KöMaL 2009 A477

Let $G$ be the complete graf with 2$n$ vertices, and suppose that $S_1$,...,$S_k$ are subgraphs of $G$ with the following properties: ($a$) Every $S_i$ is a complete bipartite graph; ($b$) Every edge of $G$ is contained by an odd number of subgraphs $S_i$. Show that $k \ge n$.