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International Mathematical Arhimede Contest (IMAC)
2010 IMAC Arhimede
6
IMAC inequality
IMAC inequality
Source:
June 18, 2010
inequalities
inequalities unsolved
algebra
Problem Statement
Consider real numbers
a
,
b
,
c
≥
0
a, b ,c \ge0
a
,
b
,
c
≥
0
with
a
+
b
+
c
=
2
a+b+c=2
a
+
b
+
c
=
2
. Prove that:
b
c
3
a
2
+
4
4
+
c
a
3
b
2
+
4
4
+
a
b
3
c
2
+
4
4
≤
2
∗
3
4
3
\frac{bc}{\sqrt[4]{3a^2+4}}+\frac{ca}{\sqrt[4]{3b^2+4}}+\frac{ab}{\sqrt[4]{3c^2+4}} \le \frac{2*\sqrt[4] {3}}{3}
4
3
a
2
+
4
b
c
+
4
3
b
2
+
4
c
a
+
4
3
c
2
+
4
ab
≤
3
2
∗
4
3
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