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Nordic
2019 Nordic
1
1
Part of
2019 Nordic
Problems
(1)
meaningful
Source: Nordic 2019, P1
4/7/2019
A set of different positive integers is called meaningful if for any finite nonempty subset the corresponding arithmetic and geometric means are both integers.
a
)
a)
a
)
Does there exist a meaningful set which consists of
2019
2019
2019
numbers?
b
)
b)
b
)
Does there exist an infinite meaningful set? Note: The geometric mean of the non-negative numbers
a
1
,
a
2
,
⋯
,
a_1, a_2,\cdots,
a
1
,
a
2
,
⋯
,
a
n
a_n
a
n
is defined as
a
1
a
2
⋯
a
n
n
.
\sqrt[n]{a_1a_2\cdots a_n} .
n
a
1
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⋯
a
n
.
algebra
number theory