A generalization of a classical zero-sum problem

In this note, we supply the details of the proof of the fact that if a1,…,an+Ω(n)a1,…,an+Ω(n) are integers, then there exists a subset M⊂{1,…,n+Ω(n)}M⊂{1,…,n+Ω(n)} of cardinality n such that the equation∑i∈Maixi≡0(modn)admits a solution (xi)i∈M∈(U(Z/nZ))n(xi)i∈M∈(U(Z/nZ))n, where U(Z/nZ)U(Z/nZ) stands for the multiplicative group modulo n.