According to Pappus, Apollonius developed a system of number systems larger than Archimedes' The Sand Reckoner, and according to Proclus, Apollonius measured the value of pi more precisely than Archimedes, and (I don't remember the source) he considered Archimedes to be his academic rival, so I wondered if he had solved any problems that Archimedes could not solve.
If there was such a problem, what was it?
I accidentally found a reference which probably inspired your question: this is the book by Reviel Netz,
A new history of Greek mathematics. Cambridge University Press, Cambridge, 2022.
On p. 158 he mentions that some (unnamed) commenters in the late antiquity mentioned that Apollonius found a closer approximation to $\pi$, but they do not state this approximation explicitly. Neither they explain how it was found.
Let me add that Archimedes (and even Euclid) perfectly understood how to obtain arbitrarily close approximations to $\pi$, so his concrete approximation was just an example.
An improvement by Apollonius of Archimedes calculus from the Sand Reconer is mentioned by Pappus, but the original book of Apollonius is lost and Pappus does not give any details.
Netz concludes that Apollonius competed with Archimedes:-)
But I repeat that if some later mathematician improves on your result, this does not mean that you could not prove this.
