_k ∈ {1, i}
Source:
October 7, 2010
trigonometry
complex numbers
algebra unsolved
algebra
Problem Statement
Let
a
∈
R
a \in \mathbb R
a
∈
R
and let
z
1
,
z
2
,
…
,
z
n
z_1, z_2, \ldots, z_n
z
1
,
z
2
,
…
,
z
n
be complex numbers of modulus
1
1
1
satisfying the relation
∑
k
=
1
n
z
k
3
=
4
(
a
+
(
a
−
n
)
i
)
−
3
∑
k
=
1
n
z
k
‾
\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}
k
=
1
∑
n
z
k
3
=
4
(
a
+
(
a
−
n
)
i
)
−
3
k
=
1
∑
n
z
k
Prove that
a
∈
{
0
,
1
,
…
,
n
}
a \in \{0, 1,\ldots, n \}
a
∈
{
0
,
1
,
…
,
n
}
and
z
k
∈
{
1
,
i
}
z_k \in \{1, i \}
z
k
∈
{
1
,
i
}
for all
k
.
k.
k
.
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