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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2002 China Second Round Olympiad
2
2
Part of
2002 China Second Round Olympiad
Problems
(1)
Minimum Value of Fraction
Source: 2002 China Second Round Olympiad
8/30/2014
There are real numbers
a
,
b
a,b
a
,
b
and
c
c
c
and a positive number
λ
\lambda
λ
such that
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
f(x)=x^3+ax^2+bx+c
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
has three real roots
x
1
,
x
2
x_1, x_2
x
1
,
x
2
and
x
3
x_3
x
3
satisfying
(
1
)
x
2
−
x
1
=
λ
(1) x_2-x_1=\lambda
(
1
)
x
2
−
x
1
=
λ
(
2
)
x
3
>
1
2
(
x
1
+
x
2
)
(2) x_3>\frac{1}{2}(x_1+x_2)
(
2
)
x
3
>
2
1
(
x
1
+
x
2
)
. Find the maximum value of
2
a
3
+
27
c
−
9
a
b
λ
3
\frac{2a^3+27c-9ab}{\lambda^3}
λ
3
2
a
3
+
27
c
−
9
ab
algebra unsolved
algebra