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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2022 China Second Round A2
4
4
Part of
2022 China Second Round A2
Problems
(1)
NT about sequence again
Source: 2022 China Second Round A2
12/22/2022
k
>
2
k>2
k
>
2
is an integer.
a
0
,
a
1
,
.
.
.
a_0,a_1,...
a
0
,
a
1
,
...
is an integer sequence such that
a
0
=
0
a_0=0
a
0
=
0
,
a
n
+
1
=
k
a
n
−
a
n
−
1
a_{n+1}=ka_n-a_{n-1}
a
n
+
1
=
k
a
n
−
a
n
−
1
. Prove that for any positive integer
m
m
m
,
(
2
m
)
!
∣
a
1
a
2
.
.
.
a
3
m
(2m)!|a_1a_2...a_{3m}
(
2
m
)!
∣
a
1
a
2
...
a
3
m
.
number theory
Sequence and Series