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National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2022 Vietnam TST
1
Vietnam TST #1
Vietnam TST #1
Source: Vietnam TST 2022 P1
April 26, 2022
function
algebra
phi function
functional equation
Problem Statement
Given a real number
α
\alpha
α
and consider function
φ
(
x
)
=
x
2
e
α
x
\varphi(x)=x^2e^{\alpha x}
φ
(
x
)
=
x
2
e
αx
for
x
∈
R
x\in\mathbb R
x
∈
R
. Find all function
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
that satisfy:
f
(
φ
(
x
)
+
f
(
y
)
)
=
y
+
φ
(
f
(
x
)
)
f(\varphi(x)+f(y))=y+\varphi(f(x))
f
(
φ
(
x
)
+
f
(
y
))
=
y
+
φ
(
f
(
x
))
forall
x
,
y
∈
R
x,y\in\mathbb R
x
,
y
∈
R
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