From child prodigy to chief scientist

After leaving the examination room, most students followed their coach's instructions and remained silent about the content of today's exam.

However, a few students couldn't resist the temptation and couldn't help but discuss it with their teammates.

"Hey, how did you draw the auxiliary lines for the first problem?"

"In that algebra problem, the periodicity of the sequence is valid, right?"

Is the number of matrices in question 2(p!)²?

He Muzhou hadn't originally intended to check the answers, but after hearing people around him discussing the exam questions, he couldn't resist in the end.

After being rejected by several teammates, He Muzhou had no choice but to try asking Tao Kewei, and to his surprise, Tao Kewei actually agreed.

Tao Kewei did not refuse mainly because he had enough confidence in his answer and believed that he was sure he was right.

However, after hearing Tao Kewei's answer, He Muzhou's expression suddenly turned regretful.

"Oh no... I think I overlooked the number of permutations of {cj} and {dj} in the matrix problem..."

At first, He Muzhou was still fantasizing about whether he was the one who did the right thing.

But after careful consideration, he realized the oversight he had made.

This is a very basic mistake. If we disregard the matrix knowledge that is not taught in normal high school, this is a mistake that no ordinary high school student should make.

Perhaps because He Muzhou was so eager to prove himself in this exam, he focused all his attention on thinking about how to solve the problems, and ended up failing on the easier parts.

Because of this oversight, He Muzhou's score on this question could be reduced by half.

Seeing He Muzhou's reaction, Tao Kewei also regretted giving him the answer.

"I knew I shouldn't have argued with you; I didn't want it to affect your exam tomorrow."

"No, it's not your fault, I insisted on asking."

Thinking about how he had lost points he shouldn't have lost on the first day of the exam, He Muzhou lost his appetite and only ate a few bites of food from his plate.

When He Muzhou returned to the room with his teammates, he found Xu Xiao sitting at her desk, studying very attentively.

When He Muzhou came out, he did not see Xu Xiao.

It seems that Xu Xiao submitted his paper early, and it appears that today's questions did not cause him any trouble.

"Xu Xiao...you really are better."

Even though Tao Kewei basically won most of the first places during the training camp, He Muzhou still felt that Xu Xiao was more capable.

He Muzhou decided to stop worrying about today's exam and quickly threw himself into studying, just like Xu Xiao, to make final preparations for tomorrow's exam.

…………

The following morning at eight o'clock, the final day of the national finals began.

Based on past patterns, the difficulty of the questions on the second day of the national finals is much higher than that on the first day.

Sure enough, after receiving the exam papers, most of the students' expressions became extremely serious.

The fourth question is a combinatorial geometry problem. The question is only two lines long, shorter than yesterday's first question, which could easily lead people to mistakenly think it is a simple problem.

Define the minima between two points A(x₁, y₁) and B(x₂, y₂) on a plane as d(A, B) = min{|x₁ - x₂|, |y₁ - y₂|}. Find the largest real number r such that there exist four points on the plane where the minima between any two points is greater than r.

However, after thinking for a long time, most of the test takers still couldn't find the right approach to solving the problem.

Even Xu Xiao couldn't find a solution immediately and kept searching for various possible ways to solve the problem.

"Is this question... really as difficult as the fourth one?"

Normally, the difficulty of the fourth question should be similar to that of the third question. Although it is more difficult than the medium-difficulty questions, it is not as exaggerated as the final question.

Even when Xu Xiao is doing these kinds of questions normally, he doesn't get stuck for this long.

However, Xu Xiao did not panic and continued to calm down and think carefully.

After trying several methods repeatedly, a new idea suddenly flashed into Xu Xiao's mind.

"By simplifying the problem using translation invariance and constructing a four-point configuration that meets the conditions, the value of r can be calculated. The proof should be sufficient by contradiction..."

It took Xu Xiao a little over an hour to finally finish the fourth question.

This timeframe was significantly longer than Xu Xiao had planned, since the latter two questions should theoretically be more difficult and require more time.

He felt that the fourth question was already at the level of the final question, but he didn't know if this was just his own feeling.

Without thinking too much, Xu Xiao immediately started looking at the combinatorial number theory problems below.

The fifth question also perfectly matches the difficulty of the final question, requiring proof that there are infinitely many prime numbers that satisfy the conditions of a certain equation and inequality.

This is equivalent to putting two completely different problems together, resulting in a larger number of questions than usual.

Fortunately, this time, Xu Xiao determined the core idea faster than before, without wasting too much time on thinking.

"By utilizing the number theory properties of quadratic residues and primitive roots, and combining them with combinatorial construction, it can be proven."

Xu Xiao first constructed a new concept, and then carried out two proofs based on it.

It took more than forty lines of writing to finally solve the problem.

After finishing the fifth question, Xu Xiao glanced at the time and was somewhat surprised to find that it was already past eleven o'clock.

"Oh no... I forgot to check the time. I should have stopped and rested earlier."

According to Xu Xiao's time allocation plan, he should stop and rest for ten minutes at 10:30 to ensure that he can maintain the best condition for the rest of the exam.

However, because he spent more time on the fourth question, his rhythm was somewhat disrupted.

In addition, I was so focused on doing the questions that I didn't even notice the time.

Seeing that there was not much time left, Xu Xiao wanted to immediately get to the last question, but she still felt a severe exhaustion of energy.

"No...we have to stop and rest for a while."

At this moment, Xu Xiao felt as exhausted as if he had just finished a 1,000-meter physical fitness test. If he didn't rest for a while, it would be absolutely impossible for him to answer the sixth question in this state.

Forcing herself to clear her mind, Xu Xiao took out a piece of chocolate from her pocket, broke it off, and put it in her mouth.

This is the energy supplement that Ma Junfeng prepared for each team member before the competition. If they feel physically exhausted during the test, they can suck on a piece of chocolate to quickly replenish their energy.