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7
2022 Team 7 f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))
2022 Team 7 f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))
Source:
March 14, 2022
algebra
functional
Problem Statement
Find, with proof, all functions
f
:
R
−
{
0
}
→
R
f : R - \{0\} \to R
f
:
R
−
{
0
}
→
R
such that
f
(
x
)
2
−
f
(
y
)
f
(
z
)
=
x
(
x
+
y
+
z
)
(
f
(
x
)
+
f
(
y
)
+
f
(
z
)
)
f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))
f
(
x
)
2
−
f
(
y
)
f
(
z
)
=
x
(
x
+
y
+
z
)
(
f
(
x
)
+
f
(
y
)
+
f
(
z
))
for all real
x
,
y
,
z
x, y, z
x
,
y
,
z
such that
x
y
z
=
1
xyz = 1
x
yz
=
1
.
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