I am a rising sophomore who is going to take the APUSH exam and class next year.
The colleges I really want to apply for is, Ivy Leagues, UCs (mostly Berkeley and LA), Caltech, and Stanford. Which ones out of these require you to submit AP scores?
Short answer- NONE of them.
Long answers- Whether you submit none, 1 or all your APs to a particular school is completely up to your discretion. So if you have a list of 20 schools, in theory, you can take different approaches to submit your AP test score on an a la carte basis. The College Board AP ordering form allows you to send whatever AP test you want to whatever school you want. And the Common App allows you to self-report your AP test scores.
Ivy Leagues generally give only college credit or placement options if you get a 5 on your AP test. Brown doesn't give you AP credit. So you might only want to reserve your 4s and 5s for Ivys and Stanford. And I'm pretty sure that Caltech only gives 5s credit on a few limited non-STEM APs as well. UC Schools, depending on the campus may give college credit for an AP with a 3. So I would check the UC AP credit schedule for each of the UC schools you are applying to prior to self-reporting them to that school.
If you are considering applying to Ivys and Stanford, keep in mind the sweet spot for APs is somewhere between 9 and 11 these days as more and more applicants have stronger course rigor than in prior years. So plan out your AP course load now, not next year. I recommend most high achieving applicants to take 1-2 sophomore year, 4-5 junior year, and 3-4 senior year. You don't want to wait until senior year to take a bunch o APs because your 10th and 11th grades are most important.
If you put your scores as a free score send, then the colleges you put on your list will receive them when you get your scores. Otherwise, you can self-report which scores you want colleges to see.
I don't believe any college in the United States requires you to submit AP scores, especially since many schools are still test-optional. Like @cpark277 mentioned, many colleges allow you to self-report scores (I think all the colleges you mentioned except Cornell have this option), but if you decide to submit an official score report, all of your scores will be on there. As for the UCs, they don't even consider SAT or ACT scores, so I highly doubt they would even look at your AP scores.
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I have a program which because of schedule, i take multiple dual enrollment classes and honors classes but the side effect is that I can only take around 5-6 ap classes throughout my highschool career.