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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.)

[![enter image description
here](https://i.stack.imgur.com/7JgXo.png)](https://i.stack.imgur.com/7JgXo.png)

> 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.

[![enter image description
here](https://i.stack.imgur.com/8YG5m.jpg)](https://i.stack.imgur.com/8YG5m.jpg)

> 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.

[![enter image description
here](https://i.stack.imgur.com/FG8KR.png)](https://i.stack.imgur.com/FG8KR.png)

> 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](http://ncm.gu.se/media/kursplaner/andralander/Japanese_COS2008Math.pdf)
in Japan (for reference, the [school years in Japan](https://web-
japan.org/kidsweb/explore/schools/#:%7E:text=The%20basic%20school%20system%20in,go%20on%20to%20high%20school.)),
is not required for a middle schooler. The formula of binomial expansion is in
fact in the [syllabus](https://www.futureschool.com/japan-
curriculum/#552f669b758c5:%7E:text=Unordered%20selections%20%E2%80%93-,combinations,-23)
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.)


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


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.

* * *

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](https://en.wikipedia.org/wiki/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.


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answer