  Let X be a projective cubic hypersurface of dimension 11 or more, which is
defined over the rationals. In this paper it is shown that X contains rational
points provided that the cubic form defining X can be written as the sum of two
forms that share no common variables.
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  This paper features an appendix "Groupe de Brauer non ramifi\'e des
hypersurfaces cubiques singuli\`eres (d'apr\`es P. Salberger)", by J.-L.
Colliot-Th\'l\`ene.
