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Polynomial division

Source: USAMO 1992

October 27, 2005
algebrapolynomialinductioncomplex numbersalgebra solved

Problem Statement

Let  P(z) \, P(z) \, be a polynomial with complex coefficients which is of degree  1992 \, 1992 \, and has distinct zeros. Prove that there exist complex numbers  a1,a2,…,a1992 \, a_1, a_2, \ldots, a_{1992} \, such that  P(z) \, P(z) \, divides the polynomial (⋯((z−a1)2−a2)2⋯−a1991)2−a1992. \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}.
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