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Thailand TST Selection Test
2018 Thailand TSTST
1
f(x+y)=f(x)+f(y), f(P(x))=f(x)
f(x+y)=f(x)+f(y), f(P(x))=f(x)
Source: 2017 Thailand October Camp 3.1
February 25, 2022
functional equation
algebra
Problem Statement
Let
P
P
P
be a given quadratic polynomial. Find all functions
f
:
R
→
R
f : \mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
and
f
(
P
(
x
)
)
=
f
(
x
)
for all
x
,
y
∈
R
.
f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
and
f
(
P
(
x
))
=
f
(
x
)
for all
x
,
y
∈
R
.
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