XII OBM

Count the number $N$ of pairs $(F,e)$, where $F$ is a face and $e$ an edge belonging to $F$. Each edge belongs to two faces, so $N$ is even. Hence the number of faces with an odd number of edges must be even. The total number of faces is odd, so the number of faces with an even number of edges must be odd. In particular, there is at least one.