MathDB

3

Part of 1999 Irish Math Olympiad

Problems(2)

prove an identity

Source: Ireland 1999

7/4/2009
If AD AD is the altitude, BE BE the angle bisector, and CF CF the median of a triangle ABC ABC, prove that AD,BE, AD,BE, and CF CF are concurrent if and only if: a^2(a\minus{}c)\equal{}(b^2\minus{}c^2)(a\plus{}c), where a,b,c a,b,c are the lengths of the sides BC,CA,AB BC,CA,AB, respectively.
geometryangle bisectorgeometry proposed
easy

Source: Ireland 1999

7/4/2009
The sum of positive real numbers a,b,c,d a,b,c,d is 1 1. Prove that: \frac{a^2}{a\plus{}b}\plus{}\frac{b^2}{b\plus{}c}\plus{}\frac{c^2}{c\plus{}d}\plus{}\frac{d^2}{d\plus{}a} \ge \frac{1}{2}, with equality if and only if a\equal{}b\equal{}c\equal{}d\equal{}\frac{1}{4}.
inequalitiesinequalities proposed