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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2010 National Olympiad First Round
31
31
Part of
2010 National Olympiad First Round
Problems
(1)
Turkey NMO 2010 1st Round - P31 (Algebra)
Source:
7/27/2012
For which pair
(
A
,
B
)
(A,B)
(
A
,
B
)
,
x
2
+
x
y
+
y
=
A
y
y
−
x
=
B
x^2+xy+y=A \\ \frac{y}{y-x}=B
x
2
+
x
y
+
y
=
A
y
−
x
y
=
B
has no real roots?
<
s
p
a
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c
l
a
s
s
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′
l
a
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x
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b
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>
(
A
)
<
/
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p
a
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(
1
/
2
,
2
)
<
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(
B
)
<
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(
−
1
,
1
)
<
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c
l
a
s
s
=
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l
a
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x
−
b
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>
(
C
)
<
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p
a
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>
(
2
,
2
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
a
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>
(
1
,
1
/
2
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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>
(
E
)
<
/
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p
a
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>
(
2
,
2
/
3
)
<span class='latex-bold'>(A)</span>\ (1/2,2) \qquad<span class='latex-bold'>(B)</span>\ (-1,1) \qquad<span class='latex-bold'>(C)</span>\ (\sqrt 2, \sqrt 2) \qquad<span class='latex-bold'>(D)</span>\ (1,1/2) \qquad<span class='latex-bold'>(E)</span>\ (2,2/3)
<
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(
A
)
<
/
s
p
an
>
(
1/2
,
2
)
<
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=
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a
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x
−
b
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>
(
B
)
<
/
s
p
an
>
(
−
1
,
1
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
p
an
>
(
2
,
2
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
(
1
,
1/2
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
(
2
,
2/3
)