I was a top student in America.

In Li Ao's notebook, he noted the two difficult points he had encountered that morning: the decomposition of two-dimensional weights on grid points, and error control in the discrete averaging of convex functions.

If you try to figure these two things out on your own, you can probably understand them in ten days or half a month.

But asking the professor directly is obviously the easiest way.

In the last class, Li Ao's approach of using p-advanced numbers to solve problems impressed Carlson quite a bit.

It is indeed rare for a high school student to master such profound tools of algebraic number theory.

When Carlson saw the child take the initiative to ask a question, he was taken aback for a moment, but then his mood brightened.

It seems that even mathematical geniuses can encounter times when they are stumped by competition problems.

This gave him, a Princeton professor, the perfect opportunity to showcase his abilities.

He took the notebook and softened his tone: "Actually, math competitions aren't as simple as you think. Even if you go through all the past IMO exam questions, you'll still encounter question types you've never seen before on the actual exam."

"Come on, let me help you organize your thoughts."

As he spoke, he looked down at the contents of his notebook.

"Consider two-dimensional weights w_ij on a finite grid set, satisfying the summation condition and the centroid condition. Prove that for any convex function f, there exists a two-sided estimate between the discrete average and the continuous convex hull average... and give the optimal constant for the boundary term when the weight rounding error does not exceed the first-order grid scale..."

Carlson stopped reading halfway through, his brows furrowing.

This is not a competition problem at all, but a standard advanced mathematics problem, the core of which lies in the discrete average of convex functions.

He suddenly looked up at Li Ao: "You've already started working on this kind of challenging discrete averaging problem for convex functions?"

Because it was so unexpected, he couldn't control his volume, and this sudden shout immediately drew the attention of the entire class.

The classroom fell silent immediately, and everyone stopped writing and turned to look at the podium.

In the eyes of the other team members, even if Li Ao was assisting Professor Philip, he would at most be organizing materials and checking literature.

But what exactly is the "discrete average problem of convex functions"?

Kevin, sitting in his seat, didn't react much. He was already used to Li Ao's unpredictable and surprising antics and thought the professor was making a fuss over nothing.

Benjamin, who was not far away, curled his lip and muttered to himself.

No wonder this kid skips class every day; he's aiming too high and is working on a university research project.

If you focus your energy on this before the IMO main competition, it's a miracle if you can perform well in the exam.

Ignoring the stares of those around him, Li Ao nodded to Carlson:

"Yes, I've just started working on the small project Professor Philip gave me."

Carlson cleared his throat, trying to cover up his earlier lapse in composure.

He picked up a pencil and began to explain as he scribbled on the paper:

"This problem combines convex analysis and discrete lattice construction, which is indeed difficult to handle. Even senior undergraduate students in mathematics departments would find it challenging."

He paused for a moment, then said, "Once you've learned measure theory or convex optimization, these kinds of two-dimensional weighting problems will become much simpler. Essentially, it's about transforming the averaging over a continuous convex hull into a decomposition problem of a finitely supported probability measure."

Li Ao had already done a lot of preparation for this topic, and now, as he listened to Carlson's train of thought, his previous sense of unease immediately dissipated.

He stopped focusing on directly estimating the error and instead changed his approach.

"Professor Carlson, I think I understand."

Before the words were even out of his mouth, Li Ao had already taken the pencil from Carlson's hand and started quickly working out the derivation on the blank space of the draft paper.

First, the two-dimensional weights are processed using centroid decomposition; then, on each small triangle, the discrete average is transformed into Jensen estimates at the three vertices; finally, the grid rounding error is used to control the boundary terms.

"Understood?"

Carlson stared blankly at the rapidly forming proof steps on the paper.

If he was only surprised by Li Ao's learning progress before, now he was truly impressed by the other party's ability to apply what he had learned to other situations.

With just one hint, a complete proof framework can be built immediately.

After he followed Li Ao's steps to verify the information and confirmed the logical problems, his gaze towards Li Ao had changed.

