VN-TST 2016 Problem 6

Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$ . For each polynomial $P$ , denote $$V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}).$$ Prove that there is a monic polynomial $Q$ , $\deg Q=8$ satisfying:

i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$ .

ii) $Q$ has $8$ real roots (including multiplicity).