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Romania - Local Contests
Grigore Moisil Intercounty
1989 Grigore Moisil Intercounty
1989 Grigore Moisil Intercounty
Part of
Grigore Moisil Intercounty
Subcontests
(1)
1
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Complex Numbers
Let
n
≥
2
n \ge 2
n
≥
2
be an integer and
z
1
,
z
2
,
z
3
z_1,~z_2,~z_3
z
1
,
z
2
,
z
3
three complex numbers such that
∣
z
i
∣
=
1
,
|z_i|=1,
∣
z
i
∣
=
1
,
arg
z
i
∈
[
0
,
2
π
n
)
\arg z_i \in \left[0, \frac{2 \pi}{n} \right)
ar
g
z
i
∈
[
0
,
n
2
π
)
∀
i
∈
{
1
,
2
,
3
}
\forall i \in \{1,2,3\}
∀
i
∈
{
1
,
2
,
3
}
and
z
1
n
+
z
2
n
+
z
3
n
=
0.
z_1^n+z_2^n+z_3^n=0.
z
1
n
+
z
2
n
+
z
3
n
=
0.
Find
∣
z
1
+
z
2
+
z
3
∣
.
|z_1+z_2+z_3|.
∣
z
1
+
z
2
+
z
3
∣.
[hide=My solution] I've used the fact that
z
1
n
,
z
2
n
,
z
3
n
z_1^n,~z_2^n,~z_3^n
z
1
n
,
z
2
n
,
z
3
n
are the affixes of an equilateral triangle, and finally I've got
∣
z
1
+
z
2
+
z
3
∣
=
3
+
4
cos
2
π
3
n
+
2
cos
4
π
3
n
.
|z_1+z_2+z_3|= \sqrt{3+4 \cos \frac{2 \pi}{3n}+2 \cos \frac{4 \pi}{3n}}.
∣
z
1
+
z
2
+
z
3
∣
=
3
+
4
cos
3
n
2
π
+
2
cos
3
n
4
π
.