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Rioplatense Mathematical Olympiad, Level 3
2019 Rioplatense Mathematical Olympiad, Level 3
4
Infinite triples
Infinite triples
Source: Rioplatense Olympiad L3 2019
December 10, 2019
number theory
Problem Statement
Prove that there are infinite triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers
a
,
b
,
c
>
1
a,b,c>1
a
,
b
,
c
>
1
,
g
c
d
(
a
,
b
)
=
g
c
d
(
b
,
c
)
=
g
c
d
(
c
,
a
)
=
1
gcd(a,b)=gcd(b,c)=gcd(c,a)=1
g
c
d
(
a
,
b
)
=
g
c
d
(
b
,
c
)
=
g
c
d
(
c
,
a
)
=
1
such that
a
+
b
+
c
a+b+c
a
+
b
+
c
divides
a
b
+
b
c
+
c
a
a^b+b^c+c^a
a
b
+
b
c
+
c
a
.
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