Prove that if $T$ is tournament that is not transitive then $T$ has at least three hamiltonian paths.
Proof: I am not sure how to start this proof. If it is not transitive then I think the tournament will be a hamiltonian. So that if $(a,b),(b,c),(c,a)$ if $a,b,c$ are the vertex in a cycle. So then you would have $3$ hamiltonian path. And a Hamiltoian path start and one vertex and ends at another and contains all the vertices.