MathDB

4

Part of 2013 European Mathematical Cup

Problems(2)

A good looking inequality

Source: European Mathematical Cup 2013, Junior Division, P4

7/3/2014
Let a,b,ca,b,c be positive reals satisfying : a1+b+c+b1+c+a+c1+a+b≥ab1+a+b+bc1+b+c+ca1+c+a \frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a} Then prove that : a2+b2+c2ab+bc+ca+a+b+c+2≥2(ab+bc+ca) \frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})
Proposed by Dimitar Trenevski
Inequalitythree variable inequality
Problem 4

Source: 2nd European Mathematical Cup

12/16/2013
Given a triangle ABCABC let DD, EE, FF be orthogonal projections from AA, BB, CC to the opposite sides respectively. Let XX, YY, ZZ denote midpoints of ADAD, BEBE, CFCF respectively. Prove that perpendiculars from DD to YZYZ, from EE to XZXZ and from FF to XYXY are concurrent.
geometrycircumcirclegeometry unsolved