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Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2005 Switzerland - Final Round
3
3
Part of
2005 Switzerland - Final Round
Problems
(1)
sum ka_k <= {n \choose 2} + sum a_k^k
Source: Switzerland - 2005 Swiss MO Final Round p3
12/26/2022
Prove for all
a
1
,
.
.
.
,
a
n
>
0
a_1, ..., a_n > 0
a
1
,
...
,
a
n
>
0
the following inequality and determine all cases in where the equaloty holds:
∑
k
=
1
n
k
a
k
≤
(
n
2
)
+
∑
k
=
1
n
a
k
k
.
\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.
k
=
1
∑
n
k
a
k
≤
(
2
n
)
+
k
=
1
∑
n
a
k
k
.
inequalities
algebra