Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.
$2015=5\cdot13\cdot35$ and so there are $8$ positive divisors of $2015$ and $16$ divisors including negative numbers. Clearly for each root both $p\left(x\right),\ q\left(x\right)$ must take the value of a divisor of $2015$. As there are at least $33$ such roots, by the pigeon hole principle both $p\left(x\right),\ q\left(x\right)$ must take some value at least $3$ times. It is well known that neither a non constant linear function nor quadratic can take the same value $3$ times, and thus both $p\left(x\right),\ q\left(x\right)$ must have degrees of at least $3$