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Caucasus Mathematical Olympiad
2018 Caucasus Mathematical Olympiad
8
Easy but nice inequality
Easy but nice inequality
Source: III Caucasus Mathematical Olympiad
March 17, 2018
inequalities
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the lengths of sides of a triangle. Prove the inequality
(
a
+
b
)
a
b
+
(
a
+
c
)
a
c
+
(
b
+
c
)
b
c
≥
(
a
+
b
+
c
)
2
/
2.
(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.
(
a
+
b
)
ab
+
(
a
+
c
)
a
c
+
(
b
+
c
)
b
c
≥
(
a
+
b
+
c
)
2
/2.
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