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National and Regional Contests
Turkey Contests
National Olympiad First Round
2005 National Olympiad First Round
35
35
Part of
2005 National Olympiad First Round
Problems
(1)
P35 [Algebra] - Turkish NMO 1st Round - 2005
Source:
11/2/2013
If for every real
x
x
x
,
a
x
2
+
b
x
+
c
≥
0
ax^2 + bx+c \geq 0
a
x
2
+
b
x
+
c
≥
0
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are reals such that
a
<
b
a<b
a
<
b
, what is the smallest value of
a
+
b
+
c
b
−
a
\dfrac{a+b+c}{b-a}
b
−
a
a
+
b
+
c
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
5
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
7
2
<span class='latex-bold'>(A)</span>\ \dfrac{5}{\sqrt 3} \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ \dfrac{\sqrt 5}2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \dfrac{\sqrt 7}2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
5
<
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
>
2
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2
7