"Leo, are you interested in studying at Princeton? Our math department can definitely provide you with top-notch resources."

Li Ao didn't respond immediately; he was looking at the system panel that had popped up in his field of vision.

[A new conclusion has been reached... Intelligence +0.1]

【Intelligence: 119.4】

Hearing Carlson's words, he snapped out of his reverie and answered truthfully:

"Professor, I haven't decided which school to apply to yet. I plan to finish the IMO first."

Carlson expressed some regret, but also said he understood.

With a talent of this caliber, it's no surprise that coaches from other schools have already extended numerous offers to him.

Li Ao took back his notebook and sat back down. Seizing the opportunity, he opened the materials Professor Philip had given him and tried to apply the solution he had just learned to a similar discrete average problem.

After class, he and his teammates went to the Illini Union Restaurant for a meal.

In the evening, I went into the discussion room in the math building again, working through the national team's practice problems with Kevin, Michael, and the others.

The following training camp days were basically a busy cycle of going back and forth between three points.

Li Ao spent increasingly longer periods of time in the discussion room.

Whenever he had free time, he would go to the coach's office to ask Professor Philip or Professor Carlson for help with number theory, geometry, and convex functions.

As his intelligence improved and his vocabulary became stronger, his understanding of discrete measure and convex analysis became more and more thorough.

When faced with complex problems, he can quickly break them down, understand the underlying mathematical structure, find the corresponding methods, and apply them flexibly.

Just before departing for Annam, this research topic was finally largely resolved.

The remaining task is to organize my thoughts and complete the final draft of the paper.

After finishing that, I can submit another newsletter.

The optimization problem of convex functions has a wide range of applications, and can be connected with machine learning, information theory and even the financial field.

Compared to the previous purely theoretical proof of the two-sided estimation of the Hermite-Hadamard inequality, this work is more practical and will likely have a shorter review period.

During this time, Li Ao also received some good news.

His previously submitted paper on the Hermite-Hadamard inequality has been officially accepted.

If all goes well, it will be published online in a few days, making it available for download and reading by mathematics professionals worldwide.

This put Li Ao at ease.

His resume will finally have one more truly impressive scientific research achievement to show for it.

At this rate, by the time he returns from the competition in Annam, he'll probably have two formally published papers in his hands.

The professors leading the team, Philip, Carlson, and Jensen, observed all of this and had some doubts in their hearts.

For the past two weeks, they have witnessed Li Ao's astonishing learning speed every day.

While coping with the intense IMO training, he was also able to steadily advance his research projects.

What's even more remarkable is that he takes each step very steadily, without any of the impetuousness or eagerness for quick success that young people often have.

The one under the most pressure is teaching assistant David.

With his own thesis still not finished, Li Ao, this high school student, is about to finish another one.

Now, whenever he bumps into his mentor, Professor Jensen, he can't help but feel uneasy.

He had originally thought that once Li Ao joined the national team, he wouldn't have the mind to do research anymore, given the intense competition and training.

I can take advantage of this free time to polish my thesis so that my supervisor will be impressed.

Who would have thought that in less than half a month, under Professor Philip's guidance, they had already solved the problem of two-dimensional discrete averaging of convex functions.

Comparing the two sides, David felt that he had simply wasted the past two weeks.

Friday, October 13th.

The national team's intensive training camp has finally come to an end, and the team will soon be flying to Annam.

There is less than a week left before the IMO main competition begins.

Meanwhile, in the university library, Li Ao looked at the system panel in front of him and finally reached a new hurdle in his intelligence attribute.

[Under intense study conditions... Intelligence +0.1]

He stopped writing and focused his attention on the changes after the breakthrough.

【Intelligence: 120.0】

[Attribute Breakthrough! Unlocking Keywords - Model Analysis]

[Entry: Model Analysis]

[Effect: Your ability to deconstruct complex structures is greatly enhanced. You can break down highly difficult problems into simpler model modules and solve them one by one. This ability increases with your Intelligence attribute.]

……