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National and Regional Contests
Turkey Contests
National Olympiad First Round
2003 National Olympiad First Round
4
4
Part of
2003 National Olympiad First Round
Problems
(1)
P04 [Algebra] - Turkish NMO 1st Round - 2003
Source:
5/6/2014
How many pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
are there such that the roots of polynomial
x
2
−
a
x
−
b
x^2-ax-b
x
2
−
a
x
−
b
are not greater than
5
5
5
?
<
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p
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c
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a
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b
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′
>
(
A
)
<
/
s
p
a
n
>
40
<
s
p
a
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c
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a
s
s
=
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a
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x
−
b
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′
>
(
B
)
<
/
s
p
a
n
>
50
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
65
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
75
<
s
p
a
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c
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a
s
s
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 40 \qquad<span class='latex-bold'>(B)</span>\ 50 \qquad<span class='latex-bold'>(C)</span>\ 65 \qquad<span class='latex-bold'>(D)</span>\ 75 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
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a
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x
−
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d
′
>
(
A
)
<
/
s
p
an
>
40
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
50
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
65
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
75
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
algebra
polynomial