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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2008 National Olympiad First Round
15
15
Part of
2008 National Olympiad First Round
Problems
(1)
Turkey NMO 2008 1st Round - P15 (Algebra)
Source:
8/25/2012
Let the sequence
(
a
n
)
(a_n)
(
a
n
)
be defined as
a
1
=
1
3
a_1=\frac 13
a
1
=
3
1
and
a
n
+
1
=
a
n
1
+
13
a
n
2
a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}
a
n
+
1
=
1
+
13
a
n
2
a
n
for every
n
≥
1
n\geq 1
n
≥
1
. If
a
k
a_k
a
k
is the largest term of the sequence satisfying
a
k
<
1
50
a_k < \frac 1{50}
a
k
<
50
1
, what is
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
>
194
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
193
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
192
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
191
<
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 above
<span class='latex-bold'>(A)</span>\ 194 \qquad<span class='latex-bold'>(B)</span>\ 193 \qquad<span class='latex-bold'>(C)</span>\ 192 \qquad<span class='latex-bold'>(D)</span>\ 191 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
194
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
193
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
192
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
191
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above