MathDB

1

Part of 1997 China Team Selection Test

Problems(2)

China TST polynomial inequality involving bionomials

Source: China TST 1997, problem 4

5/22/2005
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.
algebrapolynomialinequalitiesVietainductionalgebra unsolvedPolynomials
Extend sides and find the locus of a point

Source: China TST 1997, problem 1

5/22/2005
Given a real number λ>1\lambda > 1, let PP be a point on the arc BACBAC of the circumcircle of △ABC\bigtriangleup ABC. Extend BPBP and CPCP to UU and VV respectively such that BU=λBABU = \lambda BA, CV=λCACV = \lambda CA. Then extend UVUV to QQ such that UQ=λUVUQ = \lambda UV. Find the locus of point QQ.
geometrycircumcircletrigonometrygeometric transformationhomothetyrotationgeometry solved