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Spain Mathematical Olympiad
2023 Spain Mathematical Olympiad
3
Divisibility with sum of squares
Divisibility with sum of squares
Source: Spain MO 2023 P3
March 10, 2023
number theory
Problem Statement
Find all quadruples
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
of positive integers satisfying that
a
2
+
b
2
=
c
2
+
d
2
a^2+b^2=c^2+d^2
a
2
+
b
2
=
c
2
+
d
2
and such that
a
c
+
b
d
ac+bd
a
c
+
b
d
divides
a
2
+
b
2
a^2+b^2
a
2
+
b
2
.
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