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Vietnam Contests
Vietnam National Olympiad
2011 Vietnam National Olympiad
1
Prove the inequality for positive real x - [VMO 2011]
Prove the inequality for positive real x - [VMO 2011]
Source:
January 11, 2011
inequalities
inequalities proposed
Problem Statement
Prove that if
x
>
0
x>0
x
>
0
and
n
∈
N
,
n\in\mathbb N,
n
∈
N
,
then we have
x
n
(
x
n
+
1
+
1
)
x
n
+
1
≤
(
x
+
1
2
)
2
n
+
1
.
\frac{x^n(x^{n+1}+1)}{x^n+1}\leq\left(\frac {x+1}{2}\right)^{2n+1}.
x
n
+
1
x
n
(
x
n
+
1
+
1
)
≤
(
2
x
+
1
)
2
n
+
1
.
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