We have $n$ people and $n$ jobs. Assume that each person is able to do $k$ jobs $0<k<n$ and each job can be done by $k$ people. Proof that each job can be done at the same time
My try
Ok, I see that I can mark vertexs on two colors - so I have 2-chromatic graph. Ok, So I can represent graph as:
$$ \text{People} \hspace{3cm} \text{Jobs} \\ \bullet \hspace{4cm} \bullet \\ \bullet \hspace{4cm} \bullet \\ \bullet \hspace{4cm} \bullet \\ \bullet \hspace{4cm} \bullet \\ \cdots \\ \bullet \hspace{4cm} \bullet \\ \bullet \hspace{4cm} \bullet \\ $$ and each person and job has $k$ edges. But there I stucked :(