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All-Russian Olympiad
1977 All Soviet Union Mathematical Olympiad
251
251
Part of
1977 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 251 All Soviet Union MO 1977 P(Q(x))=Q(P(x))
Source:
7/6/2019
Let us consider one variable polynomials with the senior coefficient equal to one. We shall say that two polynomials
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
commute, if
P
(
Q
(
x
)
)
=
Q
(
P
(
x
)
)
P(Q(x))=Q(P(x))
P
(
Q
(
x
))
=
Q
(
P
(
x
))
(i.e. we obtain the same polynomial, having collected the similar terms). a) For every a find all
Q
Q
Q
such that the
Q
Q
Q
degree is not greater than three, and
Q
Q
Q
commutes with
(
x
2
−
a
)
(x^2 - a)
(
x
2
−
a
)
. b) Let
P
P
P
be a square polynomial, and
k
k
k
is a natural number. Prove that there is not more than one commuting with
P
P
P
k
k
k
-degree polynomial. c) Find the
4
4
4
-degree and
8
8
8
-degree polynomials commuting with the given square polynomial
P
P
P
. d)
R
R
R
and
Q
Q
Q
commute with the same square polynomial
P
P
P
. Prove that
Q
Q
Q
and
R
R
R
commute. e) Prove that there exists a sequence
P
2
,
P
3
,
.
.
.
,
P
n
,
.
.
.
P_2, P_3, ... , P_n, ...
P
2
,
P
3
,
...
,
P
n
,
...
(
P
k
P_k
P
k
is
k
k
k
-degree polynomial), such that
P
2
(
x
)
=
x
2
−
2
P_2(x) = x^2 - 2
P
2
(
x
)
=
x
2
−
2
, and all the polynomials in this infinite sequence pairwise commute.
polynomial
algebra