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Austrian MO National Competition
2021 Austrian MO National Competition
1
Simple inequality
Simple inequality
Source: Austria Mathematical Olympiad 2021？
May 31, 2021
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
≥
0
a,b,c\geq 0
a
,
b
,
c
≥
0
and
a
+
b
+
c
=
1.
a+b+c=1.
a
+
b
+
c
=
1.
Prove that
a
2
a
+
1
+
b
3
b
+
1
+
c
6
c
+
1
≤
1
2
.
\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.
2
a
+
1
a
+
3
b
+
1
b
+
6
c
+
1
c
≤
2
1
.
(Marian Dinca)
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