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Japan TST
2016 Japan TST
5
Pair of rational functions
Pair of rational functions
Source: Japan TST 2016 P5
January 25, 2021
function
algebra
rational function
Problem Statement
Find all pairs of functions
f
,
g
:
Q
ā
Q
f,g:\mathbb{Q}\rightarrow\mathbb{Q}
f
,
g
:
Q
ā
Q
such that for all rational
x
,
y
x,y
x
,
y
,
f
(
f
(
x
)
+
y
g
(
x
)
)
=
(
x
+
1
)
g
(
y
)
+
f
(
y
)
.
f(f(x)+yg(x))=(x+1)g(y)+f(y).
f
(
f
(
x
)
+
y
g
(
x
))
=
(
x
+
1
)
g
(
y
)
+
f
(
y
)
.
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