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SOLVED
Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?
Győri proved this with $1.03n^5$, which has been improved by Füredi. The answer is yes, as proved independently by Grzesik [Gr12] and Hatami, Hladky, Král, Norine, and Razborov [HHKNR13].
Additional thanks to: Casey Tompkins, Tuan Tran