Bulgarian/USSR Math Circles https://en.wikipedia.org/wiki/Math_circle and problems they work on are in this book https://bookstore.ams.org/mcl-25

Regular school you would solve problems in combinatorics, algebra and geometry together in class by offering small ideas or strategies then trying each one to point out what works and doesn't work. You do enough of these everyday in class then reasoning about abstract sets in analysis isn't a huge leap. Same reason we used to read Euclid in class and make students demonstrate countless proofs.

CMU Discrete Math course has kind of the same techniques where he solves problems with the whole class https://m.youtube.com/watch?v=0K540qqyJJU

The best students then joined the school math circle with professional mathematicians and became TAs like in university helping their regular class.

They tried and failed the soviet curriculum with New Math https://simple.wikipedia.org/wiki/New_Math

Singapore high school students learn the equivalent of AP Calculus in Grade 10 then either spend years 11/12 at a trade school or stay to earn an associates degree to prep for university covering engineering electives like linear algebra and differential equations.

>> Same reason we used to read Euclid in class and make students demonstrate countless proofs.

Key phrase: "Read Euclid". American schools do not read about math. They are taught to do math. They do not read anything written by the people who created it. They are not given context. They are given set problems and taught how to solve them using prescribed methods. More often than not, this results in lackluster understanding.

I did an AP course at school where we had to write a paper on a subject. Not a problem set. Not a proof. An actual paper about a mathematical concept. I did mine on the mathematics behind various methods for counting cards.