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Reciprocal Roots Lead to Integer Coefficients

Source: Indian RMO 2013 Mumbai Region Problem 6

February 1, 2014

number theorypolynomialrootsfactorialgreatest common divisor

Problem Statement

Let

P(x)=x^3+ax^2+b

and

Q(x)=x^3+bx+a

, where

a

and

b

are nonzero real numbers. Suppose that the roots of the equation

P(x)=0

are the reciprocals of the roots of the equation

Q(x)=0

. Prove that

a

and

b

are integers. Find the greatest common divisor of

P(2013!+1)

and

Q(2013!+1)