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Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia TST
2012 Indonesia TST
1
Complex Numbers Inequality
Complex Numbers Inequality
Source: 2012 Indonesia Round 2 TST 1 Problem 1
February 26, 2012
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
,
c
∈
C
a,b,c \in \mathbb{C}
a
,
b
,
c
∈
C
such that
a
∣
b
c
∣
+
b
∣
c
a
∣
+
c
∣
a
b
∣
=
0
a|bc| + b|ca| + c|ab| = 0
a
∣
b
c
∣
+
b
∣
c
a
∣
+
c
∣
ab
∣
=
0
. Prove that
∣
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
∣
≥
3
3
∣
a
b
c
∣
|(a-b)(b-c)(c-a)| \ge 3\sqrt{3}|abc|
∣
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
∣
≥
3
3
∣
ab
c
∣
.
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