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For a trip that's just a month and a half, its probably not worth buying. You'll likely spend at least a week of your time just dealing with the purchase, sale, and paperwork.

Its also likely not financially feasable. Put simply, $1000 isn't going to buy a car that is roadworthy for a cross country trip. I wouldn't even consider a car that costs less than $2000 for such a trip, and really the $3-4000 range is more realistic as a starting point. You might get about half of that back at the end, but even that could be optimistic. Not to mention, its entirely possible that you'd have to spend close to $1000 for taxes, registration, and insurance.

Throw in the additional challenges which make it nearly impossible to get registration and insurance without a full fledge visa (assuming you'd be traveling on the wiaver program), and renting really will be your best bet.