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I read a lot of prompts from different schools and it seems while many suggest a persuasive topic, many leave the topic selection up to the applicant. Is it understood that we should only write a personal statement that displays our ability to pick one side and argue for it?

Not necessarily. Just convince them that you're awesome or, if you're not, that they nevertheless want you at their school.

yes, unless your numbers speak for themselves.

you should persuade them to accept you. that's the only persuasive argument you should make.