IMO ShortList 2008, Algebra problem 7
Source: IMO ShortList 2008, Algebra problem 7, German TST 4, P3, 2009, Exam set by Christian Reiher
July 9, 2009
inequalitiesalgebraIMO Shortlist
Problem Statement
Prove that for any four positive real numbers , , , the inequality
\frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0
holds. Determine all cases of equality.
Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany