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3
Geometric Inequality - [UKRMO 2009 Grade 11]
Geometric Inequality - [UKRMO 2009 Grade 11]
Source:
January 23, 2011
inequalities
trigonometry
function
algebra
domain
geometry unsolved
geometry
Problem Statement
Point
O
O
O
is inside triangle
A
B
C
ABC
A
BC
such that
∠
A
O
B
=
∠
B
O
C
=
∠
C
O
A
=
12
0
∘
.
\angle AOB = \angle BOC = \angle COA = 120^\circ .
∠
A
OB
=
∠
BOC
=
∠
CO
A
=
12
0
∘
.
Prove that
A
O
2
B
C
+
B
O
2
C
A
+
C
O
2
A
B
≥
A
O
+
B
O
+
C
O
3
.
\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.
BC
A
O
2
+
C
A
B
O
2
+
A
B
C
O
2
≥
3
A
O
+
BO
+
CO
.
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