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VMEO = Vietnam Mathematical E-Olympiad
VMEO IV 2015
11.3
VMEO IV p3/11
VMEO IV p3/11
Source: VMEO IV
January 10, 2016
number theory
abstract algebra
Problem Statement
Find all positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying
(
a
,
b
)
=
(
b
,
c
)
=
(
c
,
a
)
=
1
(a,b)=(b,c)=(c,a)=1
(
a
,
b
)
=
(
b
,
c
)
=
(
c
,
a
)
=
1
and
{
a
2
+
b
∣
b
2
+
c
b
2
+
c
∣
c
2
+
a
\begin{cases} a^2+b\mid b^2+c\\ b^2+c\mid c^2+a \end{cases}
{
a
2
+
b
∣
b
2
+
c
b
2
+
c
∣
c
2
+
a
and none of prime divisors of
a
2
+
b
a^2+b
a
2
+
b
are congruent to
1
1
1
modulo
7
7
7
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