1) Squares of reals are non-negative;
2) integers cannot be strictly between 0 and 1.
Indeed, these two facts, masqueraded in various tricky ways, form the basis for transcendental number theory.
Most high-school students grinding for the national Olympiads are doing it to get into top universities later, but have no idea about the applications of what they're working on, they just grind away.
It's also one of the reason I never like complex math in school, because I never saw the point for it, so early in our education. Only later in life when I had to write SW for vehicle dynamics I understood the need but it's not something you run into while in school.
Alternative view, based on personal experience - not intending to over-generalize:
1. They love maths. The subject is beautiful to them. The reasoning involved makes sense. They enjoy knowing how things work. They take pleasure in seeing how the reasoning comes together.
2. Their minds are like sponges. They are bored with school. They easily surpass their colleagues, even the relatively strong ones. It is difficult to find any challenge related to coursework (which is a major minus). Any class where there is a reason to learn something deeper or more complicated is a joy. Their interests frequently become very broad because there is no depth available.
3. Competition is interesting, but I think it is primarily because of #2.
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Are there kids who "grind for glory"? Sure. I've known those, too. The people I have described above are the ones who stand out, though.
I do agree that presenting a use for complex thinking is a good thing. I actually think that creating larger programming projects is an excellent introduction to a kind of managing complexity that is challenging (and world-expanding) for many people. It's not exactly the same as competition maths, but it's not in a different world, either.
Years later, I've realized they helped me to get some sort of mathematical intuition of things that was enough for me to finish an Engineering degree without having to struggle too much with math. All my colleagues who participated in the contests felt the same.
To this day I believe those puzzles educate the Engineer in you, or if you want to finish with a Math degree they can be complementary.
Speaking as a PhD in math, Olympiad problems are very different than ordinary research math problems. Olympiad math problems in particular don't require creating new theory, but rather brilliance and knowing a bunch of standard tricks and how to apply them in insightful ways.
Becoming good at solving Olympiad problems won't really give you much skill in solving creative research problems, but it does give some indication of pure ingenuity as one dimension of mathematical intelligence.
... reminds me of when after uni i was trying to write games with a group of friends and had to rush back to the differential equations lectures to understand what's involved in vehicle simulation.
Edit: Oh and speaking of motivation as a student, your reward for qualifying for any national olympiad used to be being treated like a priceless ancient vase by every other teacher except the one supervising your grinding.
Of course, if you're national level on one subject you're probably pretty good at the others too, but it helps if they don't require as much work of you.
So you can grind at something you like anyway (I made the national computer science olympiad the last two high school years) and get an easier time (and like a month off) for the other subjects.
Intrinsic motivation ("I'm doing something because I enjoy the process"), though, dwells on autonomy ("I'm doing something because it aligns with my values and beliefs"). And for that, you need to have your own mental model of why these things are important for the world, or you, or people you care about.
I think most students wouldn't appreciate theoretical applications of some math problems. Saying "hey, you know what, a slight variation of this problem leads to a famous problem in non-commutative topology" or whatever only makes people even more confused.
Also, math is by far not the only thing you study that may have no "applications".
Does anyone in "the real world" care about Shakespeare, or Emily Dickinson? It's certainly not part of most jobs.
What about biology? Sure, knowing what a plant is and what animals are can be useful (for example, if you're a vegan), but do most jobs require knowing that "mitochondria is the powerhouse of the cell"? Nope.
Or, think history. Yes, knowing something about 20th century history can be useful (though not necessary), but do you really need to know about Mesopotamia for most practical jobs? Again, no.
School doesn't just teach stuff that everyone really needs in their everyday life. It also teaches
1) stuff that could be useful for some careers, and
2) stuff that becomes part of your background and subtly teaches other useful lessons.
If school were to be focused on the most common jobs (in the US: salespeople, cashiers, waiters, office clerks), you would see a syllabus made of:
- basic math, up to percentages and maybe powers. That's it
- a lot of Excel (for salespeople and office clerks)
- enough English to be able to write a few emails and confidently talk to customers. No literature, no 1,000 word essays.
- how to carry at least two, if not three plates at a time, as well as basic knowledge about hamburgers, meat and cocktails (for waiters)
Would this syllabus prove useful? Certainly. Would this system teach people something more than what they need to live their daily life and get a paycheck (no history, no geography, no literature, no trigonometry or biology or chemistry, and so on)? No, it probably wouldn't.
It's a good reflex when you see the sum of two squares to think at the Circle.
Considering the fact that n is natural is easy to find the solutions: n=0 and n=1
