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National and Regional Contests
North Macedonia Contests
Macedonia National Olympiad
2013 Macedonia National Olympiad
4
Problem 4
Problem 4
Source: Macedonian National Olympiad 2013
April 6, 2013
inequalities proposed
inequalities
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
4
+
b
4
+
c
4
=
3
a^4+b^4+c^4=3
a
4
+
b
4
+
c
4
=
3
. Prove that
9
a
2
+
b
4
+
c
6
+
9
a
4
+
b
6
+
c
2
+
9
a
6
+
b
2
+
c
4
≤
a
6
+
b
6
+
c
6
+
6
\frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6
a
2
+
b
4
+
c
6
9
+
a
4
+
b
6
+
c
2
9
+
a
6
+
b
2
+
c
4
9
≤
a
6
+
b
6
+
c
6
+
6
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