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Regional Mathematical Olympiad
2008 India Regional Mathematical Olympiad
3
An Inequality
An Inequality
Source: Excursion
June 25, 2020
inequalities
Inequality
positive integers
real number
Problem Statement
Prove that for every positive integer
n
n
n
and a non-negative real number
a
a
a
, the following inequality holds:
n
(
n
+
1
)
a
+
2
n
⩾
4
a
(
1
+
2
+
⋯
+
n
)
.
n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).
n
(
n
+
1
)
a
+
2
n
⩾
4
a
(
1
+
2
+
⋯
+
n
)
.
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