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Austrian-Polish
2006 Austrian-Polish Competition
5
Classical
Classical
Source: APMC 2006, Problem 5
September 9, 2006
inequalities
inequalities proposed
Problem Statement
Prove that for all positive integers
n
n
n
and all positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
the following inequality holds:
a
n
+
1
a
n
+
a
n
−
1
b
+
…
+
b
n
+
b
n
+
1
b
n
+
b
n
−
1
c
+
…
+
c
n
+
c
n
+
1
c
n
+
c
n
−
1
a
+
…
+
a
n
≥
a
+
b
+
c
n
+
1
\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}}\\ \ge \frac{a+b+c}{n+1}
a
n
+
a
n
−
1
b
+
…
+
b
n
a
n
+
1
+
b
n
+
b
n
−
1
c
+
…
+
c
n
b
n
+
1
+
c
n
+
c
n
−
1
a
+
…
+
a
n
c
n
+
1
≥
n
+
1
a
+
b
+
c
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