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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2003 National Olympiad First Round
32
32
Part of
2003 National Olympiad First Round
Problems
(1)
P32 [Algebra] - Turkish NMO 1st Round - 2003
Source:
5/31/2014
The function
f
f
f
satisfies
f
(
x
)
+
3
f
(
1
−
x
)
=
x
2
f(x)+3f(1-x)=x^2
f
(
x
)
+
3
f
(
1
−
x
)
=
x
2
for every real
x
x
x
. If
S
=
{
x
∣
f
(
x
)
=
0
}
S=\{x \mid f(x)=0 \}
S
=
{
x
∣
f
(
x
)
=
0
}
, which one is true?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
<span class='latex-bold'>(A)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
S
S
S
is an infinite set.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
<span class='latex-bold'>(B)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
{
0
,
1
}
⊂
S
\{0,1\} \subset S
{
0
,
1
}
⊂
S
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
<span class='latex-bold'>(C)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
S
=
ϕ
S=\phi
S
=
ϕ
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
<span class='latex-bold'>(D)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
S
=
{
(
3
+
3
)
/
2
,
(
3
−
3
)
/
2
}
S = \{(3+\sqrt 3)/2, (3-\sqrt 3)/2\}
S
=
{(
3
+
3
)
/2
,
(
3
−
3
)
/2
}
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
<span class='latex-bold'>(E)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
function