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2010 BAMO
5
an inequality
an inequality
Source: 2010 BAMO 12 problem 5
March 8, 2016
Inequality
cyclic quadrilateral
algebra
BAMO
inequalities
Problem Statement
Let
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
be positive real numbers such that
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
. Prove that
1
/
[
(
1
/
2
+
a
+
a
b
+
a
b
c
)
1
/
2
]
+
1
/
[
(
1
/
2
+
b
+
b
c
+
b
c
d
)
1
/
2
]
+
1
/
[
(
1
/
2
+
c
+
c
d
+
c
d
a
)
1
/
2
]
+
1
/
[
1
(
1
/
2
+
d
+
d
a
+
d
a
b
)
1
/
2
]
1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]
1/
[(
1/2
+
a
+
ab
+
ab
c
)
1/2
]
+
1/
[(
1/2
+
b
+
b
c
+
b
c
d
)
1/2
]
+
1/
[(
1/2
+
c
+
c
d
+
c
d
a
)
1/2
]
+
1/
[
1
(
1/2
+
d
+
d
a
+
d
ab
)
1/2
]
is greater than or equal to
2
1
/
2
2^{1/2}
2
1/2
.
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