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Thailand Contests
Thailand National Olympiad
2019 Thailand Mathematical Olympiad
5
5
Part of
2019 Thailand Mathematical Olympiad
Problems
(1)
Asymmetric Inequality with abc=1
Source: 2019 Thailand Mathematical Olympiad P5
5/22/2019
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals such that
a
b
c
=
1
abc=1
ab
c
=
1
. Prove the inequality
4
a
−
1
(
2
b
+
1
)
2
+
4
b
−
1
(
2
c
+
1
)
2
+
4
c
−
1
(
2
a
+
1
)
2
⩾
1.
\frac{4a-1}{(2b+1)^2} + \frac{4b-1}{(2c+1)^2} + \frac{4c-1}{(2a+1)^2}\geqslant 1.
(
2
b
+
1
)
2
4
a
−
1
+
(
2
c
+
1
)
2
4
b
−
1
+
(
2
a
+
1
)
2
4
c
−
1
⩾
1.
algebra
inequalities
2019