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National and Regional Contests
Turkey Contests
National Olympiad First Round
2012 National Olympiad First Round
15
15
Part of
2012 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 2012 Problem - 15 {Algebra}
Source:
7/1/2012
If
x
4
+
8
x
3
+
18
x
2
+
8
x
+
a
=
0
x^4+8x^3+18x^2+8x+a = 0
x
4
+
8
x
3
+
18
x
2
+
8
x
+
a
=
0
has four distinct real roots, then the real set of
a
a
a
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
(
−
9
,
2
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
(
−
9
,
0
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
[
−
9
,
0
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
[
−
8
,
1
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
(
−
8
,
1
)
<span class='latex-bold'>(A)</span>\ (-9,2) \qquad <span class='latex-bold'>(B)</span>\ (-9,0) \qquad <span class='latex-bold'>(C)</span>\ [-9,0) \qquad <span class='latex-bold'>(D)</span>\ [-8,1) \qquad <span class='latex-bold'>(E)</span>\ (-8,1)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
(
−
9
,
2
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
(
−
9
,
0
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
[
−
9
,
0
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
[
−
8
,
1
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
(
−
8
,
1
)