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National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2019 Iran MO (3rd Round)
1
Inequality
Inequality
Source: Iran MO 3rd round 2019 mid-terms - Algebra P1
August 1, 2019
inequalities
Iran
Problem Statement
a
,
b
a,b
a
,
b
and
c
c
c
are positive real numbers so that
∑
cyc
(
a
+
b
)
2
=
2
∑
cyc
a
+
6
a
b
c
\sum_{\text{cyc}} (a+b)^2=2\sum_{\text{cyc}} a +6abc
∑
cyc
(
a
+
b
)
2
=
2
∑
cyc
a
+
6
ab
c
. Prove that
∑
cyc
(
a
−
b
)
2
≤
∣
2
∑
cyc
a
−
6
a
b
c
∣
.
\sum_{\text{cyc}} (a-b)^2\leq\left|2\sum_{\text{cyc}} a -6abc\right|.
cyc
∑
(
a
−
b
)
2
≤
2
cyc
∑
a
−
6
ab
c
.
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