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National and Regional Contests
China Contests
(China) National High School Mathematics League
1999 China Second Round Olympiad
2
Find the norm of complex number
Find the norm of complex number
Source:
October 31, 2010
trigonometry
complex numbers
algebra unsolved
algebra
Problem Statement
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers. Let
z
1
z_{1}
z
1
,
z
2
z_{2}
z
2
,
z
3
z_{3}
z
3
be complex numbers such that
∣
z
k
∣
=
1
|z_{k}|=1
∣
z
k
∣
=
1
(
k
=
1
,
2
,
3
)
(k=1,2,3)
(
k
=
1
,
2
,
3
)
~
and
~
z
1
z
2
+
z
2
z
3
+
z
3
z
1
=
1
\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1
z
2
z
1
+
z
3
z
2
+
z
1
z
3
=
1
Find
∣
a
z
1
+
b
z
2
+
c
z
3
∣
|az_{1}+bz_{2}+cz_{3}|
∣
a
z
1
+
b
z
2
+
c
z
3
∣
.
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