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National and Regional Contests
Iran Contests
Iran Team Selection Test
2013 Iran Team Selection Test
11
Inequality about sides of a triangle
Inequality about sides of a triangle
Source: Iran TST 2013:TST 2,Day 2,Problem 2
April 25, 2013
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be sides of a triangle such that
a
≥
b
≥
c
a\geq b \geq c
a
≥
b
≥
c
. prove that:
a
(
a
+
b
−
a
b
)
+
b
(
a
+
c
−
a
c
)
+
c
(
b
+
c
−
b
c
)
≥
a
+
b
+
c
\sqrt{a(a+b-\sqrt{ab})}+\sqrt{b(a+c-\sqrt{ac})}+\sqrt{c(b+c-\sqrt{bc})}\geq a+b+c
a
(
a
+
b
−
ab
)
+
b
(
a
+
c
−
a
c
)
+
c
(
b
+
c
−
b
c
)
≥
a
+
b
+
c
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