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Austrian-Polish
1993 Austrian-Polish Competition
8
8
Part of
1993 Austrian-Polish Competition
Problems
(1)
x + Q(y + P(x))= y + Q(x + P(y)) when Q(0)= 0
Source: Austrian - Polish 1993 APMC
5/3/2020
Determine all real polynomials
P
(
z
)
P(z)
P
(
z
)
for which there exists a unique real polynomial
Q
(
x
)
Q(x)
Q
(
x
)
satisfying the conditions
Q
(
0
)
=
0
Q(0)= 0
Q
(
0
)
=
0
,
x
+
Q
(
y
+
P
(
x
)
)
=
y
+
Q
(
x
+
P
(
y
)
)
x + Q(y + P(x))= y + Q(x + P(y))
x
+
Q
(
y
+
P
(
x
))
=
y
+
Q
(
x
+
P
(
y
))
for all
x
,
y
ā
R
x,y \in R
x
,
y
ā
R
.
polynomial
functional
functional equation
algebra