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Coefficients of a polynomial

Source: Greece National MO 2015

March 4, 2015

algebrapolynomial

Problem Statement

Let

P(x)=ax^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

be polynomials of

x

with

a,b,c

non-zero real numbers and

b>0

.If

P(x)

has three distinct real roots

x_0,x_1,x_2

which are also roots of

Q(x)

then: A)Prove that

abc>28

, B)If

a,b,c

are non-zero integers with

b>0

,find all their possible values.

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