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National and Regional Contests
Turkey Contests
National Olympiad First Round
2010 National Olympiad First Round
22
Turkey NMO 2010 1st Round - P22 (Number Theory)
Turkey NMO 2010 1st Round - P22 (Number Theory)
Source:
July 27, 2012
Problem Statement
How many pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
are there such that
x
y
+
7
+
y
x
+
7
=
1
\frac{x}{y+7}+\frac{y}{x+7}=1
y
+
7
x
+
x
+
7
y
=
1
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
18
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
17
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
15
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
14
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
11
<span class='latex-bold'>(A)</span>\ 18 \qquad<span class='latex-bold'>(B)</span>\ 17 \qquad<span class='latex-bold'>(C)</span>\ 15 \qquad<span class='latex-bold'>(D)</span>\ 14 \qquad<span class='latex-bold'>(E)</span>\ 11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
18
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
17
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
14
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
11
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