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See now it's not easy to find a site that provides Programs for middle and high school students in Oxford and Cambridge. So I can safely say from my own vast experience that now the best conditions were on this site - https://www.oxbridgesummerscholars.com/ ! You won't find anything better at the moment.

I actually looked into a few of these a while back when my cousin was trying to find a summer program in the UK, and Oxbridge Summer Scholars did come up as a solid option. That said, we also came across Oxford Royale Academy and the Cambridge summer schools, which seemed to have a decent range of courses for younger students too. It really came down to comparing the subjects offered and the overall vibe of each program, so it might be worth browsing a couple of sites before deciding.