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Balkan MO Shortlist
2014 Balkan MO Shortlist
A4
A4
Part of
2014 Balkan MO Shortlist
Problems
(1)
BMO 2014 SL A4
Source: Balkan MO 2014 Shorltist
10/1/2016
A
4
\boxed{A4}
A
4
Let
m
1
,
m
2
,
m
3
,
n
1
,
n
2
m_1,m_2,m_3,n_1,n_2
m
1
,
m
2
,
m
3
,
n
1
,
n
2
and
n
3
n_3
n
3
be positive real numbers such that
(
m
1
−
n
1
)
(
m
2
−
n
2
)
(
m
3
−
n
3
)
=
m
1
m
2
m
3
−
n
1
n
2
n
3
(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3
(
m
1
−
n
1
)
(
m
2
−
n
2
)
(
m
3
−
n
3
)
=
m
1
m
2
m
3
−
n
1
n
2
n
3
Prove that
(
m
1
+
n
1
)
(
m
2
+
n
2
)
(
m
3
+
n
3
)
≥
8
m
1
m
2
m
3
(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3
(
m
1
+
n
1
)
(
m
2
+
n
2
)
(
m
3
+
n
3
)
≥
8
m
1
m
2
m
3
inequalities
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