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National and Regional Contests
China Contests
China Northern MO
2006 China Northern MO
2
Sequence and Divisibility
Sequence and Divisibility
Source: China North MO
August 14, 2006
induction
number theory unsolved
number theory
Problem Statement
p
p
p
is a prime number that is greater than
2
2
2
. Let
{
a
n
}
\{ a_{n}\}
{
a
n
}
be a sequence such that
n
a
n
+
1
=
(
n
+
1
)
a
n
−
(
p
2
)
4
na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}
n
a
n
+
1
=
(
n
+
1
)
a
n
−
(
2
p
)
4
. Show that if
a
1
=
5
a_{1}=5
a
1
=
5
, the
16
∣
a
81
16 \mid a_{81}
16
∣
a
81
.
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