This isn't a trick question, nor do I have a particular answer in mind.
Tomorrow, all of your memories are going to be wiped. There is a crucial piece of information that you need to make sure you remember, and more specifically, you need to be very confident you were the one that sent this message and not a third party pretending to be you.
How do you go about transmitting, "signing", and verifying such a message*?
--edit: I should have clarified that one of the assumptions is that some malicious third party can/will be attempting to send you false information from "yourself" and you need to distinguish between that and what's really you.
--edit2: this may be formally impossible, I don't actually know. If anyone can demonstrate this I'd be very appreciative.
--edit3: I don't have a particular universal definition for the term "memory wipe" in mind, mainly because I didn't want to pigeonhole the discussion. I think this pretty closely mimics reality. So I think it's totally fine to say, "If you retain this type of memory, then I'd do X."