Let $G$ be a triangle-free simple graph whose minimum degree is $> 2n/5$. Assume that $G$ is not a $5$-cycle. Prove that $G$ is bipartite.
darij grinberg's note: This is claimed to be a result by Andrásfai, Erdős & Sós (1974) in the Wikipedia, but the reference (Andrásfai, B.; Erdős, P.; Sós, V. T. (1974), "On the connection between chromatic number, maximal clique and minimal degree of a graph", Discrete Mathematics, 8 (3): 205–218, doi:10.1016/0012-365X(74)90133-2) is not very readable.
I have been trying to prove this but don't have an intuition on how to start I think it can be proved by contradiction.