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Sequences a_n, b_n and P_k(x)=x^2+a_kx+b_k

Source: Vietnamese National Mathematical Olympiad 2012-P2

February 8, 2012
quadraticsalgebrapolynomialarithmetic sequencealgebra proposed

Problem Statement

Let ⟨an⟩\langle a_n\rangle and ⟨bn⟩ \langle b_n\rangle be two arithmetic sequences of numbers, and let mm be an integer greater than 2.2. Define Pk(x)=x2+akx+bk, k=1,2,⋯ ,m.P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m. Prove that if the quadratic expressions P1(x),Pm(x)P_1(x), P_m(x) do not have any real roots, then all the remaining polynomials also don't have real roots.
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