ISL 2002 A3

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.

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