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National and Regional Contests
China Contests
China National Olympiad
1996 China National Olympiad
3
Functional Equation involving cubes
Functional Equation involving cubes
Source: Chinese MO 1996
April 25, 2010
geometry
3D geometry
function
algebra unsolved
algebra
Problem Statement
Suppose that the function
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
satisfies
f
(
x
3
+
y
3
)
=
(
x
+
y
)
(
f
(
x
)
2
−
f
(
x
)
f
(
y
)
+
f
(
y
)
2
)
f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)
f
(
x
3
+
y
3
)
=
(
x
+
y
)
(
f
(
x
)
2
−
f
(
x
)
f
(
y
)
+
f
(
y
)
2
)
for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
. Prove that
f
(
1996
x
)
=
1996
f
(
x
)
f(1996x)=1996f(x)
f
(
1996
x
)
=
1996
f
(
x
)
for all
x
∈
R
x\in\mathbb{R}
x
∈
R
.
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