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Taiwan Mathematics Olympiad
2023 Taiwan Mathematics Olympiad
5
Numbers with given means of powers
Numbers with given means of powers
Source: 2023 Taiwan Mathematics Olympiad
February 8, 2023
Taiwan
Problem Statement
Let
m
m
m
be a positive integer, and real numbers
a
1
,
a
2
,
…
,
a
m
a_1, a_2,\ldots , a_m
a
1
,
a
2
,
…
,
a
m
satisfy
1
m
∑
i
=
1
m
a
i
=
1
,
\frac{1}{m}\sum_{i=1}^{m}a_i = 1,
m
1
i
=
1
∑
m
a
i
=
1
,
1
m
∑
i
=
1
m
a
i
2
=
11
,
\frac{1}{m}\sum_{i=1}^{m}a_i ^2= 11,
m
1
i
=
1
∑
m
a
i
2
=
11
,
1
m
∑
i
=
1
m
a
i
3
=
1
,
\frac{1}{m}\sum_{i=1}^{m}a_i ^3= 1,
m
1
i
=
1
∑
m
a
i
3
=
1
,
1
m
∑
i
=
1
m
a
i
4
=
131.
\frac{1}{m}\sum_{i=1}^{m}a_i ^4= 131.
m
1
i
=
1
∑
m
a
i
4
=
131.
Prove that
m
m
m
is a multiple of
7
7
7
.Proposed by usjl
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