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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2005 National Olympiad First Round
31
31
Part of
2005 National Olympiad First Round
Problems
(1)
P31 [Algebra] - Turkish NMO 1st Round - 2005
Source:
11/2/2013
If the equation system
f
(
x
)
+
g
(
x
)
=
0
f
(
x
)
−
(
g
(
x
)
)
3
=
0
\begin{array}{rcl} f(x) + g(x) &=& 0 \\ f(x)-(g(x))^3 &=& 0 \end{array}
f
(
x
)
+
g
(
x
)
f
(
x
)
−
(
g
(
x
)
)
3
=
=
0
0
has more than one real roots, where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are reals and
f
(
x
)
=
x
2
+
a
x
+
b
f(x)=x^2 + ax+b
f
(
x
)
=
x
2
+
a
x
+
b
,
g
(
x
)
=
x
2
+
c
x
+
d
g(x)=x^2 + cx + d
g
(
x
)
=
x
2
+
c
x
+
d
, at most how many distinct real roots can the equation
f
(
x
)
g
(
x
)
=
0
f(x)g(x) = 0
f
(
x
)
g
(
x
)
=
0
have?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
4
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4