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National and Regional Contests
Turkey Contests
National Olympiad First Round
2010 National Olympiad First Round
2
2
Part of
2010 National Olympiad First Round
Problems
(1)
Turkey NMO 2010 1st Round - P02 (Number Theory)
Source:
7/27/2012
How many ordered pairs of positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
are there such that
y
2
−
x
2
=
2
y
+
7
x
+
4
y^2-x^2=2y+7x+4
y
2
−
x
2
=
2
y
+
7
x
+
4
?
<
s
p
a
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c
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a
s
s
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′
l
a
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e
x
−
b
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′
>
(
A
)
<
/
s
p
a
n
>
3
<
s
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c
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a
s
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=
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>
(
B
)
<
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a
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>
2
<
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a
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s
=
′
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a
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e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
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>
Infinitely many
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ 0 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
Infinitely many
linear algebra
matrix