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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 17
Prove this inequality on two functions
Prove this inequality on two functions
Source: 2009 Jozsef Wildt International Mathematical Competition
April 26, 2020
function
inequalities
Problem Statement
If
a
a
a
,
b
b
b
,
c
>
0
c>0
c
>
0
and
a
b
c
=
1
abc=1
ab
c
=
1
,
α
=
m
a
x
{
a
,
b
,
c
}
\alpha = max\{a,b,c\}
α
=
ma
x
{
a
,
b
,
c
}
;
f
,
g
:
(
0
,
+
∞
)
→
R
f,g : (0, +\infty )\to \mathbb{R}
f
,
g
:
(
0
,
+
∞
)
→
R
, where
f
(
x
)
=
2
(
x
+
1
)
2
x
f(x)=\frac{2(x+1)^2}{x}
f
(
x
)
=
x
2
(
x
+
1
)
2
and
g
(
x
)
=
(
x
+
1
)
(
1
x
+
1
)
2
g(x)= (x+1)\left (\frac{1}{\sqrt{x}}+1\right )^2
g
(
x
)
=
(
x
+
1
)
(
x
1
+
1
)
2
, then
(
a
+
1
)
(
b
+
1
)
(
c
+
1
)
≥
m
i
n
{
{
f
(
x
)
,
g
(
x
)
}
∣
x
∈
{
a
,
b
,
c
}
\
{
α
}
}
(a+1)(b+1)(c+1)\geq min\{ \{f(x),g(x) \}\ |\ x\in\{a,b,c\} \backslash \{ \alpha \}\}
(
a
+
1
)
(
b
+
1
)
(
c
+
1
)
≥
min
{{
f
(
x
)
,
g
(
x
)}
∣
x
∈
{
a
,
b
,
c
}
\
{
α
}}
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