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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2013 National Olympiad First Round
23
23
Part of
2013 National Olympiad First Round
Problems
(1)
P23 [Algebra] - Turkish NMO 1st Round - 2013
Source:
4/20/2013
If the conditions
f
(
2
x
+
1
)
+
g
(
3
−
x
)
=
x
f
(
(
3
x
+
5
)
/
(
x
+
1
)
)
+
2
g
(
(
2
x
+
1
)
/
(
x
+
1
)
)
=
x
/
(
x
+
1
)
\begin{array}{rcl} f(2x+1)+g(3-x) &=& x \\ f((3x+5)/(x+1))+2g((2x+1)/(x+1)) &=& x/(x+1) \end{array}
f
(
2
x
+
1
)
+
g
(
3
−
x
)
f
((
3
x
+
5
)
/
(
x
+
1
))
+
2
g
((
2
x
+
1
)
/
(
x
+
1
))
=
=
x
x
/
(
x
+
1
)
hold for all real numbers
x
≠
1
x\neq 1
x
=
1
, what is
f
(
2013
)
f(2013)
f
(
2013
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1007
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
4021
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
6037
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4029
5
<
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 above
<span class='latex-bold'>(A)</span>\ 1007 \qquad<span class='latex-bold'>(B)</span>\ \dfrac {4021}{3} \qquad<span class='latex-bold'>(C)</span>\ \dfrac {6037}7 \qquad<span class='latex-bold'>(D)</span>\ \dfrac {4029}{5} \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1007
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
4021
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
7
6037
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
5
4029
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
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