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National and Regional Contests
Poland Contests
Poland - Second Round
2006 Poland - Second Round
3
classical - symmetrical with 3 variables
classical - symmetrical with 3 variables
Source: Polish second round 2006
February 24, 2006
inequalities
inequalities proposed
Problem Statement
Positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
a
b
+
b
c
+
c
a
=
a
b
c
ab+bc+ca=abc
ab
+
b
c
+
c
a
=
ab
c
. Prove that:
a
4
+
b
4
a
b
(
a
3
+
b
3
)
+
b
4
+
c
4
b
c
(
b
3
+
c
3
)
+
c
4
+
a
4
c
a
(
c
3
+
a
3
)
≥
1
\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1
ab
(
a
3
+
b
3
)
a
4
+
b
4
+
b
c
(
b
3
+
c
3
)
b
4
+
c
4
+
c
a
(
c
3
+
a
3
)
c
4
+
a
4
≥
1
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