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Balkan MO Shortlist
2018 Balkan MO Shortlist
A4
Simple inequality
Simple inequality
Source: Shortlist BMO 2018, A4
May 3, 2019
inequalities
Balkan
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
1.
abc = 1.
ab
c
=
1.
Prove that:
2
(
a
2
+
b
2
+
c
2
)
(
1
a
2
+
1
b
2
+
1
c
2
)
≥
3
(
a
+
b
+
c
+
a
b
+
b
c
+
c
a
)
.
2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).
2
(
a
2
+
b
2
+
c
2
)
(
a
2
1
+
b
2
1
+
c
2
1
)
≥
3
(
a
+
b
+
c
+
ab
+
b
c
+
c
a
)
.
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