MathDB

2

Part of 2008 Spain Mathematical Olympiad

Problems(2)

Spanish inequality for 0<a,b<1

Source:

2/11/2011
Let aa and bb be two real numbers, with 0<a,b<10<a,b<1. Prove that ab2+a2b+(1−a)(1−b)2+(1−a)2(1−b)<2\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}
inequalitiestrigonometrySpain
Two fixed points no r such that (CM)(DN) is constant

Source:

2/11/2011
Given a circle, two fixed points AA and BB and a variable point PP, all of them on the circle, and a line rr, PAPA and PBPB intersect rr at CC and DD, respectively. Find two fixed points on rr, MM and NN, such that CM⋅DNCM\cdot DN is constant for all PP.
geometrygeometric transformationreflectiongeometry proposed