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2017 Tournament Of Towns
3
3
Part of
2017 Tournament Of Towns
Problems
(1)
Inequalities on sums of powers
Source: Tournament of Towns Spring 2017 Junior A-level
1/30/2018
From given positive numbers, the following infinite sequence is defined:
a
1
a_1
a
1
is the sum of all original numbers,
a
2
a_2
a
2
is the sum of the squares of all original numbers,
a
3
a_3
a
3
is the sum of the cubes of all original numbers, and so on (
a
k
a_k
a
k
is the sum of the
k
k
k
-th powers of all original numbers). a) Can it happen that
a
1
>
a
2
>
a
3
>
a
4
>
a
5
a_1 > a_2 > a_3 > a_4 > a_5
a
1
>
a
2
>
a
3
>
a
4
>
a
5
and
a
5
<
a
6
<
a
7
<
…
a_5 < a_6 < a_7 < \ldots
a
5
<
a
6
<
a
7
<
…
? (4 points) b) Can it happen that
a
1
<
a
2
<
a
3
<
a
4
<
a
5
a_1 < a_2 < a_3 < a_4 < a_5
a
1
<
a
2
<
a
3
<
a
4
<
a
5
and
a
5
>
a
6
>
a
7
>
…
a_5 > a_6 > a_7 > \ldots
a
5
>
a
6
>
a
7
>
…
? (4 points)(Alexey Tolpygo)
inequalities
algebra