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Vietnam TST 2016 Problem 6

Source: Vietnam TST 2016

July 26, 2016
algebrapolynomiallinear algebra

Problem Statement

Given 1616 distinct real numbers α1,α2,...,α16\alpha_1,\alpha_2,...,\alpha_{16}. For each polynomial PP, denote V(P)=P(α1)+P(α2)+...+P(α16). V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). Prove that there is a monic polynomial QQ, deg⁡Q=8\deg Q=8 satisfying:
i) V(QP)=0V(QP)=0 for all polynomial PP has deg⁡P<8\deg P<8.
ii) QQ has 88 real roots (including multiplicity).
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