Question 200
Let $a_{1}, a_{2}, \ldots, a_{n}$ be distinct integers. Prove that there do not exist two nonconstant integer-coefficient polynomials $p$ and $q$ such that
$$
\left(x-a_{1}\right)\left(x-a_{2}\right) \cdots\left(x-a_{n}\right)-1=p(x) q(x)
$$
for all $x$.
$$
\left(x-a_{1}\right)\left(x-a_{2}\right) \cdots\left(x-a_{n}\right)-1=p(x) q(x)
$$
for all $x$.