Chapter 32: On the Eve of the College Entrance Examination

"This problem requires proving an inequality. The core approach is to analyze how the parameter 'a' influences the number of solutions to the function f(x) = eˣ - ax = 0, or eˣ = ax, and then deduce the monotonicity and extreme points of f(x)."

"You can eliminate 'a' by dividing the equations satisfied by the two zeros, then construct a new function to analyze it."

Friday, June 2nd.

In the classroom of Class 3, Grade 12 at Dongping No. 1 High School, Xu Ming was using a pen to write down the proof steps on a piece of scratch paper, earnestly explaining them to his deskmate, Qian Hong.

Ever since he took first place in the mock exams, he had become the go-to example cited by teachers in every class across the school. He was mentioned time and again to motivate other students, a practice that extended even down to the freshman and sophomore years. It is no exaggeration to say that Zhou Junhao, who had held the top spot for over two years, saw his popularity quickly eclipsed by Xu Ming. Furthermore, at the principal's suggestion, Xu Ming had written down a set of tips and knowledge points he summarized during his own review process; once circulated, it was immediately revered by other students as a "bible," significantly boosting the study enthusiasm of the senior class.

And so, even Qian Hong, who always went on about being content with just getting into a vocational college, had been studying with rare diligence for the past month.

"I see now, I'll study it a bit more on my own," Qian Hong said, nodding repeatedly as he stared at the formulas on the scratch paper.

Xu Ming put down his pen and straightened his posture. Out of the corner of his eye, he glanced at his classmates and noticed everyone was still in a state of high tension. Their anxiety was palpable. Everyone wanted to review every subject but couldn't seem to manage their time effectively. After the "devil month" of exams in May, the pace had finally slowed down slightly. The remaining week before the college entrance exam was meant for students to identify and fill their own knowledge gaps—a final push.

He then lowered his gaze back to his scratchpad, continuing his attempt to derive the proof for the integral inequality. It wasn't until the bell rang for the end of class that he finally flexed his wrists, sighing inwardly.

"It really isn't easy trying to prove a new chain of inequalities."

Previously, while delving into the Cauchy-Schwarz inequality—the core of inner product spaces in mathematical analysis—he had encountered problems involving vectors or functions where the standard Cauchy-Schwarz inequality (a,b)² ≤ (a,a)(b,b) felt insufficient. He had tried to push the derivation further, eventually starting from the Gramian matrix, the matrix formed by pairs of vectors.

He realized that the Cauchy-Schwarz inequality is essentially about the non-negativity of the determinant of a Gramian matrix. After deeper reflection, he concluded that the principal minors of a Gramian matrix composed of multiple vectors contain a much richer set of inequality relationships. He had spent his time since then researching this, hoping to define a "weighted Gramian matrix" and establish new inequalities for it. If he could successfully prove them, they would have potential applications in statistical estimation and numerical analysis.

He had already finished the content for higher algebra, mathematical analysis, and analytic geometry, and could no longer gain mathematical experience points from them. He had originally planned to learn new knowledge after entering university, but after accidentally stumbling upon this research problem, he found that he could gain more mathematical experience by proving new inequalities than ever before. He realized that compared to simply learning existing mathematical content, solving mathematical problems and creating mathematical tools were the keys to leveling up quickly.

So, he became obsessed with it.

"Looks like I'll have to go to an internet cafe in a couple of days to find some literature and broaden my thinking."

Recalling the tradition at Dongping No. 1 High School where students were given two days off before the college entrance exam to relax, he murmured, finding a plan.

Once his attention drifted from the scratchpad, he habitually checked the subject information on the panel before him.

[Mathematics: Lv 1 (463/500)]

"Finally, I'm almost at level 2 in mathematics."

Seeing that he only needed about thirty more experience points to level up, his mood improved significantly. If all went well, as long as he could push his proof of the new inequality regarding the weighted Gramian matrix a few more steps forward, he should be able to gain enough experience to reach level 2, which would unlock new traits.

Just as he was thinking this, his train of thought was interrupted by Qian Hong's voice.

"Xu Ming, are you sleeping in the dorm tonight?"

"Yes, I'm going back," Xu Ming replied, turning his head.

Given his mastery and understanding of high school-level knowledge, mindless repetition of practice problems was meaningless—it would only be a waste of time. Therefore, with the principal's special approval and his homeroom teacher's consent, he had skipped the school-wide exams. The school had even prepared a room for him in the complex building where he could study and sleep, which allowed him to quickly finish higher algebra, mathematical analysis, and analytic geometry.

As it turned out, even though he hadn't gone through the "devil month," his position as the top student in the grade remained unshakable. In the third city-wide mock exam a few days ago, his score had shown an upward trend compared to the last one.

When the scores were announced, both his homeroom teacher and the school leadership were beaming, becoming increasingly confident that the school would secure the top science student in the region this year.

Qian Hong said happily, "That's great." With the college entrance exam drawing nearer, opportunities to sleep in the same dorm were dwindling, and he naturally cherished these moments.

Seeing the look on Qian Hong's face, Xu Ming couldn't help but feel a touch of melancholy, wondering how many more times they would see each other after graduation.

When they returned to the dorm, they heard the others chatting away.

"After we take our graduation photos tomorrow, we're officially on break. Then we come back on the 6th to check the exam rooms."

"Why does it feel like time is passing so fast? Senior year is almost over in a blink."

"What are your plans for summer vacation?"

"That goes without saying, I'm going out to have fun. I'm planning to stay at my brother-in-law's place for a while."

Li Wenhui lay on his bed, listening quietly without joining the conversation. It wasn't until he saw Xu Ming enter that he chimed in.

"Xu Ming, what are you planning to do during the summer break?"

Ever since he was woken up by Xu Ming before 4:00 AM once and gave up after just one day, he had developed a deep sense of admiration and even reverence for Xu Ming. Sometimes, in the classroom, the cafeteria, or the dorm, he would proactively greet him and humbly ask for help with math problems.

"Yeah, Xu Ming, with your grades, you're a lock for Yenching University or Tsinghua University. Do you have any plans for the summer?" the boy who had mentioned his brother-in-law's place added.

Facing this sudden question, Xu Ming didn't hesitate for even half a second. He sat down on the edge of his bed and said, "The vacation is so long, I'm definitely going to get a head start on university-level studies."

As soon as he finished speaking, the room fell into a dead silence. Beside him, Qian Hong silently raised his signature thumbs-up.

For others, studying might be a painful task, but Xu Ming found that he enjoyed the sense of accomplishment it brought.