MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN A Problems
53
53
Part of
PEN A Problems
Problems
(1)
A 53
Source:
5/25/2007
Suppose that
x
,
y
,
x, y,
x
,
y
,
and
z
z
z
are positive integers with
x
y
=
z
2
+
1
xy=z^2 +1
x
y
=
z
2
+
1
. Prove that there exist integers
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
and
d
d
d
such that
x
=
a
2
+
b
2
x=a^2 +b^2
x
=
a
2
+
b
2
,
y
=
c
2
+
d
2
y=c^2 +d^2
y
=
c
2
+
d
2
, and
z
=
a
c
+
b
d
z=ac+bd
z
=
a
c
+
b
d
.
Divisibility Theory