Shen Qi had just finished a little over half of the third problem, an algebra question, when something that made him slightly nervous happened.
An Indian-American contestant stood up and made a gesture indicating he was turning in his paper.
The proctor walked over, placed his exam paper in a file envelope, and sealed it.
The Indian-American contestant left the exam room with an easy air, looking completely unbothered.
"Don't let that Indian-American throw you off. Let him do his thing; the breeze will brush the mountain ridge."
Shen Qi turned his attention back to his exam paper and finished the infinite sequence of standard orthogonal elements he had yet to complete.
A few minutes later, the very handsome Russian contestant turned in his paper.
"Let him do his thing; the moon shines over the great river. There's no need to race them. I'm after the highest possible score, not the fastest time."
Shen Qi finally completed his proof. He wasn't in a hurry to turn in his paper, though. He checked it again and again, waiting until the full four and a half hours had passed before handing it in.
The first day of the IMO was over. All the contestants' papers were sealed away, and wouldn't be opened until this time tomorrow.
The six members of the Chinese math team returned to the hotel and gathered to share what they knew. The deputy team leader and the subject coaches acted as strategists, helping them plan their approach.
"How did you find today's competition?" the deputy team leader asked the contestants.
"Hard."
"Pretty hard."
"Harder than last year."
"It was definitely a bit hard. I've done nearly five years' worth of past IMO problems. Leaving aside today's first logic problem, the analytic geometry and algebra problems that followed were the hardest in those five years." Shen Qi gave his honest assessment, then added, "I also noticed that a few contestants from certain countries are formidable on their own. For example, that Indian-American on the U.S. team and the left-handed contestant on the Russian team."
The deputy team leader pulled out the seating chart and took a look. "The Indian-American on the U.S. team is called Al-Razi. This is his second time competing in the IMO. Last year, he scored full marks and tied for the individual championship. Right, that's it. I remember the Indian kid. I led the team last year too. China beat the U.S. by two points to take the team championship. Although our six contestants brought home five golds and one silver, not one of them got full marks."
"That Indian kid is that good?" Shen Qi thought the Indian contestant he'd been competing alongside looked unremarkable, even a little shifty. He didn't look like a math genius at all.
That handsome young Russian, on the other hand, had made Shen Qi feel a lot of pressure. The main reason was that Russian contestant was so good-looking: vivid blue eyes, a straight nose, and a head of silky blond hair that was almost unfair. Shen Qi knew he couldn't match the guy's looks or his flashy hairstyle, and it was stressing him out.
The deputy team leader said, "India's mathematics is also highly regarded worldwide. Shen Qi, you've probably heard of the Ramanujan Prize. It's one of the world's most prestigious awards, second only to the Fields Medal, and is named after the Indian mathematician Ramanujan."
Shen Qi nodded. "I've heard of it. Ramanujan was incredible. The Ramanujan Conjecture is pretty advanced stuff. I haven't mastered all of it myself."
"Shen Qi, the Russian contestant in your room is called Peter Vladislavsky. Don't be fooled by his looks. This handsome young Russian is a widely recognized math genius. He's competed in the IMO three times in a row and scored full marks in his first two appearances. Peter has already been admitted to ENS Paris, one of the world's top three institutions for mathematics. He hasn't gone to France to enroll yet, so he's still eligible to compete in this year's IMO."
"Without a doubt, Peter intends to win full marks for the third time in a row in his final IMO. Compared to the U.S., which is a multinational force, I actually have more faith in this year's Russian team. If the other Russian contestants perform well and back up their star player, Peter, they might very well snatch the team championship from us." As the deputy team leader spoke, his expression grew increasingly grim.
Shen Qi's expression was solemn as well. "The Russians are strong in mathematics. Of the seven Millennium Prize Problems, the only one that's been solved is the Poincaré conjecture, and it was solved by the Russian mathematician Perelman. In fact, I often use Lobachevsky's theory of Non-Euclidean Geometry. I never received formal instruction in the subject, since I learned math outside the usual system. Lobachevsky was the one who introduced me to Non-Euclidean Geometry. I have a lot of respect for him, even though he died more than a century ago."
"With Peter and Al-Razi competing, it won't be easy for any of you to win the individual championship," the deputy team leader said to the six contestants. "So we have to stick together and do our best individually, while making sure we win the team championship."
