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South East Mathematical Olympiad
2022 South East Mathematical Olympiad
7
the construction of the sequence
the construction of the sequence
Source: 2022 China Southeast Grade 11 P7
August 3, 2022
number theory
Problem Statement
Prove that for any positive real number
λ
\lambda
λ
,there are
n
n
n
positive numbers
a
1
,
a
2
,
⋯
,
a
n
(
n
≥
2
)
a_1,a_2,\cdots,a_n(n\geq 2)
a
1
,
a
2
,
⋯
,
a
n
(
n
≥
2
)
,so that
a
1
<
a
2
<
⋯
<
a
n
<
2
n
λ
a_1<a_2<\cdots<a_n<2^n\lambda
a
1
<
a
2
<
⋯
<
a
n
<
2
n
λ
and for any
k
=
1
,
2
,
⋯
,
n
k=1,2,\cdots,n
k
=
1
,
2
,
⋯
,
n
we have
gcd
(
a
1
,
a
k
)
+
gcd
(
a
2
,
a
k
)
+
⋯
+
gcd
(
a
n
,
a
k
)
≡
0
(
m
o
d
a
k
)
\gcd(a_1,a_k)+\gcd(a_2,a_k)+\cdots+\gcd(a_n,a_k)\equiv 0\pmod{a_k}
g
cd
(
a
1
,
a
k
)
+
g
cd
(
a
2
,
a
k
)
+
⋯
+
g
cd
(
a
n
,
a
k
)
≡
0
(
mod
a
k
)
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