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National and Regional Contests
Romania Contests
Romania - Local Contests
Nicolae Coculescu
2007 Nicolae Coculescu
3
Another cyclic logarithmic inequality
Another cyclic logarithmic inequality
Source:
December 13, 2019
inequalities
Problem Statement
Show that for any three numbers
a
,
b
,
c
∈
(
1
,
∞
)
,
a,b,c\in (1,\infty ) ,
a
,
b
,
c
∈
(
1
,
∞
)
,
the following inequality is true:
log
a
b
c
+
log
b
c
a
+
log
c
a
b
≥
l
o
g
a
2
b
c
b
c
+
l
o
g
b
2
c
a
c
a
+
l
o
g
c
2
a
b
a
b
\log_{ab} c +\log_{bc} a +\log_{ca} b\ge log_{a^2bc} bc +log_{b^2ca} ca +log_{c^2ab} ab
lo
g
ab
c
+
lo
g
b
c
a
+
lo
g
c
a
b
≥
l
o
g
a
2
b
c
b
c
+
l
o
g
b
2
c
a
c
a
+
l
o
g
c
2
ab
ab
Costel Anghel
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