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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2004 National Olympiad First Round
36
36
Part of
2004 National Olympiad First Round
Problems
(1)
P36 [Algebra] - Turkish NMO 1st Round - 2004
Source:
11/16/2013
If the function
f
f
f
satisfies the equation
f
(
x
)
+
f
(
1
1
−
x
3
3
)
=
x
3
f(x) + f\left ( \dfrac{1}{\sqrt[3]{1-x^3}}\right ) = x^3
f
(
x
)
+
f
(
3
1
−
x
3
1
)
=
x
3
for every real
x
≠
1
x \neq 1
x
=
1
, what is
f
(
−
1
)
f(-1)
f
(
−
1
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
−
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
7
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of above
<span class='latex-bold'>(A)</span>\ -1 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 14 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 12 \qquad<span class='latex-bold'>(D)</span>\ \dfrac 74 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
−
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
function