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Is it appropriate to send targeted mailings to other offices of firms that allow you only to interview with 1 office at OCI?

Let's say, hypothetically, I was bidding Latham LA and the firm allowed applicants only to bid on one office at OCI. Would it be counterproductive to an offer to then mail Latham DC and Latham NY?

I would say yes. I may be wrong, but most of the firms I've seen that only let you bid on one office intend for you to just inform the person interviewing you that you're interested in multiple offices. Though if you are currently living in DC and you're from DC, who knows, maybe it's worth asking if you could interview while you're in town.