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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2015 Greece National Olympiad
2
2
Part of
2015 Greece National Olympiad
Problems
(1)
Coefficients of a polynomial
Source: Greece National MO 2015
3/4/2015
Let
P
(
x
)
=
a
x
3
+
(
b
−
a
)
x
2
−
(
c
+
b
)
x
+
c
P(x)=ax^3+(b-a)x^2-(c+b)x+c
P
(
x
)
=
a
x
3
+
(
b
−
a
)
x
2
−
(
c
+
b
)
x
+
c
and
Q
(
x
)
=
x
4
+
(
b
−
1
)
x
3
+
(
a
−
b
)
x
2
−
(
c
+
a
)
x
+
c
Q(x)=x^4+(b-1)x^3+(a-b)x^2-(c+a)x+c
Q
(
x
)
=
x
4
+
(
b
−
1
)
x
3
+
(
a
−
b
)
x
2
−
(
c
+
a
)
x
+
c
be polynomials of
x
x
x
with
a
,
b
,
c
a,b,c
a
,
b
,
c
non-zero real numbers and
b
>
0
b>0
b
>
0
.If
P
(
x
)
P(x)
P
(
x
)
has three distinct real roots
x
0
,
x
1
,
x
2
x_0,x_1,x_2
x
0
,
x
1
,
x
2
which are also roots of
Q
(
x
)
Q(x)
Q
(
x
)
then: A)Prove that
a
b
c
>
28
abc>28
ab
c
>
28
, B)If
a
,
b
,
c
a,b,c
a
,
b
,
c
are non-zero integers with
b
>
0
b>0
b
>
0
,find all their possible values.
algebra
polynomial