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China TST polynomial inequality involving bionomials

Source: China TST 1997, problem 4

May 22, 2005
algebrapolynomialinequalitiesVietainductionalgebra unsolvedPolynomials

Problem Statement

Find all real-coefficient polynomials f(x)f(x) which satisfy the following conditions:
i. f(x)=a0x2n+a2x2n−2+⋯+a2n−2x2+a2n,a0>0f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0; ii. ∑j=0na2ja2n−2j≤(2nn)a0a2n\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}; iii. All the roots of f(x)f(x) are imaginary numbers with no real part.
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