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Korea Junior Mathematics Olympiad
2012 Korea Junior Math Olympiad
1
An easy symmetric inequality
An easy symmetric inequality
Source: kjmo 2012 pr 1
September 21, 2017
Inequality
algebra
Problem Statement
Prove the following inequality where positive reals
a
a
a
,
b
b
b
,
c
c
c
satisfies
a
b
+
b
c
+
c
a
=
1
ab+bc+ca=1
ab
+
b
c
+
c
a
=
1
.
a
+
b
a
b
(
1
−
a
b
)
+
b
+
c
b
c
(
1
−
b
c
)
+
c
+
a
c
a
(
1
−
c
a
)
≤
2
a
b
c
\frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc}
ab
(
1
−
ab
)
a
+
b
+
b
c
(
1
−
b
c
)
b
+
c
+
c
a
(
1
−
c
a
)
c
+
a
≤
ab
c
2
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