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Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2013 JBMO TST - Macedonia
3
ineq
ineq
Source: Macedonian JBMO TST 2013
May 26, 2013
inequalities
inequalities proposed
Problem Statement
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
and
a
b
c
=
1
abc=1
ab
c
=
1
. Prove that
1
2
(
a
+
b
+
c
)
+
1
1
+
a
+
1
1
+
b
+
1
1
+
c
≥
3
\frac{1}{2}\ (\sqrt{a}\ +\sqrt{b}\ + \sqrt{c}\ ) +\frac{1}{1+a}\ + \frac{1}{1+b}\ + \frac{1}{1+c}\ge\ 3
2
1
(
a
+
b
+
c
)
+
1
+
a
1
+
1
+
b
1
+
1
+
c
1
≥
3
. ( The official problem is with
a
b
c
=
1
abc = 1
ab
c
=
1
but it can be proved without using it. )
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