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China TST 1996 sum inequalities

Source: China TST 1996, problem 5

May 17, 2005
inequalitiesalgebraSums and ProductsReal RootsChina

Problem Statement

Let α1,α2,…,αn\alpha_1, \alpha_2, \dots, \alpha_n, and β1,β2,…,βn\beta_1, \beta_2, \ldots, \beta_n, where n≥4n \geq 4, be 2 sets of real numbers such that ∑i=1nαi2<1and∑i=1nβi2<1.\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1. Define \begin{align*} A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\ B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\ W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2. \end{align*} Find all real numbers λ\lambda such that the polynomial xn+λ(xn−1+⋯+x3+Wx2+ABx+1)=0,x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0, only has real roots.
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