MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand TST Selection Test
2019 Thailand TSTST
2
Two inequality mixed together
Two inequality mixed together
Source: 2018 Thailand October Camp 2.2
February 19, 2022
inequalities
Problem Statement
Let
a
,
b
,
c
∈
(
0
,
4
3
)
a,b,c\in(0,\frac{4}{3})
a
,
b
,
c
∈
(
0
,
3
4
)
and
a
+
b
+
c
=
3
a + b + c = 3
a
+
b
+
c
=
3
. Prove that
4
a
b
c
(
a
+
b
)
(
a
+
c
)
+
(
a
+
b
)
2
+
(
a
+
c
)
2
(
a
+
b
)
+
(
a
+
c
)
≤
∑
c
y
c
1
a
2
(
3
b
+
3
c
−
5
)
.
\frac{4abc}{(a+b)(a+c)}+\frac{(a+b)^2+(a+c)^2}{(a+b)+(a+c)}\leq\sum_{cyc}\frac{1}{a^2(3b+3c-5)}.
(
a
+
b
)
(
a
+
c
)
4
ab
c
+
(
a
+
b
)
+
(
a
+
c
)
(
a
+
b
)
2
+
(
a
+
c
)
2
≤
cyc
∑
a
2
(
3
b
+
3
c
−
5
)
1
.
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