A genius? I just love studying.

Chapter 142 A Unique City, The Pigeonhole Principle
9 month 9 day,
To be a stranger in a foreign country alone, think of love every time during the festive season.

At Li Zehan's urging, Chen Hui, unusually, did not study and instead went with him to downtown Paris.

Walking along Boulevard Saint-Michel, smelling the stench of the Seine, and stepping on the rain-soaked cobblestones, a raised cobblestone suddenly sprang into action, splashing water onto his sneakers and chilling his toes. Chen Hui felt a mix of emotions.

Fortunately, Paris in September wasn't cold; in fact, it gave Chen Hui a slightly cool feeling, though his soaked shoes made him a little uncomfortable.

"Look!"

Li Zehan suddenly shouted.

Chen Hui looked up and saw the rising sun illuminating the Eiffel Tower, its steel frame glowing against the violet sky like a proof of the golden ratio.

Walking from Boulevard Michel to the Champs-Élysées, along the way, you'll see international students playing the Marseillaise on the erhu, gentlemen in suits playing the piano, and others playing the pipa and guzheng...

In a distance of only two or three kilometers, Chen Hui saw four or five such street performances.

A crowd surrounded them, and children danced to music in the open space.

As the music ends, some people leave, and new people join in, as if it were an endless party where you could join in and revel at any time.

Are you here to participate in the IMO competition?

While watching the erhu performance of Marseillaise, an elderly French woman in the crowd suddenly asked a question.

She recognized the badges the three of them were wearing, which were issued by the IMO organizing committee.

"Yes!"

"Oh, what an excellent young man!"

The old lady winked at them and hurried away, saying, "Good luck, little mathematicians!"

Chen Hui's feelings became even more complicated; this was truly a unique city.

While he has many flaws, you can't deny that he possesses a unique charm.

Li Zehan and Chen Ling'er chattered excitedly. For these high school students who usually do nothing but practice problems, not having to do practice problems was enough to excite them.

However, they soon quieted down, and their mouths were dry as they looked around, hoping to find a convenience store.

After some effort, they found the familiar Carrefour. Although none of the three spoke French, they recognized the logo. A quick search revealed that Carrefour was actually a chain supermarket originating from France.

Li Zehan vowed that he would never buy anything at Carrefour again, and that he would only buy things at Hongqi. Then, without hesitation, he walked into the Carrefour and bought three bottles of what looked like purified water.

When it was time to pay, the cashier was a fair-skinned young woman with long brown hair. She smiled and told Li Zehan the price in English, "7.5 euros."

"?"

Li Zehan stood there stunned, somewhat doubting his own existence, and went back to where the mineral water was kept to look at the price. He found it hard to imagine how three bottles of mineral water, each costing 1.9 euros, could be calculated to have such an outrageous price of 7.5 euros.

“One point nine euros per bottle?”

"Yeah!"

Seeing the girl's serious expression, Li Zehan was somewhat overwhelmed. He suddenly remembered that he seemed to have seen some self-media posts before, saying that Westerners were ridiculously bad at arithmetic, but he never thought that it could be visualized to such an extent.

"Even if you calculate it at 2 euros per bottle, three bottles should be 6 euros, how could it be 7.5 euros?"

李泽翰决定教这个小姑娘一道简单的算术题，“最后的结果应该是2*3-0.1*3=5.7才对。”

He felt that calculating 1.9*3 directly was too difficult for the young girl, so he simplified it for her.

But the little girl suddenly became wary, and her hand had already reached under the cash register.

"This is 7.5 euros, thank you."

Chen Hui took out some change and led Li Zehan out of Carrefour. He also decided that he would not go to Carrefour again after returning to China.

He could tell that the young girl wasn't deliberately trying to scam them, but Carrefour was definitely going to take the blame.

"Isn't this too outrageous?"

"How could Westerners be so bad at mathematics?"

Li Zehan held the water he had just bought, forgetting even that he was thirsty, his small eyes filled with confusion.

Chen Hui unscrewed the cap and took a sip. Unlike domestic mineral water, it was mint and cucumber flavored, but it wasn't unpleasant to drink. On the contrary, it had a refreshing taste.

"You Chinese are really good at math, but you don't have many mathematicians."

Although the speech was broken, the other person was indeed speaking Chinese. Li Zehan, who was just complaining, turned around in surprise and saw that the person who had spoken was a blond, blue-eyed Westerner who was about the same age as them and had a face covered with freckles.

He was wearing a badge issued by the IMO organizing committee on his chest, which clearly indicated that they were also there to participate in the IMO. These people were probably IMO contestants who were out for sightseeing, just like them.

