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Iran MO (3rd Round)
2017 Iran MO (3rd round)
2
Inequality
Inequality
Source: Iran 3rd round 2017 first Algebra exam
August 7, 2017
inequalities
mixing
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
and
d
d
d
be positive real numbers such that
a
2
+
b
2
+
c
2
+
d
2
≥
4
a^2+b^2+c^2+d^2 \ge 4
a
2
+
b
2
+
c
2
+
d
2
≥
4
. Prove that
(
a
+
b
)
3
+
(
c
+
d
)
3
+
2
(
a
2
+
b
2
+
c
2
+
d
2
)
≥
4
(
a
b
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
)
(a+b)^3+(c+d)^3+2(a^2+b^2+c^2+d^2) \ge 4(ab+bc+cd+da+ac+bd)
(
a
+
b
)
3
+
(
c
+
d
)
3
+
2
(
a
2
+
b
2
+
c
2
+
d
2
)
≥
4
(
ab
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
)
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