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IMO ShortList 2002, algebra problem 3

Source: IMO ShortList 2002, algebra problem 3

September 28, 2004

algebrapolynomialIMO Shortlistequationinfinitely many solutions

Problem Statement

Let

P

be a cubic polynomial given by

P(x)=ax^3+bx^2+cx+d

, where

a,b,c,d

are integers and

a\ne0

. Suppose that

xP(x)=yP(y)

for infinitely many pairs

x,y

of integers with

x\ne y

. Prove that the equation

P(x)=0

has an integer root.