Are there elite schools or private schools in Japan that teach advanced knowledge for their students' age?

Question

At around 14:00 in the first episode of Puella Magi Madoka Magica, there are 3 math questions in the blackboard. (The following pictures are for the convenience for reference.)

Question: Let p be a prime number, n an arbitrary fixed natural number, prove that (1+n)^p - n^p - 1 can be divided by p.

Question: An integer divided by 14 will have an integer quotient with a remainder between 0 and 13. a is an integer with a remainder of 6 when divided by 14, b is an integer with a remainder of 1 when divided by 14. The quadratic equation x^2-2ax+b=0 can be solved for an integer solution. Find the remainder when this integer solution is divided by 14.

Question: Find all the integer solutions (a,b) for the equation a^3+a^2-1=(a-1)b.

In the anime, the characters are just middle school students, but the questions are not in a normal middle school level. The solution of the first question above uses the formula of binomial expansion, which, according to the middle school math syllabus in Japan (for reference, the school years in Japan), is not required for a middle schooler. The formula of binomial expansion is in fact in the syllabus of the high school. The other two questions are barely in the syllabus of middle school math, yet their solutions are quite tricky, which no one would expect from a middle schooler.

I wonder if the MCs are attending an elite school, which expects more from its students. Are there elite schools or private schools in Japan that teach advanced knowledge for their students' age, say high school or even university math? (It's said that the USSR used to have this kind of schools for talented students.)

Answer 1

Speaking simply about the reality, I believe the short answer is no.

The basic compulsory Japanese education is 6 years (primary) + 3 years (middle). But practically most children proceed to high school (3 years).

"Elite schools" typically (especially near Tokyo) combines middle and high school education, which means that children enters those schools at 12 and stay there for 6 years. The schools' primary objective is to prepare those children for entrance exams for universities. So, in some schools, children are taught the full contents (i.e., what is supposed to be taught in 6 years) in 5 years, and spend the last year practicing for tests. In this sense, some schools teach stuff earlier/quicker than normal, but not like teaching high school stuff in middle school (at least not fully).

That said, obviously, there are pupils in those middle schools who can solve high school problems, just as some children skip grades in the US or elsewhere. Some of those elite schools are known for producing math Olympiad medalists. But those pupils just learn on their own or learn them as extra-curricula activities, not as part of the standard teaching.

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There are private prep schools whose target is pupils of elite schools and that teach 6-year stuff in 3 years (i.e., children who finished the curriculum know everything required for uni entrance at 15). But this is a little extreme case, and they are not exactly "schools" (in the sense that they are outside formal education).

Another thing is that there is Kumon (where mostly primary school pupils gets extra education, like prep school). I've heard that able children under 12 learn differentials or integrals in some cases. But as far as I know, it is about mechanically following calculations and not exactly teaching what those math concepts really are.