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2018 Polish MO Finals
6
Polish MO Finals 2018, Problem 6
Polish MO Finals 2018, Problem 6
Source:
April 19, 2018
number theory
Poland
TST
combinatorics
Problem Statement
A prime
p
>
3
p>3
p
>
3
is given. Let
K
K
K
be the number of such permutations
(
a
1
,
a
2
,
…
,
a
p
)
(a_1, a_2, \ldots, a_p)
(
a
1
,
a
2
,
…
,
a
p
)
of
{
1
,
2
,
…
,
p
}
\{ 1, 2, \ldots, p\}
{
1
,
2
,
…
,
p
}
such that
a
1
a
2
+
a
2
a
3
+
…
+
a
p
−
1
a
p
+
a
p
a
1
a_1a_2+a_2a_3+\ldots + a_{p-1}a_p+a_pa_1
a
1
a
2
+
a
2
a
3
+
…
+
a
p
−
1
a
p
+
a
p
a
1
is divisible by
p
p
p
. Prove
K
+
p
K+p
K
+
p
is divisible by
p
2
p^2
p
2
.
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