KöMaL 2009 A481

Prove that there are infinitely many $n$, for which there exist simple graphs $S_1$,...,$S_n$ with the following properties: ($a$) each $S_i$ is a complete bipartite graph; ($b$) the union of the graphs $S_1$,...,$S_n$ is a complete graph on 2$n$ vertices; ($c$) each edge of this complete graph is contained in an odd number of the graphs $S_i$.