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Kvant 2022
M2687
M2687
Part of
Kvant 2022
Problems
(1)
Labeling the vertices of an n-gon
Source: Kvant Magazine No. 2 2022 M2687
3/8/2023
We have a regular
n
n{}
n
-gon, with
n
⩾
4
n\geqslant 4
n
⩾
4
. We consider the arrangements of
n
n{}
n
numbers on its vertices, each of which is equal to 1 or 2. For each such arrangement
K
K{}
K
, we find the number of odd sums among all sums of numbers in several consecutive vertices. This number is denoted by
α
(
K
)
\alpha(K)
α
(
K
)
.[*]Find the largest possible value of
α
(
K
)
\alpha(K)
α
(
K
)
. [*]Find the number of arrangements for which
α
(
K
)
\alpha(K)
α
(
K
)
takes this largest possible value.Proposed by P. Kozhevnikov
combinatorics
Kvant