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239 Open Math Olympiad
2012 239 Open Mathematical Olympiad
4
Three variable inequality with sum of one
Three variable inequality with sum of one
Source: 239 2012 J4
July 30, 2020
inequalities
Problem Statement
For positive real numbers
a
a
a
,
b
b
b
, and
c
c
c
with
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
, prove that:
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
1
−
27
a
b
c
2
.
(a-b)^2 + (b-c)^2 + (c-a)^2 \geq \frac{1-27abc}{2}.
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
2
1
−
27
ab
c
.
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