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International Contests
Czech-Polish-Slovak Match
1995 Czech and Slovak Match
2
2
Part of
1995 Czech and Slovak Match
Problems
(1)
f ,g : Z -> Z, f (g(x)+y) = g( f (y)+x) , g 1-1
Source: Czech and Slovak Match 1995 P2
10/1/2017
Find all pairs of functions
f
,
g
:
Z
ā
Z
f ,g : Z \rightarrow Z
f
,
g
:
Z
ā
Z
that satisfy
f
(
g
(
x
)
+
y
)
=
g
(
f
(
y
)
+
x
)
f (g(x)+y) = g( f (y)+x)
f
(
g
(
x
)
+
y
)
=
g
(
f
(
y
)
+
x
)
for all integers
x
,
y
x,y
x
,
y
and such that
g
(
x
)
=
g
(
y
)
g(x) = g(y)
g
(
x
)
=
g
(
y
)
only if
x
=
y
x = y
x
=
y
.
functional equation in Z
algebra