Each voter in an election puts $n$ names of candidates on the ballot. There are $n + 1$ at the polling station urn. After the elections it turned out that each ballot box contained at least at least one ballot, for every choice of the $(n + 1)$-th ballot, one from each ballot box, there is a candidate whose surname appears in each of the selected ballots. Prove that in at least one ballot box all ballots contain the name of the same candidate.