Top Student at Their Peak
Chapter 85 - 70: Is this final difficult mainly because of the attention to detail?

Chapter 85: Chapter 70: Is this final difficult mainly because of the attention to detail?

After roughly understanding the rules, Qiao Yu directly opened the browser, entered the finals website, and logged into his account.

The competition isn’t like college entrance exams, being early by a minute or late by a minute doesn’t matter. It doesn’t gain any advantage because the system will automatically time, anyway, everyone only has eight hours to solve the problems.

And it’s a continuous eight hours.

In other words, once the timer starts, it can’t be stopped.

Everything in between, like eating, drinking water, going to the bathroom, is counted in the answering time.

Fortunately, for a group of young people, this isn’t a big deal.

Whether middle school students or high school students, they may not run fast, but most can sit still.

...

Soon, Qiao Yu saw the finals questions, and the first one made him very happy.

To be honest, if it was the Qiao Yu before solving Professor Xue Song’s problem, he would probably be headaches with such questions.

It’s not that these questions are too difficult; mainly, they cover many concepts, and deep understanding is required. For example, the definition of subrings, understanding of matrix rings, the concept of lattice, isomorphic classification of modules, and understanding of finite generation, and so on...

But the current Qiao Yu is truly terrifyingly strong.

For instance, based on the given conditions, Qiao Yu immediately judged that the matrix shape provided in the question can be written as:

Obviously, this type of matrix forms a subring with a special algebraic structure, which can be set as R.

Then it becomes simple, the core of the proof is merely to determine how many different R-lattices there are.

With a rough idea of the solution in mind, Qiao Yu didn’t rush to start answering, but quickly scanned across, the second question, simple; the third question, not difficult either. Until the fourth question, he paused a bit.

Well then, it’s a problem asking for an equation with no integer solutions. (The illustration quota for today is used up, can’t show you the question, those interested can check the Easter egg Chapter.)

To be honest, for others, Qiao Yu thought it might indeed be quite difficult. But now he finds out that by carefully reviewing the question, this type of proof is truly not hard. It merely involves introducing unit roots and polynomial expressions, then simplifying the equation and analyzing the algebraic number theory background.

Even at this point, Qiao Yu could already see that there are no integer solutions to this equation.

Because at the equation simplification step, the left side of the equation can be considered as some polynomial’s factorization form, and each factor is related to the real part of the p-th root of unity. These factors correspond to Chebyshev polynomials or symmetric polynomials related to unit roots.

And these types of polynomials usually have non-integer coefficients, so it can basically be inferred that the roots of these polynomials won’t be integers.

Of course, the specific situation still needs to be proven.

But as long as further formalization through mod p arithmetic is done, that’s enough.

So Qiao Yu thinks this problem isn’t that difficult either.

The fifth question is a linear algebra type, merely involves some concepts in topological groups, there is difficulty, but it happens to be Qiao Yu’s strong point. The key is merely choosing infinite subsequences and analyzing uniform convergence.

Simply put, Qiao Yu believes the problem setter of this question is likely just testing the contestant’s understanding of the generation of matrix groups, the behavior of matrix sequence products, and convergence issues under matrix multiplication.

The sixth question, the main point is probably the direct sum decomposition in group representation theory, tensor product operations, and the proof of isomorphism and uniqueness of modules. The difficulty lies in how to analyze the structure of finitely generated modules under the action of p-groups.

So Qiao Yu thinks that as long as one understands how to establish isomorphic relations between different modules, this question isn’t too hard.

The seventh question, oh, no more... only six questions.

Just these six questions, giving a full eight hours of time, Qiao Yu thought this somewhat feels like an underestimation.

Of course, it’s not underestimating him, mainly thinking that the question setters kind of underestimate those master’s and doctoral students from prestigious universities or something.

After all, these people are not like Teacher Lan.

Their research direction is number theory, getting full marks on these questions probably isn’t a problem.

This made Qiao Yu a bit concerned, if everyone gets full marks, will the committee have enough gold medals and prizes to go around?

So maybe this year’s finals are mainly about details?

Qiao Yu has mentally pictured a scenario where a bunch of big-shot professors are nitpicking the solving process, then settled his mood and seriously clicked on the answer interface.

Once realizing that details are very important, then this situation isn’t so difficult.

...

After the contestants entered the exam hall, Lan Jie was led by the staff to the designated rest area.

The competition is set to last for eight hours.

Of course, this doesn’t mean the participants inside need to stay in the prepared hall all the time.

Things like eating and going to the bathroom in between are permissible. It’s just that the time lost won’t be replenished.

This is actually the same as taking the exam at home.

Within twenty-four hours, choosing eight continuous hours to answer. Log into the exam interface to start timing, after the time is up, you can no longer modify answers. So being able to endure longer than other participants is considered an advantage.

Of course, these accompanying teachers or guardians don’t need to endure with them.

Just like during breakfast.

Some grouped together chatting, some were engrossed in playing with their phones.

Lan Jie, sitting on a sofa by the window, opened his phone and flipped through this year’s preliminary examination questions, not finding it too boring.

Mathematics is like this, encountering something that can never be understood clearly, truly just looking at it for a few moments makes you want to sleep. But encountering interesting problems that can just be tackled, the more you solve, the more interesting it becomes, even to the point of neglecting sleep and meals.