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Turkey MO (2nd round)
2018 Turkey MO (2nd Round)
5
5
Part of
2018 Turkey MO (2nd Round)
Problems
(1)
Four Numbers with Divisibility Property
Source: Turkey National Mathematical Olympiad 2018
12/2/2018
Let
a
1
,
a
2
,
a
3
,
a
4
a_1,a_2,a_3,a_4
a
1
,
a
2
,
a
3
,
a
4
be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let
i
,
j
,
k
∈
{
1
,
2
,
3
,
4
}
i,j,k\in\{1,2,3,4\}
i
,
j
,
k
∈
{
1
,
2
,
3
,
4
}
with
i
≠
j
i \neq j
i
=
j
,
j
≠
k
j\neq k
j
=
k
, and
k
≠
i
k\neq i
k
=
i
. Determine the maximum number of triples
(
i
,
j
,
k
)
(i,j,k)
(
i
,
j
,
k
)
for which
(
g
c
d
(
a
i
,
a
j
)
)
2
∣
a
k
.
({\rm gcd}(a_i,a_j))^2|a_k.
(
gcd
(
a
i
,
a
j
)
)
2
∣
a
k
.
number theory