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2001 Mongolian Mathematical Olympiad
Problem 2
inequality in triangle with medians
inequality in triangle with medians
Source: Mongolia MO 2001 Grade 10 P2
April 12, 2021
geometric inequality
Inequality
geometry
Problem Statement
In an acute-angled triangle
A
B
C
ABC
A
BC
,
a
,
b
,
c
a,b,c
a
,
b
,
c
are sides,
m
a
,
m
b
,
m
c
m_a,m_b,m_c
m
a
,
m
b
,
m
c
the corresponding medians,
R
R
R
the circumradius and
r
r
r
the inradius. Prove the inequality
a
2
+
b
2
a
+
b
⋅
b
2
+
c
2
b
+
c
⋅
a
2
+
c
2
a
+
c
≥
16
R
2
r
m
a
a
⋅
m
b
b
⋅
m
c
c
.
\frac{a^2+b^2}{a+b}\cdot\frac{b^2+c^2}{b+c}\cdot\frac{a^2+c^2}{a+c}\ge16R^2r\frac{m_a}a\cdot\frac{m_b}b\cdot\frac{m_c}c.
a
+
b
a
2
+
b
2
⋅
b
+
c
b
2
+
c
2
⋅
a
+
c
a
2
+
c
2
≥
16
R
2
r
a
m
a
⋅
b
m
b
⋅
c
m
c
.
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