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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2009 National Olympiad First Round
35
35
Part of
2009 National Olympiad First Round
Problems
(1)
When a_2 a_3 ... a_k > 3
Source: 0
4/28/2009
For every
n
≥
2
n \ge 2
n
≥
2
, a_n \equal{} \sqrt [3]{n^3 \plus{} n^2 \minus{} n \minus{} 1}/n. What is the least value of positive integer
k
k
k
satisfying
a
2
a
3
⋯
a
k
>
3
a_2a_3\cdots a_k > 3
a
2
a
3
⋯
a
k
>
3
?
<
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a
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a
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>
(
A
)
<
/
s
p
a
n
>
100
<
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a
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=
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(
B
)
<
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>
102
<
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l
a
s
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=
′
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a
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x
−
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>
(
C
)
<
/
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a
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>
104
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
106
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None
<span class='latex-bold'>(A)</span>\ 100 \qquad<span class='latex-bold'>(B)</span>\ 102 \qquad<span class='latex-bold'>(C)</span>\ 104 \qquad<span class='latex-bold'>(D)</span>\ 106 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
<
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p
an
c
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a
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a
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−
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>
(
A
)
<
/
s
p
an
>
100
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
102
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
104
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
106
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None
geometry
3D geometry