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National and Regional Contests
Greece Contests
Greece JBMO TST
1999 Greece JBMO TST
2
(a+b+c)/2-ab/(a+b)-bc/(b+c)-ca/(c+a)\ge 0, a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}
(a+b+c)/2-ab/(a+b)-bc/(b+c)-ca/(c+a)\ge 0, a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}
Source: Greece JBMO TST 1999
May 25, 2019
inequalities
three variable inequality
algebra
Problem Statement
For
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
, prove that (i)
a
+
b
+
c
2
−
a
b
a
+
b
−
b
c
b
+
c
−
c
a
c
+
a
≥
0
\frac{a+b+c}{2}-\frac{ab}{a+b}-\frac{bc}{b+c}-\frac{ca}{c+a}\ge 0
2
a
+
b
+
c
−
a
+
b
ab
−
b
+
c
b
c
−
c
+
a
c
a
≥
0
(ii)
a
(
1
+
b
)
+
b
(
1
+
c
)
+
c
(
1
+
a
)
≥
6
a
b
c
a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}
a
(
1
+
b
)
+
b
(
1
+
c
)
+
c
(
1
+
a
)
≥
6
ab
c
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