You know the answer: this is a specific cup, thus it must have "the"
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BelieverThank you, Marius and Milky.
Tell me if my understanding is wrong. When you have some phrase in genitive??, then almost always the article the is for the noun in genitive and not the noun that comes/follows after the noun in genitive.
the president's cup (here the is for "president" and not for "cup.")
Jane's cup (here the