Doing My Part

"The difficulty of Shen Qi's question lies not in providing the answer, but in proving the answer," said Vice President Tan.

Cadre Liu continued, "That's right. The two-dimensional case of this problem is very simple. We can consider the outermost 4n-1 points. If there is no line x=n or y=n, then each line passes through at most two of these 4n-1 points, so at least 2n lines are needed. If there is a line x=n or y=n, it's not difficult to find the answer, which is still 2n. Therefore, the final result in the two-dimensional case is 2n."

"However, problems arise when we extend this method to three dimensions. In three dimensions, we are covering planes instead of lines. Planes have three free variables, and it is not easy to select landmark points for examination. Nevertheless, we can still quickly determine that the final result in three dimensions must be 3n. Any high school student with competition experience can intuitively deduce the answer of 3n. However, it is very difficult to rigorously prove 3n planes." Cadre Liu looked at the blackboard and shared his views.

"The conclusion is obvious, but proving it is incredibly difficult." Cadre Kong walked to the blackboard, took off his reading glasses, and re-examined the problem. "Everyone, how do you think we can prove that at least 3n planes are needed?"

"A logic game," Shen Qi said with a smile, picking up the chalk again to prepare a proof.

Cadre Liu rolled up his sleeves, seemingly already having an idea. "The key to this problem is transforming what appears to be a combinatorial calculation into a pure algebraic problem. Once you break free from this mental limitation, the possibilities open up."

"As expected of Cadre Liu, it seems there's nothing for me to do." Shen Qi wouldn't forget that Cadre Liu was his algebra coach when he was in the math Olympiad training team.

"Shen Qi, we'll accept this question. Thank you. It's an excellent math Olympiad problem," said Vice President Tan.

"Alright, I'll take my leave now. There's an exchange meeting upstairs that I need to attend." Shen Qi left the conference room and went to the academic lecture hall upstairs.

Several young mathematicians presented their latest research findings, and Shen Qi, as one of the guests, provided commentary.

After the academic conference ended, Shen Qi strolled downstairs and found Vice President Tan, Cadre Liu, and Cadre Kong still in the conference room.

The blackboard was covered with mathematical symbols. Shen Qi quickly scanned it and found that the math problem he had set had been perfectly proven.

Shen Qi praised it highly, "A very elegant proof. Representing each plane as a ternary linear polynomial, and then multiplying these polynomials together, only one point needs to be proven: the resulting polynomial has at least degree 3n. Well done, Cadre Liu."

However, Cadre Liu tapped his watch with a worried look on his face. "Old Tan, Old Kong, and I spent two hours completing the proof. We don't believe any high school student could prove 3n within the allotted time."

"You don't know until you try it, and once you do, you're shocked. This problem is at the IMO's final level, so it's not really suitable for the CMO competition," Vice President Tan summarized.

"If no Chinese math Olympiad contestant can prove this problem in the national finals, it would be a huge blow to the enthusiasm of young math enthusiasts." Cadre Kong felt the problem was far too difficult and suggested lowering the difficulty level.

"You guys just said you'd accept this question." Shen Qi was speechless. You guys really underestimate contemporary Chinese high school students. You can't even imagine the terrifying strength of those born in the 2000s.

"Shen Qi, after thoroughly proving this problem, I realized it's full of traps!" Cadre Liu frantically rummaged in his pocket for something.

Shen Qi exclaimed in surprise, "Cadre Liu, are you having another heart attack? Okay, okay, let's change the question, okay? Let's lower the difficulty a bit!"

As a result, Cadre Liu pulled out an electronic cigarette, took a couple of puffs, and felt much better. "Since you went abroad, my heart condition hasn't flared up again, and it's been under quite good control."

"What are you saying, Lao Liu? In your mind, I'm probably just a walking illness, right?" Shen Qi chuckled wryly.

After some thought, Vice President Tan proposed a compromise. "The problem set by Shen Qi is indeed of a very high level. Only by pushing mathematical skills to the limit and breaking conventional thinking patterns can high school students possibly provide a correct proof. Let's do this: in accordance with the principle of selecting top-level math Olympiad contestants, we will include this math Olympiad problem in the CMO national finals. No points will be deducted for incorrect answers, and points will be added for correct answers."

"Feasible."

"Safe."

Shen Qi, Cadre Liu, and Cadre Kong all agreed with Vice President Tan's proposal, so this seemingly simple but actually extremely difficult question was listed as an optional question for this year's CMO national finals.

Shen Qi made a bet with Lao Tan, Lao Liu, and Lao Kong: if fewer than five high school students in all of China could solve this optional question, then Shen Qi would consider himself to have lost.

"You're very likely to lose, let's bet!" Old Tan, Old Liu, and Old Kong accepted the challenge, the stakes being a meal.

With things mostly settled in the capital, Shen Qi left for Nangang City. His other identity, despite his busy schedule, was a specially appointed professor of physics at Nangang University of Technology.

Having been given a nominal position, Shen Qi was required to spend some time at Nangang University of Technology throughout the year. He went to the Nangang University of Technology and approached Academician Shi: "Academician Shi, I'll be staying in China for another ten days or so this time. Is there anything your Physics Department needs me to do?"

Academician Shi, who is also the dean of the School of Physics, said, "You can't accomplish anything big in just over ten days. Our university doesn't have any strict performance indicators for you. Why don't you go around and point out some flaws in our School of Physics, School of Mathematics, or other schools? Give us some suggestions for improvement regarding shortcomings in research, teaching, logistics, and all other aspects."

"No problem offering suggestions for improvement, Academician Shi. May I borrow some equipment from our Academy of Physics' astronomical observatory?"

"Of course," Academician Shi immediately agreed, then added, "But recently, the main equipment at the Physics Institute's Astronomical Observatory has been undergoing maintenance. If you can stay for a few more months, that would be fine."

"What a pity, it seems I'll have to wait until next time." Shen Qi had originally planned to play with the Radio Telescope at Nangang University of Technology, but it seems he won't be able to this time.

Then let's offer some suggestions for improvement to the School of Physics at Nangang University of Technology. We should at least contribute to the field of physics at this university; otherwise, what if the system doesn't recognize the task as completed?

Shen Qi arrived at the university town wearing a mask. With the flu rampant recently, quite a few people were wearing masks.

Most universities in Nangang City have campuses in the University Town. Shen Qi arrived at the Nangang University of Technology campus in the University Town, entered the teaching building, sat in the last row, and listened to a class on "Quantum Mechanics" for physics majors.

After attending only half a class, Shen Qi discovered many problems, which he wrote down one by one and pondered solutions.