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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova National Olympiad
2006 Moldova National Olympiad
12.5
maybe easy
maybe easy
Source:
November 5, 2011
inequalities
inequalities proposed
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_{1},a_{2},...,a_{n}
a
1
,
a
2
,
...
,
a
n
be real positive numbers and
k
>
m
,
k
,
m
k>m, k,m
k
>
m
,
k
,
m
natural numbers. Prove that
(
n
−
1
)
(
a
1
m
+
a
2
m
+
.
.
.
+
a
n
m
)
≤
a
2
k
+
a
3
k
+
.
.
.
+
a
n
k
a
1
k
−
m
+
a
1
k
+
a
3
k
+
.
.
.
+
a
n
k
a
2
k
−
m
+
.
.
.
+
a
1
k
+
a
2
k
+
.
.
.
+
a
n
−
1
k
a
n
k
−
m
(n-1)(a_{1}^m +a_{2}^m+...+a_{n}^m)\leq\frac{a_{2}^k+a_{3}^k+...+a_{n}^k}{a_{1}^{k-m}}+\frac{a_{1}^k+a_{3}^k+...+a_{n}^k}{a_2^{k-m}}+...+\frac{a_{1}^k+a_{2}^k+...+a_{n-1}^k}{a_{n}^{k-m}}
(
n
−
1
)
(
a
1
m
+
a
2
m
+
...
+
a
n
m
)
≤
a
1
k
−
m
a
2
k
+
a
3
k
+
...
+
a
n
k
+
a
2
k
−
m
a
1
k
+
a
3
k
+
...
+
a
n
k
+
...
+
a
n
k
−
m
a
1
k
+
a
2
k
+
...
+
a
n
−
1
k
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