Need proofs for these problems.

A) Let G be a simple graph on 2k vertices containing no triangles. Prove, by induction on k, that G has at most k² edges, and give an example of a graph for which this upper bound is achieved. (This result is often called Turán’s extremal theorem.)

B) Prove that, if two distinct cycles of a graph G each contain an edge e, then G has a cycle that does not contain e.

C) Let G be a simple graph with 2n vertices and n² edges.  If G has no triangles, then G is the complete bipartite graph K_n,n.