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239 Open Math Olympiad
2015 239 Open Mathematical Olympiad
6
Inequality with Wired Condition
Inequality with Wired Condition
Source: 239 2015 S P6
May 14, 2020
inequalities
Problem Statement
Positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
2
a
3
b
+
2
b
3
c
+
2
c
3
a
=
a
2
b
2
+
b
2
c
2
+
c
2
a
2
.
2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2.
2
a
3
b
+
2
b
3
c
+
2
c
3
a
=
a
2
b
2
+
b
2
c
2
+
c
2
a
2
.
Prove that
2
a
b
(
a
−
b
)
2
+
2
b
c
(
b
−
c
)
2
+
2
c
a
(
c
−
a
)
2
≥
(
a
b
+
b
c
+
c
a
)
2
.
2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2 \geq(ab+bc+ca)^2.
2
ab
(
a
−
b
)
2
+
2
b
c
(
b
−
c
)
2
+
2
c
a
(
c
−
a
)
2
≥
(
ab
+
b
c
+
c
a
)
2
.
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