I was a top student in America.

"In a math competition, some of the participants were friends with each other."

"Friendship is a two-way street."

"If any two participants in a group are friends, then that group is called a 'clique'."

"It is known that the number of people in the largest 'group' is even. Prove that all participants can be assigned to two rooms such that the number of people in the largest 'group' is equal in each room."

Li Ao quickly read the question, feeling slightly surprised.

The third question surprisingly tested combinatorial graph theory.

Typically, questions of this level are scheduled for the next day.

This question is significantly more difficult than the previous two.

"It seems the test creators aren't planning to make things too easy for us test takers on the first day," Li Ao muttered to himself.

However, this question still did not stump him.

He drew diagrams on draft paper and quickly found a breakthrough: transforming the "room allocation problem" into a "problem of changing the number of clusters in the graph".

Following this line of thought, he used the minimal counterexample to continuously compress the difference between the two sides by moving the vertices one by one until the difference completely disappeared.

Next, Li Ao wrote down a standard proof of the least counterexample method in one go.

After finishing, he was still not satisfied. After thinking for a while, he began to add a second solution in the blank space.

In this solution, he treats all partitioning methods as a discrete state space.

Next, he defined the absolute difference between the maximum number of people in the two rooms and proved that if the difference is not zero, there must exist a way for the vertices to move so that the difference can be reduced.

This approach utilizes the extreme value process method from university discrete mathematics.

Unlike conventional competitive proofs, it derives a stronger and more general conclusion.

Under this broader conclusion, the original question is merely a special case.

This is something he learned while studying the "discrete average of convex functions".

In this context, it can serve as a more concise and elegant supplementary solution.

Of course, Li Ao did not forget the IMO's scoring standards.

Content beyond the curriculum at the university level should not be graded as a formal answer.

For this reason, he made a note next to the second solution, indicating that this was only a supplementary approach and not a formal solution.

"call."

After completing the entire test paper and carefully checking it from beginning to end, Li Ao put down his pen and let out a long breath.

He was very satisfied with his new solution to the third problem, and looking at the elegant conclusion, he felt an indescribable sense of satisfaction.

Then, I looked up at the clock on the wall.

There is still more than half an hour left before the exam ends.

Although it wasn't as early as during summer camp, it was still worthwhile mainly because a lot of time was spent supplementing new solutions.

Since there were no problems, he didn't plan to sit there and wait another half hour, so he raised his hand to ask to hand in his paper early.

"I want to hand in my paper."

Upon hearing the sound, a few candidates in the examination room looked up at him, then lowered their heads and continued working on the questions, without causing much of a commotion.

In the IMO, submitting your paper early does not mean you've finished; it could also mean you've given up early.

Kevin, who was also in the exam room, glanced at the sound, then looked down at his draft paper filled with derivations.

"Leo is truly amazing. He managed to hand in such a ridiculously difficult test early. He's really impressive." Kevin thought to himself, "However, compared to his feat of handing in his test an hour or two early during summer camp, he must have spent a lot of time on the third question this time."

"It seems that diverting his energy to scientific research projects and forum presentations has slightly affected his competitive performance."

However, Benjamin, sitting not far away, could not hide the shock on his face.

I was stunned for two or three seconds before I came to my senses and continued looking at the test paper.

He had been stuck on the third question for a long time, and several methods had been tried but none of them worked.

Therefore, he knew better than anyone how difficult this problem was.

In fact, he is now on the verge of collapse.

Besides this combined problem, he hadn't had time to verify the previous inequality problem, which involved a huge amount of calculation.

If he still has no clue about the third question after another ten minutes, he will have to strategically give up and go back to check the previous steps.

However, Li Ao actually finished the entire test paper before the allotted time.

"No wonder all the coaches are vying to take him on as a student. This guy is a monster."

Benjamin watched Li Ao's departing figure, secretly wondering, "Just how high is this guy's ceiling? What kind of exam paper would it take to truly assess his abilities?"

no way.

Everyone says that Benjamin is one of the top math geniuses in the United States, but when faced with someone like Li Ao, whose limits are unknown, he truly feels a sense of powerlessness, like an ant trying to see the blue sky.

After handing in his paper, Li Ao left the examination room and went straight back to the hotel's study room.

As always, I adhered to the principle of "not wasting a single second" and continued to study intensively for 18 hours a day.

The exam schedule for the second day remained the same.

Besides the fourth problem, which is about plane geometry, there is also an extremely challenging number theory problem, and the final combinatorial problem, which is the last one to be presented.

The IMO competition phase only truly came to a close after the long two-day, nine-hour exams were completed.

Because the IMO uses a unified marking and coordination system, all exam papers are sent to the marking group for centralized marking.

Afterwards, the team leaders and coordinators of each team will jointly confirm the final scores.

For this reason, almost at the same time Li Ao finished the exam, his answers to questions three and six were singled out by the leaders of the American team and given special attention.

When Professors Jensen and Carlson saw the two completely different solutions on the exam paper, even though they had prepared themselves mentally, they could not hide the shock on their faces.

Even Professor Philip didn't expect that Li Ao not only perfectly proved the problem, but also conveniently derived a more general conclusion using discrete mathematics.

That evening, Li Ao returned to his hotel room, turned on his laptop, and logged into his email as usual.

A new email from the Elsevier editorial office is quietly lying in my inbox.

He suppressed his excitement and opened the email.

Sure enough, this was the acceptance letter for his paper on "two-dimensional discrete averaging of convex functions".

His article has been accepted by Applied Mathematics Letters.

This means that his brief paper exploring discrete averaging and two-dimensional weight estimation of convex functions will soon be published in the latest issue of the journal.

He immediately shared this good news with Professor Philip.

After all, this topic originated from the professor's suggestion.

……