MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 1
2017 Taiwan TST Round 1
2
Inequality from 2017 Taiwan TST
Inequality from 2017 Taiwan TST
Source: 2017 Taiwan TST Round 1
April 13, 2018
inequalities
Problem Statement
Given
a
,
b
,
c
,
d
>
0
a,b,c,d>0
a
,
b
,
c
,
d
>
0
, prove that:
∑
c
y
c
c
a
+
2
b
+
∑
c
y
c
a
+
2
b
c
≥
8
(
(
a
+
b
+
c
+
d
)
2
a
b
+
a
c
+
a
d
+
b
c
+
b
d
+
c
d
−
1
)
,
\sum_{cyc}\frac{c}{a+2b}+\sum_{cyc}\frac{a+2b}{c}\geq 8(\frac{(a+b+c+d)^2}{ab+ac+ad+bc+bd+cd}-1),
cyc
∑
a
+
2
b
c
+
cyc
∑
c
a
+
2
b
≥
8
(
ab
+
a
c
+
a
d
+
b
c
+
b
d
+
c
d
(
a
+
b
+
c
+
d
)
2
−
1
)
,
where
∑
c
y
c
f
(
a
,
b
,
c
,
d
)
=
f
(
a
,
b
,
c
,
d
)
+
f
(
d
,
a
,
b
,
c
)
+
f
(
c
,
d
,
a
,
b
)
+
f
(
b
,
c
,
d
,
a
)
\sum_{cyc}f(a,b,c,d)=f(a,b,c,d)+f(d,a,b,c)+f(c,d,a,b)+f(b,c,d,a)
∑
cyc
f
(
a
,
b
,
c
,
d
)
=
f
(
a
,
b
,
c
,
d
)
+
f
(
d
,
a
,
b
,
c
)
+
f
(
c
,
d
,
a
,
b
)
+
f
(
b
,
c
,
d
,
a
)
.
Back to Problems
View on AoPS