KöMaL 2007 B3982

100 freshly graduated mathematicians are looking for jobs. They consult two headhunter companies that have the same 100 positions to offer. Each company makes an offer to each applicant, a different one to each of them. Each applicant chooses from the two offers and, luckily, all positions are filled. Three months later, however, each of them decides to change for the job offered by the other company. (If it is the same as his actual job, then he decides to stay.) Show that all positions will be filled again.