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Asian Pacific Mathematics Olympiad
›
2010
›
Problem 5
Problem 5
Find all functions
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
such that for all
x
,
y
,
z
∈
R
x,y,z\in\mathbb R
x
,
y
,
z
∈
R
,
f
(
f
(
x
)
+
f
(
y
)
+
f
(
z
)
)
=
f
(
f
(
x
)
−
f
(
y
)
)
+
f
(
2
x
y
+
f
(
z
)
)
+
2
f
(
x
z
−
y
z
)
f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz)
f
(
f
(
x
)
+
f
(
y
)
+
f
(
z
))
=
f
(
f
(
x
)
−
f
(
y
))
+
f
(
2
x
y
+
f
(
z
))
+
2
f
(
x
z
−
y
z
)
.
Step 1 of 8: Constant case
Previous step
Next step
f
≡
c
⟹
c
=
0
f\equiv c\Longrightarrow c=0
f
≡
c
⟹
c
=
0
Detailed analysis
If
f
f
f
is constant with value
c
c
c
, the equation becomes
c
=
3
c
c=3c
c
=
3
c
, hence
c
=
0
c=0
c
=
0
.
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