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Romania Contests
Romania Team Selection Test
2002 Romania Team Selection Test
2
Nice inequality with frac 45
Nice inequality with frac 45
Source: Romanian selection test 2002
August 12, 2003
inequalities
algebra
romania
Problem Statement
Let
n
≥
4
n\geq 4
n
≥
4
be an integer, and let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers such that
a
1
2
+
a
2
2
+
⋯
+
a
n
2
=
1.
a_1^2+a_2^2+\cdots +a_n^2=1 .
a
1
2
+
a
2
2
+
⋯
+
a
n
2
=
1.
Prove that the following inequality takes place
a
1
a
2
2
+
1
+
⋯
+
a
n
a
1
2
+
1
≥
4
5
(
a
1
a
1
+
⋯
+
a
n
a
n
)
2
.
\frac{a_1}{a_2^2+1}+\cdots +\frac{a_n}{a_1^2+1} \geq \frac{4}{5}\left( a_1 \sqrt{a_1}+\cdots +a_n \sqrt{a_n} \right)^2 .
a
2
2
+
1
a
1
+
⋯
+
a
1
2
+
1
a
n
≥
5
4
(
a
1
a
1
+
⋯
+
a
n
a
n
)
2
.
Bogdan Enescu, Mircea Becheanu
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