Special properties of subgraphs I have two statements I am not sure how to prove them:
(1) Every graph with $n=|V(G)|\ge 6$ and size $\lfloor n^2/4\rfloor+1$ contains a 5-cycle $C_5$
(2) Every graph with $n=|V(G)|\ge 5$ and size $\lfloor n^2/4\rfloor+2$ contains two trinagles with exactly one vertex in common. 
I think the best way is prove it by induction. For the simple cases n=6 or n=5 it is obvious for both cases (can be verfied geometrically.)
I do not know how to proceed with the induction step, may you can help me with that. If there is an alternative way to prove this it would be also very interesting. 

Surprisingly, but I cannot find references on a proof of your first statement. As to second one, we proved it in our paper (B.V.Novikov, L.Yu.Polyakova, G.N Zholtkevich, 
A decomposition of directed graphs, Ukrainian Math. J., to appear); later I found it as an exercise in B.Bollobás, Modern Graph Theory, of course, without a proof.
