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Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2023 China National Olympiad
5
5
Part of
2023 China National Olympiad
Problems
(1)
China MO 2023 P5
Source: China MO 2023 P5
12/30/2022
Prove that there exist
C
>
0
C>0
C
>
0
, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence
a
1
,
a
2
,
a
3
,
⋯
a_1, a_2, a_3,\cdots
a
1
,
a
2
,
a
3
,
⋯
, if the greatest common divisor of
a
1
a_1
a
1
and
a
2
a_2
a
2
is squarefree, then there exists a positive integer
m
≤
C
⋅
a
2
2
m\le C\cdot {a_2}^2
m
≤
C
⋅
a
2
2
, such that
a
m
a_m
a
m
is squarefree. Note: A positive integer
N
N
N
is squarefree if it is not divisible by any square number greater than
1
1
1
.Proposed by Qu Zhenhua
number theory