Capitalization and Definite/Indefinite Article

Are the following sentences all acceptable?

1. The town opened a Community Center. (There are no other community centers in the town. Should this be "The town opened a community center" or "The town opened the Community Center")
2. The city signed a Memorandum of Agreement with its sister city (should this be "The city signed a memorandum of agreement..." or "The city signed the Memorandum of Agreement."?)