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2021 Poland - Second Round
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2021 Poland - Second Round
Problems
(1)
Number Theory
Source: Polish Math Olympiad 2021 2nd round p3 day 1
2/13/2021
Positive integers
a
,
b
,
z
a,b,z
a
,
b
,
z
satisfy the equation
a
b
=
z
2
+
1
ab=z^2+1
ab
=
z
2
+
1
. Prove that there exist positive integers
x
,
y
x,y
x
,
y
such that
a
b
=
x
2
+
1
y
2
+
1
\frac{a}{b}=\frac{x^2+1}{y^2+1}
b
a
=
y
2
+
1
x
2
+
1
number theory
Sum of Squares