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China National Olympiad
2017 China National Olympiad
1
China Mathematical Olympiad 2017 Q1
China Mathematical Olympiad 2017 Q1
Source: China Changsha ，Dec 23， 2016
November 23, 2016
algebra
number theory
Sequence
Problem Statement
The sequences
{
u
n
}
\{u_{n}\}
{
u
n
}
and
{
v
n
}
\{v_{n}\}
{
v
n
}
are defined by
u
0
=
u
1
=
1
u_{0} =u_{1} =1
u
0
=
u
1
=
1
,
u
n
=
2
u
n
−
1
−
3
u
n
−
2
u_{n}=2u_{n-1}-3u_{n-2}
u
n
=
2
u
n
−
1
−
3
u
n
−
2
(
n
≥
2
)
(n\geq2)
(
n
≥
2
)
,
v
0
=
a
,
v
1
=
b
,
v
2
=
c
v_{0} =a, v_{1} =b , v_{2}=c
v
0
=
a
,
v
1
=
b
,
v
2
=
c
,
v
n
=
v
n
−
1
−
3
v
n
−
2
+
27
v
n
−
3
v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}
v
n
=
v
n
−
1
−
3
v
n
−
2
+
27
v
n
−
3
(
n
≥
3
)
(n\geq3)
(
n
≥
3
)
. There exists a positive integer
N
N
N
such that when
n
>
N
n> N
n
>
N
, we have
u
n
∣
v
n
u_{n}\mid v_{n}
u
n
∣
v
n
. Prove that
3
a
=
2
b
+
c
3a=2b+c
3
a
=
2
b
+
c
.
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