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National and Regional Contests
Thailand Contests
Thailand TST Selection Test
2017 Thailand TSTST
5
Another Weird Inequality
Another Weird Inequality
Source: 2016 Thailand October Camp 3.5
February 28, 2022
inequalities
Problem Statement
Let
a
,
b
,
c
∈
R
+
a, b, c \in \mathbb{R}^+
a
,
b
,
c
∈
R
+
such that
a
+
b
+
c
=
3
a + b + c = 3
a
+
b
+
c
=
3
. Prove that
∑
c
y
c
(
a
3
+
1
a
2
+
1
)
≥
1
27
(
a
b
+
b
c
+
c
a
)
4
.
\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.
cyc
∑
(
a
2
+
1
a
3
+
1
)
≥
27
1
(
ab
+
b
c
+
c
a
)
4
.
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