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Balkan MO Shortlist
2015 Balkan MO Shortlist
A3
m_a (b/a-1)(c/a-1 )+m_b (a/b-1)(c/b-1)+m_c (a/c-1)(b/c-1 ) >= 0
m_a (b/a-1)(c/a-1 )+m_b (a/b-1)(c/b-1)+m_c (a/c-1)(b/c-1 ) >= 0
Source: Balkan BMO Shortlist 2015 A3
August 5, 2019
inequalities
geometric inequality
median
Problem Statement
Let a
,
b
,
c
,b,c
,
b
,
c
be sidelengths of a triangle and
m
a
,
m
b
,
m
c
m_a,m_b,m_c
m
a
,
m
b
,
m
c
the medians at the corresponding sides. Prove that
m
a
(
b
a
−
1
)
(
c
a
−
1
)
+
m
b
(
a
b
−
1
)
(
c
b
−
1
)
+
m
c
(
a
c
−
1
)
(
b
c
−
1
)
≥
0.
m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) +m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.
m
a
(
a
b
−
1
)
(
a
c
−
1
)
+
m
b
(
b
a
−
1
)
(
b
c
−
1
)
+
m
c
(
c
a
−
1
)
(
c
b
−
1
)
≥
0.
(FYROM)
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