MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2014 National Olympiad First Round
27
27
Part of
2014 National Olympiad First Round
Problems
(1)
P27 [Algebra] - Turkish NMO 1st Round - 2014
Source:
5/23/2014
Let
f
f
f
be a function defined on positive integers such that
f
(
1
)
=
4
f(1)=4
f
(
1
)
=
4
,
f
(
2
n
)
=
f
(
n
)
f(2n)=f(n)
f
(
2
n
)
=
f
(
n
)
and
f
(
2
n
+
1
)
=
f
(
n
)
+
2
f(2n+1)=f(n)+2
f
(
2
n
+
1
)
=
f
(
n
)
+
2
for every positive integer
n
n
n
. For how many positive integers
k
k
k
less than
2014
2014
2014
, it is
f
(
k
)
=
8
f(k)=8
f
(
k
)
=
8
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
45
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
120
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
165
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
180
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
215
<span class='latex-bold'>(A)</span>\ 45 \qquad<span class='latex-bold'>(B)</span>\ 120 \qquad<span class='latex-bold'>(C)</span>\ 165 \qquad<span class='latex-bold'>(D)</span>\ 180 \qquad<span class='latex-bold'>(E)</span>\ 215
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
45
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
120
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
165
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
180
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
215
function