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Japan MO Finals
2024 Japan MO Finals
1
Magical Set
Magical Set
Source: 2024 Japan MO P1
February 11, 2024
algebra
Problem Statement
Let
n
≥
2
n\geq 2
n
≥
2
be an integer. Determine all sets of real numbers
(
a
1
,
a
2
,
…
,
a
n
)
(a_1, a_2,\dots, a_n)
(
a
1
,
a
2
,
…
,
a
n
)
such that
a
1
−
2
a
2
,
a
2
−
2
a
3
,
…
,
a
n
−
1
−
2
a
n
,
a
n
−
2
a
1
a_1-2a_2, a_2-2a_3,\dots, a_{n-1}-2a_n, a_n-2a_1
a
1
−
2
a
2
,
a
2
−
2
a
3
,
…
,
a
n
−
1
−
2
a
n
,
a
n
−
2
a
1
are rearrangements of
a
1
,
a
2
,
…
,
a
n
a_1, a_2,\dots, a_n
a
1
,
a
2
,
…
,
a
n
.
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