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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2014 National Olympiad First Round
31
31
Part of
2014 National Olympiad First Round
Problems
(1)
P31 [Algebra] - Turkish NMO 1st Round - 2014
Source:
5/26/2014
a
1
=
1
a_{1}=1
a
1
=
1
and for all
n
≥
1
n \geq 1
n
≥
1
,
(
a
n
+
1
−
2
a
n
)
⋅
(
a
n
+
1
−
1
a
n
+
2
)
=
0.
(a_{n+1}-2a_{n})\cdot \left (a_{n+1} - \dfrac{1}{a_{n}+2} \right )=0.
(
a
n
+
1
−
2
a
n
)
⋅
(
a
n
+
1
−
a
n
+
2
1
)
=
0.
If
a
k
=
1
a_{k}=1
a
k
=
1
, which of the following can be equal to
k
k
k
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
10
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
12
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding