Question 6
Let $a_1,\ldots,a_{16}$ be distinct real numbers. For each real polynomial $P$, put $V(P)=\sum_{i=1}^{16}P(a_i)$. Prove there exists exactly one monic polynomial $Q$ of degree $8$ satisfying both:
1. $V(QP)=0$ for every polynomial $P$ of degree less than $8$;
2. $Q$ has $8$ real roots, counted with multiplicity.
1. $V(QP)=0$ for every polynomial $P$ of degree less than $8$;
2. $Q$ has $8$ real roots, counted with multiplicity.