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Soros Olympiad in Mathematics
VII Soros Olympiad 2000 - 01
9.5
2 parameter cubic equation (VII Soros Olympiad 2000-01 R1 9.5)
2 parameter cubic equation (VII Soros Olympiad 2000-01 R1 9.5)
Source:
July 28, 2021
parameterization
algebra
Cubic
equation
Problem Statement
For all valid values of
a
a
a
and
b
b
b
, solve the equation
x
3
(
x
−
a
)
(
x
−
b
)
+
a
3
(
a
−
b
)
(
a
−
x
)
+
b
3
(
b
−
x
)
(
b
−
a
)
=
x
2
+
a
+
b
\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b
(
x
−
a
)
(
x
−
b
)
x
3
+
(
a
−
b
)
(
a
−
x
)
a
3
+
(
b
−
x
)
(
b
−
a
)
b
3
=
x
2
+
a
+
b
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