2002 IMO Shortlist S13

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.

令 $P$ 为由 $P(x)=ax^3+bx^2+cx+d$ 给出的三次多项式，其中 $a,b,c,d$ 为整数， $a\ne0$ 。假设 $xP(x)=yP(y)$ 对于无限多个 $x,y$ 整数对 $x\ne y$ 。证明方程 $P(x)=0$ 有整数根。