Question 17
Let $P$ be a cubic polynomial given by $P(x) = ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are integers and $a \ne 0$. 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.