If OP’s son is just relatively good at math he will have huge difficulty gaining admissions at top privates in US. Keep in mind that even if he is a US citizen and makes it to USAMO he is just one out of 500+ Asian boys every year who will be vying for a spot at top schools. Most of them will be disappointed in college admissions if they don’t have anything else to show. A case in point here on CC a few years ago (the boy had three USAMOs and top stats ):
http://talk.qa.collegeconfidential.com/college-admissions/1876770-what-did-i-do-wrong-p1.html
If anything college admissions have become more holistic in recent years. I agree with Twoin18 that UK colleges may be your best bet.
Another data point, I know some kid from China who won Bronze medal at Int Math Olympia was shut out on all top schools he applied to in US and he ended up going to a top college in China.