What is a Hellenistic state?

A country which is heavily influenced or directly controlled by the Ancient Greeks. The Greeks were split into several often feuding city states and each of them founded colonies or took over other states as clients, most importantly Egypt - who got a Greek line of pharaohs.

Basically the whole of the Eastern Mediterranean could be described as Hellenistic for the last 3 centuries BCE, after which Rome conquered Greece and became the major power, but the Roman Empire itself was very influenced by Greek culture...