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National and Regional Contests
Turkey Contests
National Olympiad First Round
2012 National Olympiad First Round
23
23
Part of
2012 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 2012 Problem - 23 {Algebra}
Source:
7/1/2012
a
,
b
,
c
a,b,c
a
,
b
,
c
are distinct real roots of
x
3
−
3
x
+
1
=
0
x^3-3x+1=0
x
3
−
3
x
+
1
=
0
.
a
8
+
b
8
+
c
8
a^8+b^8+c^8
a
8
+
b
8
+
c
8
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
156
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
171
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
180
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
186
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
201
<span class='latex-bold'>(A)</span>\ 156 \qquad <span class='latex-bold'>(B)</span>\ 171 \qquad <span class='latex-bold'>(C)</span>\ 180 \qquad <span class='latex-bold'>(D)</span>\ 186 \qquad <span class='latex-bold'>(E)</span>\ 201
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
156
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
171
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
180
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
186
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
201
algebra
polynomial
Vieta