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Japan TST
2022 Japan TST
9
R FE with given f(-1)=-1
R FE with given f(-1)=-1
Source: Japan TST 2022, Day 3, P3
September 17, 2022
algebra
functional equation
Problem Statement
Find all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
, such that
f
(
x
)
f
(
y
)
−
f
(
x
+
y
)
=
y
f
(
x
)
−
f
(
x
+
f
(
y
)
)
f(x)f(y)-f(x+y)=yf(x)-f(x+f(y))
f
(
x
)
f
(
y
)
−
f
(
x
+
y
)
=
y
f
(
x
)
−
f
(
x
+
f
(
y
))
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
and
f
(
−
1
)
=
−
1
f(-1)=-1
f
(
−
1
)
=
−
1
.
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