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National and Regional Contests
Taiwan Contests
TST Round 2
2017 Taiwan TST Round 2
1
Functional equation 007
Functional equation 007
Source: 2017 Taiwan TST Round 2, Day 3, Problem 1
April 19, 2017
algebra
functional equation
Problem Statement
Determine all surjective functions
f
:
Z
ā
Z
f: \mathbb{Z} \to \mathbb{Z}
f
:
Z
ā
Z
such that
f
(
x
y
z
+
x
f
(
y
)
+
y
f
(
z
)
+
z
f
(
x
)
)
=
f
(
x
)
f
(
y
)
f
(
z
)
f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right)
f
(
x
yz
+
x
f
(
y
)
+
y
f
(
z
)
+
z
f
(
x
)
)
=
f
(
x
)
f
(
y
)
f
(
z
)
for all
x
,
y
,
z
x,y,z
x
,
y
,
z
in
Z
\mathbb{Z}
Z
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