What the deputy team leader had said was really just stating the obvious. The team rankings were based on total scores, which were simply the sum of the six contestants' individual scores. To ensure the team championship, the team couldn't have any weak links, and it needed one or two star players out in front, boosting the total score.
Although the Chinese math team had won three team championships in the past five years, the deputy team leader knew they were lacking a star contestant with truly exceptional individual ability. Yes, they had four twos, a Small Joker, and a straight. All they were missing was the Big Joker.
Back in his hotel room, Shen Qi looked up some statistics online.
The statistics told him that getting full marks at the IMO was no easy feat. The number of contestants who got full marks at any given IMO was usually in the single digits.
There was a clear pattern: contestants either didn't get full marks, or they got them year after year.
The three most famous contestants in the history of Chinese math competitions all followed this international pattern. Each competed in the IMO twice in a row, got full marks both years, and earned the title of "prodigy." Two of the three prodigies were admitted to Yenching University without taking the entrance exam, while the other was admitted to Zhejiang University.
Shen Qi would graduate from high school next year, so he could compete in the IMO at most twice. There was no way he could match the Romanian contestant who had set the world record for three consecutive years of full marks.
Shen Qi didn't want to leave himself with regrets. Besides the team championship, he wanted even more to win the gold medal and take first place—the individual championship.
The same international pattern applied here. The Indian-American on the U.S. team had scored full marks last year, while the left-handed Russian heartthrob had won the championship two years in a row. It was highly likely that both formidable opponents would get full marks again this year.
So if Shen Qi wanted to take first place in the individual rankings at this year's IMO, he had to make sure he got full marks.
The next day was the second day of the IMO competition.
Before the competition began, Shen Qi crossed paths with Peter Vladislavsky, the Russian contestant in the same exam room.
"Hey, man." Shen Qi took the initiative and greeted Peter Vladislavsky.
Peter Vladislavsky nodded. "Pru-pu-lu-ka-bu-lu." He was speaking Russian.
Shen Qi asked, "Can you speak English?"
Peter Vladislavsky shook his head to indicate that he couldn't.
Oh. So the Russian prodigy was terrible at English—worse than me, even. Shen Qi was at a loss. He didn't know any Russian either, so they couldn't communicate.
The Russian heartthrob stopped paying attention to Shen Qi. He moved a few steps away, embraced the Indian-American contestant, and chatted with him in French. They seemed quite familiar with each other.
Shen Qi wandered off, bored. The handsome one and the plain one had tied for the individual championship last year, while Shen Qi had no record at all in international competitions. They weren't going to include him.
Fine, then. I just couldn't score lower than them.
Shen Qi got himself into the right frame of mind and prepared for the challenge.
At exactly nine o'clock, the second day of the IMO began. Each contestant would face the three remaining problems.
After getting his exam paper, Shen Qi quickly scanned the three problems.
"This last problem... looks pretty insane. I'll leave it for now and tackle the first two."
Shen Qi had to plan his strategy carefully. He'd already prepared for the worst: the last problem was probably devised by some god. I can't even make sense of it!
He couldn't even hope for full marks now. The complicated international situation had caught Shen Qi completely off guard.
The psychological pressure was definitely there. Fortunately, he made good progress on the first two problems. Two hours later, Shen Qi had no choice but to face the problem that would anchor this year's IMO.
The last problem began with 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, give two different forms of mathematical interpretation. (7 points)
This pile of numbers looked like the opening scene of The Matrix. What was the pattern?
The first thing Shen Qi had to do was find the pattern, then explain it using mathematical language.
At first glance, there seemed to be some kind of pattern. Each new line had numbers in common with the previous one. But there weren't enough terms for Shen Qi to explain the array of numbers using any kind of mathematical language. It definitely wasn't Pascal's Triangle, since it had equal signs.
Shen Qi pressed hard on his temples, fretting.
Not far away, the Russian contestant Peter Vladislavsky and the Indian-American contestant Al-Razi didn't look worried at all. They were flipping through reference materials they'd brought with them.
Shen Qi had brought three reference books too. He flipped through them, but the result was disappointing. Looking through the books didn't help.
With what he knew, Shen Qi couldn't think of any book that covered an array of numbers like this. Could this thing be a brand-new research result?
Peter Vladislavsky and Al-Razi flipped through their books as they worked on the exam. Shen Qi grew even more worried. Maybe the curricula in the U.S. and Russia were different from China's. Had they studied this stuff?
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