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National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2003 Turkey MO (2nd round)
2
Geometric inequality.
Geometric inequality.
Source: Turkey MO 2004.
February 27, 2005
geometry
inequalities
parallelogram
inequalities solved
Geometric Inequalities
Problem Statement
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and
K
,
L
,
M
,
N
K,L,M,N
K
,
L
,
M
,
N
be points on
[
A
B
]
,
[
B
C
]
,
[
C
D
]
,
[
D
A
]
[AB],[BC],[CD],[DA]
[
A
B
]
,
[
BC
]
,
[
C
D
]
,
[
D
A
]
, respectively. Show that,
s
1
3
+
s
2
3
+
s
3
3
+
s
4
3
≤
2
s
3
\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}
3
s
1
+
3
s
2
+
3
s
3
+
3
s
4
≤
2
3
s
where
s
1
=
Area
(
A
K
N
)
s_1=\text{Area}(AKN)
s
1
=
Area
(
A
K
N
)
,
s
2
=
Area
(
B
K
L
)
s_2=\text{Area}(BKL)
s
2
=
Area
(
B
K
L
)
,
s
3
=
Area
(
C
L
M
)
s_3=\text{Area}(CLM)
s
3
=
Area
(
C
L
M
)
,
s
4
=
Area
(
D
M
N
)
s_4=\text{Area}(DMN)
s
4
=
Area
(
D
MN
)
and
s
=
Area
(
A
B
C
D
)
s=\text{Area}(ABCD)
s
=
Area
(
A
BC
D
)
.
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