Are the complements of $NP$-languages with only $n$ words of length $n$ also in $NP$?
Hint: Suppose we are given a word $w$ of length $n$ which is not in $L$, and let $w_1,\ldots,w_n$ be the words of length $n$ which are in $L$. A witness that $w \notin L$ is the list $w_1,\ldots,w_n$ (all different from $w$!) together with witnesses that $w_1,\ldots,w_n \in L$.