Three Methods

Shen Qi took the white paper handed over by Cadre Liu. "Alright, as someone new here, I'll do a problem to calm my nerves first."

"You have one hour. I'm going out to eat." Cadre Liu stood up and checked his watch. It was past five, time for dinner.

"I'm hungry too." Speaking of food, Shen Qi was indeed hungry.

"If you finish this problem within the time limit and get the answer correct, you'll be allowed to eat dinner," Cadre Liu said expressionlessly. He added, "This is the first rule of the Olympic Math national team."

"Cadre Liu, treating a minor like this is not permissible by law!" Shen Qi protested vehemently. Not being allowed to eat if he couldn't solve the problem? How cruel! This was a socialist country governed by law! This shock was too much; he couldn't calm down.

"Shen Qi, you are a gold medalist in math competitions, a national champion, not an ordinary minor." Cadre Liu was unyielding. He gestured to the corners of the twenty-square-meter office. "During your problem-solving, you can consult any mathematical literature in this room. Many are not for sale; the Chinese Mathematical Society acquired them through special channels. You should feel fortunate to be swimming in an ocean of knowledge."

"As a high school student from out of town, I have no relatives or friends here. What can I do? Whatever you say, Cadre Liu, you're the boss, I have no choice but to submit." Shen Qi had no other option; he was the disadvantaged party.

Cadre Liu then left, leaving behind a cold silhouette.

Shen Qi pulled over a chair, spread the white paper on the desk, and carefully examined the problem.

The first thing that caught his eye was a simple diagram: a straight line passing through an ellipse, resembling a flattened "日" character.

At the 12 o'clock, 3 o'clock, and 9 o'clock positions of the "日" character, y(x), x2, and x1 were marked respectively.

Below that was the text:

"Based on the diagram above, the unknown function appearing in the integrand causes the integral to reach a maximum or minimum. The specific description is the shape that a rotating surface must have to minimize resistance when moving axially at a constant speed. What is its mathematical description?"

The problem wasn't complicated. Everyone could recognize the diagram and the Chinese characters. However, it could be conservatively estimated that over 99% of high school students in China wouldn't understand what the problem was asking them to do.

"Heh, heh heh heh." Shen Qi laughed, like an idiot.

He laughed foolishly because he could eat dinner now.

This problem wasn't difficult. For Shen Qi, it could be summarized in one sentence: describe simple physical motion in mathematical language.

This required the solver to have a solid foundation in mathematics, while physics knowledge only needed to cover the scope of required high school textbooks, including Newton's three laws.

Shen Qi picked up his pen and wrote his mathematical description on the A4 paper.

To put it simply, the problem was like this: Cadre Liu casually tossed a ball onto a lake. The ball bounced on the surface and eventually floated. Explain this common phenomenon in mathematical language.

Shen Qi applied Euler's brachistochrone theory, which involved using a finite sum to replace the integral and finite differences to replace the derivatives in the integrand.

This not-so-profound theory had extremely wide applications. Its function was to transform an integral into a function composed of a finite number of coordinates of y(x).

Shen Qi drew an arc on the white paper with a very concise technique.

This operation wasn't difficult; any engineering student who scored eighty or ninety on linear algebra could do it.

Shen Qi continued his operation. By setting the variation of the integral to zero and transforming the resulting difference equation through a rough limiting process, he obtained the conditions that the minimized arc must satisfy.

Shen Qi's solution filled one side of the A4 paper. Done.

Checking the time, only 15 minutes had passed. Cadre Liu hadn't returned yet. Shen Qi, with nothing else to do, flipped the A4 paper over and continued writing.

This time, Shen Qi's operation was slightly more complex. Maximizing or minimizing an integral didn't necessarily require using finite differences.

Euler's method was merely introductory; Lagrange's approach was advanced.

Shen Qi sketched a crescent moon on the paper and derived the conditions that a curve with variable endpoints must satisfy for minimization. This process was more convoluted.

It took 20 minutes for Shen Qi to provide the second mathematical explanation.

Seeing that there was still time, Shen Qi casually grabbed a new sheet of paper and began deriving a third mathematical explanation.

