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Canada National Olympiad
1981 Canada National Olympiad
4
4
Part of
1981 Canada National Olympiad
Problems
(1)
Two polynomials, prove P(P(x))=Q(Q(x)) has no real solution
Source: Canadian Mathematical Olympiad - 1981 - Problem 4.
5/27/2011
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
are two polynomials such that
P
(
x
)
=
Q
(
x
)
P(x)=Q(x)
P
(
x
)
=
Q
(
x
)
has no real solution, and
P
(
Q
(
x
)
)
≡
Q
(
P
(
x
)
)
∀
x
∈
R
P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}
P
(
Q
(
x
))
≡
Q
(
P
(
x
))
∀
x
∈
R
. Prove that
P
(
P
(
x
)
)
=
Q
(
Q
(
x
)
)
P(P(x))=Q(Q(x))
P
(
P
(
x
))
=
Q
(
Q
(
x
))
has no real solution.
algebra
polynomial
function
algebra proposed