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International Contests
International Olympiad of Metropolises
2016 IOM
5
5
Part of
2016 IOM
Problems
(1)
International Olympiad of metropolises 2016 P5
Source: International Olympiad of metropolises 2016
9/7/2016
Let
r
(
x
)
r(x)
r
(
x
)
be a polynomial of odd degree with real coefficients. Prove that there exist only finitely many (or none at all) pairs of polynomials
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
with real coefficients satisfying the equation
(
p
(
x
)
)
3
+
q
(
x
2
)
=
r
(
x
)
(p(x))^3 + q(x^2) = r(x)
(
p
(
x
)
)
3
+
q
(
x
2
)
=
r
(
x
)
.
IOM
algebra