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2003 APMO
4
In ineqaulity with sides of triangle
In ineqaulity with sides of triangle
Source: APMO 2003
March 5, 2006
inequalities
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the sides of a triangle, with
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
, and let
n
≥
2
n\ge 2
n
≥
2
be an integer. Show that
a
n
+
b
n
n
+
b
n
+
c
n
n
+
c
n
+
a
n
n
<
1
+
2
n
2
.
\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<1+\frac{\sqrt[n]{2}}{2}.
n
a
n
+
b
n
+
n
b
n
+
c
n
+
n
c
n
+
a
n
<
1
+
2
n
2
.
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