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Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2022 Philippine MO
7
Fourth power inequality
Fourth power inequality
Source: Philippine MO 2022/7
March 19, 2022
inequalities
algebra
Problem Statement
Let
a
,
b
,
a, b,
a
,
b
,
and
c
c
c
be positive real numbers such that
a
b
+
b
c
+
c
a
=
3
ab + bc + ca = 3
ab
+
b
c
+
c
a
=
3
. Show that
b
c
1
+
a
4
+
c
a
1
+
b
4
+
a
b
1
+
c
4
≥
3
2
.
\dfrac{bc}{1 + a^4} + \dfrac{ca}{1 + b^4} + \dfrac{ab}{1 + c^4} \geq \dfrac{3}{2}.
1
+
a
4
b
c
+
1
+
b
4
c
a
+
1
+
c
4
ab
≥
2
3
.
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