MathDB

2

Part of 2008 IMC

Problems(2)

IMC 2008 Day 1 P2

Source:

7/26/2014
Denote by V\mathbb{V} the real vector space of all real polynomials in one variable, and let γ:V→R\gamma :\mathbb{V}\to \mathbb{R} be a linear map. Suppose that for all f,g∈Vf,g\in \mathbb{V} with γ(fg)=0\gamma(fg)=0 we have γ(f)=0\gamma(f)=0 or γ(g)=0\gamma(g)=0. Prove that there exist c,x0∈Rc,x_0\in \mathbb{R} such that \gamma(f)=cf(x_0)  \forall f\in \mathbb{V}
vectoralgebrapolynomialIMCcollege contests
IMC 2008 Day 2 P2 - Ellipses with common focus

Source: Problem 2

7/28/2008
Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.
conicsellipsesearchgeometryratioanalytic geometrytrigonometry