IMO's First Day

"Andrew Wiles, I've actually heard of him. A big name in contemporary mathematics," Shen Qi said, his gaze fixed on Wiles on the podium.

Zhou Yu'an said matter-of-factly, "Duh. A Fields Medalist who proved the super difficult Fermat's Last Theorem. Who in the math world hasn't heard of Wiles?"

"But we didn't know this old man was Wiles!" the two said in unison.

"Does someone that important need to go to the supermarket to buy salad dressing himself?" Shen Qi asked, puzzled. His father, a small business owner, never went to the supermarket; he just told his subordinates what to get.

"Hey, Shen Qi, Zhou Yu'an, what are you two whispering about? Are you guys close with Professor Wiles?" the deputy team leader asked.

Shen Qi shook his head. "Not close, but I know Professor Wiles is a math expert with profound skill. Perhaps the top expert around here."

"You don't say? Be quiet and listen to Professor Wiles read the competition rules. It's very rare for the organizing committee to invite him to be the head of the judging panel." The deputy team leader put his earplugs back in, connected to the simultaneous translation device.

"I never expected that in the same city, there would be weak chickens who can't do subtraction within 10, and also a grandmaster-level figure in the math world. The difference in strength is too vast, like the difference between a weakling and a god." Shen Qi put on his earplugs and listened to Wiles read the competition rules.

"...The competition rules have been read. Children, the last thing I want to say is, 'Where there is number, there is beauty.' We are engaged in a beautiful profession, worthy of our lifelong service." Wiles's final words were a quote from the Greek mathematician Proclus. He had quoted this at the Fields Medal ceremony in 1998 and quoted it again today.

Applause roared through the auditorium. The British, foreigners, Europeans, Americans, Asians, contestants from all countries, coaching staff, and security guards all paid their highest respects to Wiles.

People seemed to have forgotten that Wiles was not the protagonist of this IMO; he was just a judge, routinely reading the competition rules.

But did that matter?

Shen Qi was deeply moved. He realized that reaching the pinnacle of academia brought supreme glory and respect from the world. This glory and respect transcended nationality, skin color, or race.

"I really hope that one day, I can achieve great things like Professor Wiles. What is bringing honor to one's ancestors? This is truly bringing honor to one's ancestors and dominating the world."

Shen Qi felt that this trip abroad had been very rewarding. Some scenes couldn't be truly experienced without being there in person to feel the soul-stirring impact.

Professor Wiles had given a powerful dose of chicken soup to over 600 young contestants from 102 countries. Young talents from Europe, Asia, America, Africa, and Australia were sharpening their swords, ready for battle.

On the morning of July 26th, the first day of the IMO competition began at the University of South Bank, London.

Professor Wiles, the head of the judging panel, had clearly explained the competition rules and schedule. There were a total of six problems; three would be tested today and three tomorrow. Each problem was worth 7 points, for a maximum score of 42 points.

Each competition day had a time limit of 4.5 hours. Any stationery and drawing tools were allowed, but all electronic devices were forbidden in the examination hall.

Since the competition time was long, contestants could bring their own food and drinks and up to three reference books.

Right, it was an open-book exam, with food, drinks, and the ability to flip through books.

It looked like the IMO was quite relaxed, almost like a summer camp.

But was that really the case?

Clearly not.

Shen Qi's food consisted of muffins and bear cookies, his drinks were mineral water and a Red Bull bought from a local supermarket, and his three reference books were "Introduction to Analytical Methods," "The Art of Conjecture," and "Studies in Analytic Geometry and Differential Geometry."

When Shen Qi received the first day's exam paper, he found that his three reference books were basically useless. The most useful items were the bear cookies and the Red Bull.

The three problems on the first day's paper had been translated into Chinese, and Shen Qi had to crack the first one first, as it was a threshold problem.

A line of cold Chinese characters was printed on the paper: "Problem 1, Threshold Problem. Correct answer earns 7 points; incorrect answer or no answer earns -7 points."

This threshold problem was truly insane. If you answered it correctly, you'd get a maximum of 7 points. But if you got it wrong or left it blank, you thought you'd get 0 points? No way.

Look carefully: an incorrect answer or no answer on the threshold problem earns -7 points.

This meant the baseline score for this IMO exam was not 0, but -7 points.

A slight misstep could be fatal.

Traveling all the way to Great Britain only to bring back a negative score, how embarrassing would that be?

The zero egg had finally hatched; he had waited so long for this day.

Shen Qi dared not be careless. He reread the problem statement for the threshold problem for the third time:

"A mathematics professor at Oxford University has three intelligent students: Tom, Jerry, and Thomas."

"One day, the professor wanted to test which of the three students was the smartest."

"The professor placed a slip of paper with a number written on it on each student's forehead. The students could see the numbers on the foreheads of the other two students, but not their own."

"The professor said, 'The numbers on your slips are all positive integers, and the sum of any two numbers equals the third number. So Tom, I ask you, what is the number on your forehead?'"

"'I'm sorry, Professor, I don't know,' Tom replied."

"'Professor then asked Jerry, 'Hey Jerry, what about you? Do you know your number?'"

"'I'm sorry, Professor, I don't know,' Jerry replied."

"'Professor continued to ask Thomas, 'My dear Thomas, it's just you left. What is your answer?'"

"'I'm sorry, Professor, I don't know,' Thomas replied."

"'It's okay. Tom, Jerry, and Thomas, let's try again,' said the professor."

"'The second time, Tom and Jerry were still unable to state their numbers, but Thomas gave his number: 144.'"

"'The professor nodded. 'Congratulations, Thomas, your answer is correct.'"

"'If you have understood the little story above, please tell me, what are the numbers on Tom's and Jerry's foreheads, respectively? ()().'"

The above is the complete text of the first threshold problem. A correct answer earns 7 points; an incorrect answer or no answer earns -7 points.

Shen Qi knew that facing this life-or-death threshold question, Real Analysis, Stieltjes integrals, Cayley transformation matrices, Weierstrass quadratic forms, superelliptic integrals, Abel's theorem... all advanced mathematical formulas, theorems, and corollaries were useless.

This was a purely logical deduction problem, set as the threshold question by the team leaders of 102 countries.

Bringing any reference books would be useless; the most practical resource would be a junior high school math textbook.

The required mathematical knowledge reserve was addition, subtraction, multiplication, and division within 500, along with simple linear equation solving methods.

All that remained were two words: brainpower.

International rules seemed different from Chinese rules. Shen Qi knew without thinking that their Chinese team leader must have voted against this threshold logic question, but the minority obeys the majority, so voting against it was useless.

Chinese students excelled at pattern-based problem-solving, whereas internationally, more emphasis was placed on young people's logical thinking abilities.

Shen Qi fell into thought. Coming to the international stage, he realized that what made him unafraid of any opponent worldwide wasn't profound theories, but the multiplication tables and the simplest linear equations.

He had a slight idea and followed it, deducing three necessary conditions and setting up a system of linear equations.

Next, whether he would get 7 points or -7 points depended purely on a battle of wits.