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Romanian National Olympiad 2018 - Grade 9 - problem 2
Romanian National Olympiad 2018 - Grade 9 - problem 2
Source: Romania NMO - 2018
April 12, 2018
inequalities
Problem Statement
Let
a
,
b
,
c
≥
0
a,b,c \geq 0
a
,
b
,
c
≥
0
and
a
+
b
+
c
=
3.
a+b+c=3.
a
+
b
+
c
=
3.
Prove that
a
1
+
b
+
b
1
+
c
+
c
1
+
a
≥
1
1
+
b
+
1
1
+
c
+
1
1
+
a
\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \geq \frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+a}
1
+
b
a
+
1
+
c
b
+
1
+
a
c
≥
1
+
b
1
+
1
+
c
1
+
1
+
a
1
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