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Brazil EGMO TST
2023 Brazil EGMO Team Selection Test
3
Just a simple square root inequality
Just a simple square root inequality
Source: Brazil EGMO TST1 2023 #3
January 29, 2024
inequalities
Jensen
square roots
square root inequality
Problem Statement
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
+
a
2
+
⋯
+
a
n
=
1
a_1 + a_2 + \cdots + a_n = 1
a
1
+
a
2
+
⋯
+
a
n
=
1
. Prove that
a
1
1
−
a
1
+
⋯
+
a
n
1
−
a
n
≥
1
n
−
1
(
a
1
+
⋯
+
a
n
)
.
\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).
1
−
a
1
a
1
+
⋯
+
1
−
a
n
a
n
≥
n
−
1
1
(
a
1
+
⋯
+
a
n
)
.
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