MathDB
Problems
Contests
International Contests
IMO Shortlist
1981 IMO Shortlist
6
Prove that P(z) ≡ Q(z)
Prove that P(z) ≡ Q(z)
Source:
September 15, 2010
algebra
polynomial
functional equation
IMO Shortlist
Problem Statement
Let
P
(
z
)
P(z)
P
(
z
)
and
Q
(
z
)
Q(z)
Q
(
z
)
be complex-variable polynomials, with degree not less than
1
1
1
. Let
P
k
=
{
z
∈
C
∣
P
(
z
)
=
k
}
,
Q
k
=
{
z
∈
C
∣
Q
(
z
)
=
k
}
.
P_k = \{z \in \mathbb C | P(z) = k \}, Q_k = \{ z \in \mathbb C | Q(z) = k \}.
P
k
=
{
z
∈
C
∣
P
(
z
)
=
k
}
,
Q
k
=
{
z
∈
C
∣
Q
(
z
)
=
k
}
.
Let also
P
0
=
Q
0
P_0 = Q_0
P
0
=
Q
0
and
P
1
=
Q
1
P_1 = Q_1
P
1
=
Q
1
. Prove that
P
(
z
)
≡
Q
(
z
)
.
P(z) \equiv Q(z).
P
(
z
)
≡
Q
(
z
)
.
Back to Problems
View on AoPS