Coefficients of a Cubed Product

★ 6/10complex numbers algebra Binomial Theorem Polynomials Vieta's Formulas

Problem

Let $z_1, z_2, \dots, z_{673}$ be distinct complex numbers such that

(x-z_1)^3 (x-z_2)^3 \cdots (x-z_{673})^3 = x^{2019} + 20x^{2018} + 19x^{2017} + g(x),

where $g(x)$ is a polynomial with complex coefficients of degree at most $2016$ . Suppose

\left| \sum_{1 \leqslant j < k \leqslant 673} z_j z_k \right| = \frac{m}{n},

where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Answer

Solution

Difficulty	6/10
Topics	complex numbers, algebra, Binomial Theorem, Polynomials, Vieta's Formulas

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