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2022 Indonesia MO
5
5
Part of
2022 Indonesia MO
Problems
(1)
How is it not prime relatives???
Source: Indonesian MO 2022/5
10/5/2022
Let
N
≥
2
N\ge2
N
≥
2
be a positive integer. Given a sequence of natural numbers
a
1
,
a
2
,
a
3
,
…
,
a
N
+
1
a_1,a_2,a_3,\dots,a_{N+1}
a
1
,
a
2
,
a
3
,
…
,
a
N
+
1
such that for every integer
1
≤
i
≤
j
≤
N
+
1
1\le i\le j\le N+1
1
≤
i
≤
j
≤
N
+
1
,
a
i
a
i
+
1
a
i
+
2
…
a
j
≢
1
m
o
d
N
a_ia_{i+1}a_{i+2}\dots a_j \not\equiv1\mod{N}
a
i
a
i
+
1
a
i
+
2
…
a
j
≡
1
mod
N
Prove that there exist a positive integer
k
≤
N
+
1
k\le N+1
k
≤
N
+
1
such that
gcd
(
a
k
,
N
)
≠
1
\gcd(a_k, N) \neq 1
g
cd
(
a
k
,
N
)
=
1
number theory
INAMO 2022