At 6:20 PM, Shen Qi was unbearably hungry. He placed his pen on the desk, unzipped his backpack, and searched for food. "Why isn't Cadre Liu back yet? I'll have a chicken leg first."

Tearing open the sealed packaging, Shen Qi ate the chicken leg while pacing around the office.

"This truly is an ocean of knowledge. So many books! Some literature can't be found elsewhere."

"Opera-Omnia, L. Euler... What the heck is this? A book entirely in English... Huh, wait, I think I recognize the author. Euler. This is Euler's book... Damn, I can't understand any of it because it's all in English. Let's switch."

"Mei kan ni ke... Anna lai di ke... Holy crap, this one is even more impressive, a French book! The author is... Lagrange. Damn, I can't understand any of it because it's all in French. Let's switch."

"Ugh... there are even Russian versions of literature... Is Cadre Liu a foreign language expert? There are hardly any Chinese mathematical texts in this room."

Shen Qi was stunned. Cadre Liu's office was an ocean of mathematical knowledge, but he could only understand a few books. Cadre Liu's collection consisted almost entirely of foreign language reference materials.

Shen Qi was very interested in the reprints of Euler's and Lagrange's mathematical works. Such precious mathematical literature couldn't be bought in bookstores outside or on Taobao, unless you imported them yourself.

However, no matter how interested he was, it was useless if he couldn't understand foreign languages.

"Should I upgrade my English?" Shen Qi stood in front of a wall of books, gnawing on a braised chicken leg, lost in thought.

Just then, Cadre Liu returned. Seeing Shen Qi not engrossed in his work but eating a chicken leg, he was very displeased. "Shen Qi! Who allowed you to eat a chicken leg? You're violating team rules on your first day reporting to the national team! Your organizational discipline is too poor!"

"I'm hungry! I'm growing, Cadre Liu!" Shen Qi cried out, feigning innocence.

Cadre Liu said resolutely, "Hunger is no excuse. Being stupid is far more tragic than starving! Throw away the chicken leg. You're not allowed to eat until you've finished the problem!"

"Tyrant!" Shen Qi shouted. Reluctantly, he threw the half-eaten chicken leg into the trash can, muttering, "Tyrant, Dark Tyrant..."

"Wait, that's not right! I finished the problem Cadre Liu gave me, so why should I throw away the chicken leg!" Shen Qi suddenly realized.

"You finished it and didn't say anything?" Cadre Liu walked to the desk and bent over to examine the white paper with the problem.

"I was starving and lost my mind. I shouldn't have tried to farm the Dark Tyrant at level 5," Shen Qi grumbled on the ground, sulking. "This chicken leg was bought by my dad, full of fatherly love. Damn it, I feel sorry for my dad."

"What Dark Tyrant nonsense are you talking about?" Cadre Liu frowned and continued to review Shen Qi's solution. "Euler's second-order ordinary differential equation. You got it right. An answer like this within an hour is standard."

"Hmph..." Shen Qi grumbled unhappily, looking down.

"There's writing on the back too?" Cadre Liu flipped the white paper over and was instantly astonished. "Lagrange's indefinite integral formula. Shen Qi... you... you solved my problem using two methods... within an hour?"

Cadre Liu clutched his chest with his left hand, sweat beading on his forehead.

"Hmph... there's another piece of paper next to it." Shen Qi squatted on the ground, head down, drawing characters in the air with his finger. He wrote the character for "miserable."

"You... Shen Qi... you even know how to use Bernoulli's divergent series... You solved my problem using three methods, within... within an hour?" Cadre Liu panted heavily, his speech faltering.

"Who could be more miserable than me? I'm only allowed to do problems, not eat," Shen Qi muttered to himself.

Thump!

A dull thud.

Cadre Liu collapsed to the ground.

"Holy crap?" Shen Qi looked up sharply, seeing Cadre Liu lying on the floor, convulsing violently.

"Cadre Liu, what's wrong? You can't die!" Shen Qi rushed to Cadre Liu's side, stunned.

Cadre Liu's hand trembled as he pointed to a corner of the office. "Quick... go get... the fast-acting... heart medication... in the briefcase..."