Question 50
Let $n \gt 1$ and $p(x) = x^{n} + a_{n-1} x^{n-1} + \cdots + a_{0}$ be a polynomial with $n$ real roots (counted with multiplicity). Let the polynomial $q$ be defined by
$$
q(x) = \prod_{j=1}^{2015} p(x + j)
$$
We know that $p(2015) = 2015$. Prove that $q$ has at least 1970 different roots $r_{1}, \ldots, r_{1970}$ such that $\left| r_{j} \right| \lt 2015$ for all $j = 1, \ldots, 1970$.
$$
q(x) = \prod_{j=1}^{2015} p(x + j)
$$
We know that $p(2015) = 2015$. Prove that $q$ has at least 1970 different roots $r_{1}, \ldots, r_{1970}$ such that $\left| r_{j} \right| \lt 2015$ for all $j = 1, \ldots, 1970$.