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Balkan MO Shortlist
2020 Balkan MO Shortlist
A2
Inequality
Inequality
Source: Balkan MO SL 2020 A2
September 9, 2021
inequalities
algebra
Balkan
Problem Statement
Given are positive reals
a
,
b
,
c
a, b, c
a
,
b
,
c
, such that
1
a
+
1
b
+
1
c
=
3
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3
a
1
+
b
1
+
c
1
=
3
. Prove that
a
+
b
c
+
b
+
c
a
+
c
+
a
b
3
≤
a
+
b
+
c
−
1
2
\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}
3
a
+
c
b
+
b
+
a
c
+
c
+
b
a
≤
2
a
+
b
+
c
−
1
.Albania
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