Prove that for any four positive real numbers a, b, c, d 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 inequalitiesalgebraIMO Shortlist