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National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2007 Mongolian Mathematical Olympiad
Problem 4
syseq, sums of two squares
syseq, sums of two squares
Source: Mongolian MO 2007 Teachers P4
April 8, 2021
number theory
Problem Statement
If
x
,
y
,
z
ā
N
x,y,z\in\mathbb N
x
,
y
,
z
ā
N
and
x
y
=
z
2
+
1
xy=z^2+1
x
y
=
z
2
+
1
prove that there exists integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
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
,
z
=
a
c
+
b
d
z=ac+bd
z
=
a
c
+
b
d
.
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