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Azerbaijan BMO TST
2016 Azerbaijan BMO TST
1
Balkan TSTp4.1
Balkan TSTp4.1
Source: Azerbaijan Balkan TST 2016 no 4
October 20, 2016
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be nonnegative real numbers.Prove that
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)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (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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