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Soros Olympiad in Mathematics
III Soros Olympiad 1996 - 97 (Russia)
10.9
a^2/x^2 -b^2/y^2=8(y^4-x^4) , ax-by=x^4-y^4 (III Soros Olympiad 1996-97 R3 10.9)
a^2/x^2 -b^2/y^2=8(y^4-x^4) , ax-by=x^4-y^4 (III Soros Olympiad 1996-97 R3 10.9)
Source:
May 31, 2024
algebra
system of equations
Problem Statement
For any positive
a
a
a
and
b
b
b
, find positive solutions of the system
{
a
2
x
2
−
b
2
y
2
=
8
(
y
4
−
x
4
)
a
x
−
b
y
=
x
4
−
y
4
\begin{cases} \dfrac{a^2}{x^2}- \dfrac{b^2}{y^2}=8(y^4-x^4) \\ ax-by=x^4-y^4 \end{cases}
⎩
⎨
⎧
x
2
a
2
−
y
2
b
2
=
8
(
y
4
−
x
4
)
a
x
−
b
y
=
x
4
−
y
4
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