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Balkan MO Shortlist
2020 Balkan MO Shortlist
N5
N5
Part of
2020 Balkan MO Shortlist
Problems
(1)
divisible by p^n-1 but not p^n
Source: Balkan MO Shortlist N5
9/14/2021
Consider an integer
n
≥
2
n\geq 2
n
≥
2
and an odd prime
p
p
p
. Let
U
U
U
be the set of all positive integers
(
(
(
strictly
)
)
)
less than
p
n
p^n
p
n
that are not divisible by
p
p
p
, and let
N
N
N
be the number of elements of
U
U
U
. Does there exist permutation
a
1
,
a
2
,
⋯
a
N
a_1,a_2,\cdots a_N
a
1
,
a
2
,
⋯
a
N
of the numbers in
U
U
U
such that the sum
∑
k
=
1
N
a
k
a
k
+
1
\sum_{k=1}^N a_ka_{k+1}
∑
k
=
1
N
a
k
a
k
+
1
,where
a
N
+
1
=
a
1
a_{N+1}=a_1
a
N
+
1
=
a
1
, be divisible by
p
n
−
1
p^{n-1}
p
n
−
1
but not by
p
n
p^n
p
n
?
A
l
e
x
a
n
d
e
r
I
v
a
n
o
v
B
u
l
g
a
r
i
a
Alexander \ Ivanov \, Bulgaria
A
l
e
x
an
d
er
I
v
an
o
v
B
u
l
g
a
r
ia
number theory