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National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2006 Switzerland Team Selection Test
1
Swiss imo selection 06 inequality
Swiss imo selection 06 inequality
Source:
May 25, 2006
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
,
c
∈
R
+
a,b,c \in \mathbb{R^+}
a
,
b
,
c
∈
R
+
and
1
a
+
1
b
+
1
c
=
1
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1
a
1
+
b
1
+
c
1
=
1
. Show
a
b
+
c
+
b
c
+
a
+
c
a
+
b
≥
a
b
c
+
a
+
b
+
c
\sqrt{ab+c} + \sqrt{bc+a} + \sqrt{ca+b} \ge \sqrt{abc} + \sqrt{a} + \sqrt{b} + \sqrt{c}
ab
+
c
+
b
c
+
a
+
c
a
+
b
≥
ab
c
+
a
+
b
+
c
. :D
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