Did a god write this question?

Halfway through the algebra problem, Shen Qi encountered a situation that made him slightly nervous.

The Indian-American contestant stood up and made a gesture indicating he was finished.

The proctor walked over and placed his exam paper into a sealed envelope.

The Indian-American contestant left the examination hall with ease, appearing completely relaxed.

"Don't be influenced by the Indian-American. Let him be arrogant; the wind blows over the mountain."

Shen Qi refocused on his paper, completing the unfinished infinite sequence of standard orthogonal elements.

A few minutes later, the very handsome Russian contestant submitted his paper.

"Let him be; the bright moon shines over the river. There's no need to race them. I want absolute points, not speed."

Shen Qi finally completed his proof. He wasn't in a hurry to submit, instead checking and rechecking his work. He waited the full 4.5 hours before handing in his paper.

The first day of the IMO competition concluded. All contestants' papers were sealed and would not be opened until this time tomorrow.

The six members of the Chinese Math Olympiad team immediately returned to the hotel for a debrief. They shared information, with the deputy leader and coaches acting as strategists to offer advice.

"How did the first day of competition feel?" the deputy leader asked the team members.

"Difficult."

"Quite difficult."

"Harder than last year."

"It was indeed a bit difficult. I've done the past five years of IMO past papers. This year's first logic problem aside, the analytical geometry and algebra problems were the hardest in these five years," Shen Qi stated truthfully. He then added, "And I noticed that some individual contestants from certain countries have strong individual capabilities, like the Indian-American contestant from the US team, and the left-handed Russian contestant."

The deputy leader pulled out the seating chart and looked. "The Indian-American contestant from the US team is named Al-Razi. This is his second time participating in the IMO. Last year, he got a perfect score and tied for the individual championship. Yes, that's right, I remember this Indian kid. I led the team last year too. Our Chinese team narrowly won the team championship by 2 points. Although the six team members won five gold medals and one silver, none of them achieved a perfect score."

"This Indian kid is that good?" Shen Qi found the Indian contestant he competed with to be unremarkable, even a bit scruffy. He didn't look like a math prodigy at all.

The handsome Russian youngster, however, did put immense pressure on Shen Qi. Primarily because the Russian contestant was so handsome, with his sapphire-blue eyes, straight nose, and a head of soft golden hair that was breathtakingly beautiful. Shen Qi felt he couldn't compete with him in terms of looks, nor was his hairstyle as cool. The pressure was immense.

The deputy leader said, "India's level of mathematics is among the best in the world. Shen Qi, you should have heard of the Ramanujan Prize. It's one of the world's top awards, second only to the Fields Medal, named after the Indian mathematician Ramanujan."

Shen Qi nodded. "I've heard of it. Ramanujan was an incredible mathematician. The Ramanujan Conjecture series is quite advanced; I haven't even fully grasped them all."

"Shen Qi, the Russian contestant in your competition is named Peter Poluboyarinov. Don't be fooled by his appearance. This handsome Russian youngster is a recognized mathematical genius. He has participated in the IMO three consecutive times, achieving a perfect score in the previous two. Peter has already been accepted into one of the world's top three mathematics programs, ENS Paris. Since he hasn't reported to France yet, he is eligible to participate in this year's IMO."

"Undoubtedly, Peter, participating in the IMO for the last time, is aiming for a third consecutive perfect individual score. Compared to a multinational force like the United States, I actually have higher hopes for the Russian team this year. If Russia's other contestants perform well and cooperate with their ace player Peter, they might very well snatch the team championship from us," the deputy leader said, his expression becoming grim.

Shen Qi's expression also turned serious. "The Russians are brilliant at mathematics. The only one of the seven Millennium Prize Problems to be solved was the Poincare Conjecture, completed by the Russian mathematician Perelman. In fact, I often use Lobachevsky's Non-Euclidean Geometry theories. Coming from a non-traditional background, I never received systematic instruction in Non-Euclidean Geometry. Lobachevsky is my enlightenment teacher in Non-Euclidean Geometry. I respect him greatly, even though he passed away over a hundred years ago."

"Because of Peter and Al-Razi's presence, it won't be easy for any one of you to win the individual championship," the deputy leader said to the six team members. "Therefore, we must unite. While ensuring the team championship, we must fully utilize our individual strengths."

