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National and Regional Contests
Cuba Contests
Cuba MO
2012 Cuba MO
8
8
Part of
2012 Cuba MO
Problems
(1)
(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2, (a^2 + d^2)(b^2 + c^2) = (ad + bc)^2
Source: 2012 Cuba MO 2.8
9/18/2024
If the natural numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
verify the relationships:
(
a
2
+
b
2
)
(
c
2
+
d
2
)
=
(
a
b
+
c
d
)
2
(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2
(
a
2
+
b
2
)
(
c
2
+
d
2
)
=
(
ab
+
c
d
)
2
(
a
2
+
d
2
)
(
b
2
+
c
2
)
=
(
a
d
+
b
c
)
2
(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2
(
a
2
+
d
2
)
(
b
2
+
c
2
)
=
(
a
d
+
b
c
)
2
and the
g
c
d
(
a
,
b
,
c
,
d
)
=
1
gcd(a, b, c, d) = 1
g
c
d
(
a
,
b
,
c
,
d
)
=
1
, prove that
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
is a perfect square.
number theory
Perfect Square