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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2020 Vietnam National Olympiad
3
VMO 2020 -P3
VMO 2020 -P3
Source: VMO-2020-Day1-P3
December 27, 2019
Sequence
number theory
Problem Statement
Let a sequence
(
a
n
)
(a_n)
(
a
n
)
satisfy:
a
1
=
5
,
a
2
=
13
a_1=5,a_2=13
a
1
=
5
,
a
2
=
13
and
a
n
+
1
=
5
a
n
−
6
a
n
−
1
,
∀
n
≥
2
a_{n+1}=5a_n-6a_{n-1},\forall n\ge2
a
n
+
1
=
5
a
n
−
6
a
n
−
1
,
∀
n
≥
2
a) Prove that
(
a
n
,
a
n
+
1
)
=
1
,
∀
n
≥
1
(a_n, a_{n+1})=1,\forall n\ge1
(
a
n
,
a
n
+
1
)
=
1
,
∀
n
≥
1
b) Prove that:
2
k
+
1
∣
p
−
1
∀
k
∈
N
2^{k+1}|p-1\forall k\in\mathbb{N}
2
k
+
1
∣
p
−
1∀
k
∈
N
, if p is a prime factor of
a
2
k
a_{2^k}
a
2
k
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