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Vietnam National Olympiad
2015 Vietnam National Olympiad
2
Viet Nam National Math Olympiad 2015
Viet Nam National Math Olympiad 2015
Source:
January 8, 2015
inequalities
induction
calculus
derivative
inequalities proposed
Problem Statement
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are nonnegative real numbers, then
3
(
a
2
+
b
2
+
c
2
)
≥
(
a
+
b
+
c
)
(
a
b
+
b
c
+
c
a
)
+
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
(
a
+
b
+
c
)
2
.
{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}
3
(
a
2
+
b
2
+
c
2
)
≥
(
a
+
b
+
c
)
(
ab
+
b
c
+
c
a
)
+
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
(
a
+
b
+
c
)
2
.
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