Let $a_1, a_2, \dots, a_{33}$ be different integer roots of $f(x) = p(x)q(x) - 2015$. Hence, $p(a_i)q(a_i) = 2015$ for all $i = 1, 2, 3, \dots, 33$. It follows that all integers $p(a_i)$, $i = 1, 2, 3, \dots, 33$ are divisors of $2015$. Because $2015 = 5 \cdot 13 \cdot 31$, this number has $16$ distinct integer divisors, $8$ positive and $8$ negative. Therefore, by the Pigeon Hole principle, at least three of the numbers $p(a_i)$ are equal. Suppose without loss of generality that $p(a_1) = p(a_2) = p(a_3) = d$. Then $p(x) - d$ has at least three distinct roots. Hence its degree and that of the polynomial $p(x)$ is at least $3$. The same argument works for the polynomial $q(x)$.