Pentagons in triangle-free graphs

For all n≥9, we show that the only triangle-free graphs on n vertices maximizing the number 5-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erdős, and extends results by Grzesik and Hatami, Hladký, Král', Norin and Razborov, where they independently showed this same result for large n and for all n divisible by 5.