MathDB
Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1992 Irish Math Olympiad
3
complex roots
complex roots
Source:
January 17, 2014
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
be real numbers with
a
≠
0
a \neq 0
a
=
0
. Prove that if all the roots of the cubic equation
a
z
3
+
b
z
2
+
c
z
+
d
=
0
az^{3} +bz^{2} +cz+d=0
a
z
3
+
b
z
2
+
cz
+
d
=
0
lie to the left of the imaginary axis in the complex plane, then
a
b
>
0
,
b
c
−
a
d
>
0
,
a
d
>
0
ab >0, bc-ad >0, ad>0
ab
>
0
,
b
c
−
a
d
>
0
,
a
d
>
0
.
Back to Problems
View on AoPS