What the deputy leader said was essentially stating the obvious. Calculating the team ranking based on total scores, in essence, is the accumulation of the individual strengths of the six contestants. Ensuring the team championship requires that there are no weak links within the team and that one or two ace players lead the charge to boost the total score.

Although the Chinese Math Olympiad team had won three team championships in the past five years, the deputy leader knew clearly that they lacked those exceptionally strong individual ace players. Yes, they had four 2s, a joker, and a straight, but they were missing the King.

Back in his hotel room, Shen Qi went online to check some data.

The data told him that achieving a perfect score in the IMO was not easy. The number of perfect scorers in a single IMO is usually in the single digits.

There was a clear pattern: perfect scorers either didn't get a perfect score, or once they did, they kept getting them consecutively.

The three most famous contestants in the history of Chinese Math Olympiad competitions all fit this international norm. All three participated in the IMO for two consecutive years, achieving perfect scores both times, and were hailed as prodigies. Two of these prodigies were admitted to Yanjing University, and one to Zhejiang University.

Shen Qi would graduate from high school next year. He could participate in at most two more IMOs, making it impossible to match the Romanian contestant's world record of "three consecutive perfect scores."

Shen Qi didn't want to leave any regrets. Besides the team championship, he also wanted to achieve the highest gold medal, the individual championship.

Following the international norm, the Indian-American contestant from the US team achieved a perfect score last year, and the handsome left-handed Russian achieved two consecutive championships last year and the year before. These two strong rivals were highly likely to achieve perfect scores again this year.

Therefore, to become the top individual gold medalist at this IMO, Shen Qi had to ensure a perfect score.

The next day was the second day of the IMO competition.

Before the competition began, Shen Qi exchanged glances with Peter Poluboyarinov, the Russian contestant in the same examination hall.

"Hey, man," Shen Qi greeted Peter Poluboyarinov proactively.

Peter Poluboyarinov nodded and said, "Privet." He spoke in Russian.

Shen Qi asked, "Can you speak English?"

Peter Poluboyarinov shook his head, indicating he couldn't.

Oh, so the Russian prodigy was terrible at English, not even as good as me. Shen Qi was helpless. He didn't understand Russian either, so communication was impossible.

The handsome Russian ignored Shen Qi and moved a few steps away, embracing the Indian-American contestant. They conversed in French, appearing quite familiar with each other.

Shen Qi walked away, uninterested. Those two, one handsome and one scruffy, were the co-individual champions last year. Shen Qi had no achievements on the international stage, so they didn't include him.

Fine if they don't include me. I just can't score lower than you.

Shen Qi adjusted his mindset, preparing for the challenge.

At 9:00 AM sharp, the second day of the IMO competition began on time. The contestants would face the remaining three problems.

After receiving his exam paper, Shen Qi first quickly scanned the three problems.

"This last problem... seems monstrous. I'll worry about it later and focus on finishing the first two."

Shen Qi had to strategize his approach. He had already prepared for the worst-case scenario: this last problem might be set by a god, and he might not even understand it.

He didn't dare to hope for a perfect score; the complex international situation caught Shen Qi off guard.

There was definitely psychological pressure, but thankfully, the solutions to the first two problems went relatively smoothly. Two hours later, Shen Qi had to face the ultimate challenge of this IMO.

The final problem started with just a pile of numbers:

1=1

196884=196883+1

21493760=21296876+196883+1

864299970=842609326+21296876+2*196883+2*1

Based on the pattern above, please provide two different mathematical explanations. (7 points)

This pile of numbers looked like the opening scene of The Matrix; what was their pattern?

Shen Qi's first task was to find the pattern and then explain it in mathematical language.

At first glance, there seemed to be a pattern; the next line always had numbers overlapping with the previous one. However, the problem was the insufficient sample size. Shen Qi couldn't explain this array of numbers using any mathematical language. It was definitely not Pascal's Triangle; it had equals signs.

Shen Qi pressed hard on his temples, worried.

Not far away, the Russian competitor Peter Boraslavsky and the Indian-American competitor Al-Razi didn't look worried. They were flipping through their own reference materials.

Shen Qi had also brought three reference books. He flipped through them, but the results were disappointing; flipping through the books was useless.

Within Shen Qi's knowledge base, he didn't know which book recorded a similar array of numbers. Could this be the latest research finding?

Peter Boraslavsky and Al-Razi were answering their papers while flipping through books, making Shen Qi even more worried. Perhaps the curricula in the US and Russia were different from China's; had they learned about this?