National Mathematical League Finals - Second Half

"The answer to the last question, I also wrote √3/2," Shen Qi said.

"So awesome!" Qi Jianhong exclaimed excitedly.

"It's okay, Jianhong, not that awesome," Shen Qi chuckled. This kid had been blindly admiring me since the National Preliminary. Sigh, it's so troublesome. The burden on a great figure is heavy.

Qi Jianhong explained, "Oh no, no, I mean I'm so awesome."

Shen Qi was taken aback. "What the hell are you so awesome about?"

"The National Final paper, I actually got one question right. Brother, do you think I'm awesome or not? My answer is the same as yours, √3/2. That means I got it right." Qi Jianhong was quite smug.

"You're so annoying. Go away, go cool off somewhere else." Shen Qi picked up the Mathematical Herald again and continued his research.

Qi Jianhong asked again, "You must have gotten all three questions from the first half of the National Final right, right?"

"Just did them casually, depending on my mood."

"Brother, you're solid!"

Afterward, the other team members gradually returned to the dormitory. They all discussed amongst themselves, generally agreeing that the National Final paper was too difficult and that getting even one question right was cause for celebration.

Teacher Tian hadn't set any strict targets for the team members. He had heard that a participant from Zhedong Province, a strong province in math competitions, had fainted during the exam and, unable to continue answering questions, had to give up. The student cried so much that they fainted again right after waking up. The Chinese Mathematical Society had even hired a psychological counselor to intervene with this contestant.

At least the psychological fortitude of our province's team members was up to par; they weren't the so-called "glass hearts." Teacher Tian was very pleased. He addressed the team members: "This trip to the Capital is about to end. Tomorrow, there are only three questions left. I hope you will see it through to the end and gain something, leaving no regrets in your youth. No matter how well you study math, it might not make you rich and famous, but you must remember that the only language to communicate with God is mathematics."

Shen Qi raised his bottle of mineral water and spoke passionately, "Teacher Tian said it perfectly. Come, let's drink to this bowl of chicken soup!"

"Cheers!"

"Cheers! This chicken soup is all-natural, green, and non-toxic!"

The team members raised their cans of Coke and Sprite, all drinking in one gulp, their spirits soaring.

"You bunch of brats..." Teacher Tian said, a mix of amusement and exasperation. He liked this kind of atmosphere within the team, so different from a few days ago.

The stars shifted and the heavens changed, the moon set, and the sun rose in the east.

Time flies. In the blink of an eye, the second day arrived.

As mentioned earlier, the National Final was divided into two halves. Yesterday's three questions in 4.5 hours constituted the first half. Today's second half also had 4.5 hours to solve the remaining three questions.

This was a grueling battle, testing not only the contestants' mathematical prowess but also their unwavering willpower and psychological resilience. Most of these contestants were not yet legally adults.

Everything was nearing its end, and the results were about to be revealed.

The exam papers for the second half of the National Final had been distributed to each contestant. Shen Qi calmly began the test.

The exam papers from the first half of the National Final had been sealed yesterday. They would be unsealed after 4.5 hours, and together with the second-half papers, all six questions would be graded.

Shen Qi was confident that he would score highly on yesterday's three questions, but the battle was not over. He had to fight to the very end, allowing no room for carelessness.

Outside the examination hall, Shen Qi was lively by nature. But once his pen touched paper and the exam was spread out, he immediately entered a state of combat. Come on, let's fight!

The first question of the second half of the National Final asked for the complex roots of P(x+iy).

Shen Qi pondered for a moment and began to write.

Integers, fractions, irrational numbers, negative numbers, and complex numbers. The world of number systems is simple. The world of number systems is complex.

Except for complex numbers, which exist in a state of semi-reality on paper and in academic research, the other number systems are used daily by ordinary people. Mathematics, seemingly ethereal and profound, is in fact the most widely applied foundational discipline in society.

Mathematics can be used for grocery shopping, accounting, stock trading, and financial management. It can also serve as the sole language to communicate with God and explore the vast universe. It is lofty, and it permeates every corner of society.

Proof and computation based on pure number systems are the purest form of Algebra. Although most mathematicians prefer Geometry, Algebra still holds its important position.

What are the complex roots of P(x+iy)?

Where do they come from, and where do they go?

The first university textbook Shen Qi taught himself was Advanced Algebra. He liked Cauchy, but he also found Cauchy troublesome.

No matter which country's rankings of great mathematicians in history you look at, Cauchy definitely holds a place. He is absolutely a template for Level 15 reference, although the system's sample was Gauss.

The reason Shen Qi likes Cauchy is that Cauchy, single-handedly, pushed Algebra forward. His contributions to Algebra are unparalleled.

The first question of the second half of the National Final would inevitably require the use of Cauchy's Theorem.

Shen Qi quickly found the product of the differences between the two roots, known in algebraic language as the discriminant. It is a sharp blade; polynomials and linear combinations of derivatives are no match for it, crumbling before it.

P(x+iy) is a timid coward, hiding behind the polynomial of x, relying on the "discriminant is not zero" defense tower to wear down Shen Qi's minion waves.

"Heh heh, you scumbag, you think I don't dare to dive tower and kill you? Heh heh, you're too naive, P(x+iy)."

Shen Qi unleashed his ultimate move with great flourish. He charged under the "discriminant is not zero" defense tower, protected by the "d'Alembert's Rule" shield, working in conjunction with Cauchy's Theorem. He furiously tore P(x+iy) apart into u(x,y) + iv(x,y), cleanly and efficiently, retreating without a scratch.

Under Shen Qi's powerful and sharp offensive, P(x+iy) instantly lost its ability to resist. It honestly handed over its roots: a+bi.

The first question of the second half of the National Final was conquered.

Once you have the advantage, you must press it. In math competitions, you can never be soft-hearted.

After Algebra, Geometry must follow.

The second question was an Analytic Geometry problem, similar to the order of yesterday's questions.

Plane Analytic Geometry in high school is the foundational part of Coordinate Geometry. The patterns in the coordinate system look like ripples or butterflies. Symmetry has a harmonious beauty, while asymmetry has a dynamic aesthetic.

The seemingly simpler a girl is, the more difficult it often is to conquer her, because the conditions she provides are stringent.

Shen Qi spent a full hour thinking here. He could draw the conchoid, the quadratrix, and even the lemniscate. Each curve in the coordinate system represented a meaning, corresponding to an answer.

Shen Qi had to pass through the coordinate mist as quickly as possible, capturing that most beautiful and correct slender curve.

"Yes, that's right, the logarithmic spiral."

Shen Qi finally began to draw. He encountered logρ=aθ, a curve resembling both a conch and a snail. It turned and turned, spiraling round and round, eventually passing through the origin, the intersection of the x-axis and y-axis.

There are thousands of beautiful appearances, but ultimately, there is only one destination: the beautiful girls in the coordinate system – curves. No matter how demanding they are, they will eventually pass through the origin and return to their simple, unadorned beginnings.

"Done!"

Two and a half hours passed. Shen Qi had completed the solutions for the first two problems in the second half of the National Final. Including the three problems from yesterday, he had completed a total of five.

"Hoo..." Shen Qi took a deep breath and rested for a moment. A few months ago, he wouldn't have dared to imagine that he would have the chance to participate in China's top high school mathematics competition, becoming one of the TOP 60. Oh no, no, one person fainted and withdrew, so he was one of the TOP 59.

At this moment, Shen Qi was only one small step away from potentially touching his first gold medal, a national-level mathematics gold medal.