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Kosovo National Mathematical Olympiad
2011 Kosovo National Mathematical Olympiad
3
Kosovo Mathematical Olympiad, #3. (Grade 12)
Kosovo Mathematical Olympiad, #3. (Grade 12)
Source:
March 13, 2011
inequalities
inequalities proposed
Problem Statement
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are real positive numbers prove that the inequality holds:
a
3
+
b
3
a
2
+
b
2
+
b
3
+
c
3
b
2
+
c
2
+
c
3
+
a
3
c
2
+
a
2
≥
6
(
a
b
+
b
c
+
a
c
)
(
a
+
b
+
c
)
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
\frac{\sqrt{a^3+b^3}}{a^2+b^2}+\frac{\sqrt{b^3+c^3}}{b^2+c^2}+\frac{\sqrt{c^3+a^3}}{c^2+a^2} \ge \frac{6(ab+bc+ac)}{(a+b+c)\sqrt{(a+b)(b+c)(c+a)}}
a
2
+
b
2
a
3
+
b
3
+
b
2
+
c
2
b
3
+
c
3
+
c
2
+
a
2
c
3
+
a
3
≥
(
a
+
b
+
c
)
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
6
(
ab
+
b
c
+
a
c
)
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