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International Contests
Baltic Way
2021 Baltic Way
20
20
Part of
2021 Baltic Way
Problems
(1)
divisibility with condition on lcm
Source: Baltic Way 2021, Problem 20
11/15/2021
Let
n
≥
2
n\ge 2
n
≥
2
be an integer. Given numbers
a
1
,
a
2
,
…
,
a
n
∈
{
1
,
2
,
3
,
…
,
2
n
}
a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}
a
1
,
a
2
,
…
,
a
n
∈
{
1
,
2
,
3
,
…
,
2
n
}
such that
lcm
(
a
i
,
a
j
)
>
2
n
\operatorname{lcm}(a_i,a_j)>2n
lcm
(
a
i
,
a
j
)
>
2
n
for all
1
≤
i
<
j
≤
n
1\le i<j\le n
1
≤
i
<
j
≤
n
, prove that
a
1
a
2
…
a
n
∣
(
n
+
1
)
(
n
+
2
)
…
(
2
n
−
1
)
(
2
n
)
.
a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).
a
1
a
2
…
a
n
∣
(
n
+
1
)
(
n
+
2
)
…
(
2
n
−
1
)
(
2
n
)
.
number theory
number theory proposed
Divisibility
least common multiple