"Sorry."

Chen Hui apologized and quickly left with Li Zehan. It was indeed Li Zehan who had gone too far in his sweeping generalizations, and he had no one to blame but himself.

He never expected to encounter a Westerner who could speak Chinese on the streets of Paris.

The strength of China was finally tangible at this moment.

"Why do I feel like these people are so hostile towards us?"

Chen Ling'er frowned slightly and asked in confusion, "Since they checked in yesterday, they have encountered many contestants, and many of them look at them with a hint of hostility."

"What else could it be?"

"Of course it's because we're too strong!"

Li Zehan said noncommittally.

Chen Ling'er clearly disagreed with this answer, and she turned to look at Chen Hui.

"Of course I don't know."

Chen Hui shrugged. "But perhaps Li Zehan is right."

“Over the years, we have won too many team first places. When their teachers are training, they inevitably mention us as a rival. Perhaps they did not do it intentionally, but these students may have developed some hostility towards us without realizing it.”

"It doesn't matter."

It really doesn't matter. IMO is only a few days long. Soon they will return to their lives. These people may never see each other again in their lives.

When they returned to their hotel, they saw an Iranian boy kneeling on the carpet in the lobby, holding a compass and intently adjusting the direction.

"Hey buddy, what are you doing?"

Li Zehan had already forgotten his earlier displeasure, and curiosity drove him to step forward and ask. The young boy from Yilang was not annoyed by the sudden interruption; he turned around with a gentle expression and said with a smile, "I'm confirming the influence of Mecca coordinates on celestial mechanics calculations."

"Good luck!"

Li Zehan isn't always sharp-tongued.

Upon arriving at the second-floor restaurant, the South Korean and Italian teams were debating in English how many olives should be placed on a pizza to conform to the optimal combination theory, while the German player next to them was unconsciously drawing Riemann function curves on the condensed glass window with his knuckles...

After finishing his meal and returning to his room, Chen Hui took out his papers and began reading. When he left Jiangcheng University, his teacher had prepared a thick stack of papers for him, including not only mathematics but also physics. After half a month of study, he had already started reading papers related to physics.

Your physics rating has increased from level 2 (99%) to 100%.

Your physics level has been increased to level 3.

More than an hour later, the barrage of comments popped up again.

Unfortunately, and unsurprisingly, this time the subject level upgrade did not grant any free attribute points.

With a thought, he summoned the control panel.

[Host: Chen Hui
Chinese Level 2 (8%)

Mathematics Level 3 (7%)

English Level 3 (4%)

Physics Level 3 (0%)

Chemistry Level 2 (0%)

Biology Level 2 (0%)

Physical Education Level 2 (0%)

Geography Level 2 (0%)

History Level 2 (0%)

Politics Level 2 (29%)

Chen Hui was astonished by his proficiency in politics. After these subjects turned gray, there were no more bullet screen prompts for improvement. However, before coming to Paris, they were only at 25%, but in just over a day, they had increased by a full 4%, which was terrifying.

However, a problem has now arisen in his path.

Should I continue improving my math skills, or study biology and chemistry, raising both subjects to level 3 to earn free attribute points? That is the question.

Biology and chemistry are destined to be more difficult than physics. Mathematics and physics are inextricably linked, and the level of mathematics has an important impact on physics. Therefore, after improving his mathematics to level 3, he only took a little over a month to improve his physics to level 3 as well.

However, the connection between chemistry and biology and mathematics is not as strong as that between physics. It may take three months or even longer to raise both subjects to level 3.

He doesn't need these three months right now, but would he gain more by spending that time on mathematics?

Chen Hui made a decision quickly.

Let's focus on getting his math level up to 4 first. He's not really interested in chemistry and biology yet. The human eye can only see a few hundred meters. Let's do what we want to do right now!
A night of silence,
The next day, Li Zehan and the others got up very early, only to find that Chen Hui was already studying at his desk, and seemed to have been studying for quite some time.

"Boss, are you cultivating immortality?"

From CMO to IMO, the two lived together for quite some time. Li Zehan knew that Chen Hui studied very hard, but he never deliberately investigated it.

"Okay, let's go eat."

Chen Hui glanced at the time but did not answer the question.

After breakfast, the group entered Saclay University and, led by staff, arrived at a dilapidated teaching building.

Chen Hui's exam room was on the first floor. Coincidentally, the contestant who had clashed with them at the airport and the American contestant who retorted to them on the streets of Paris were in the same exam room as Chen Hui.

Both of them naturally noticed Chen Hui, and they looked at him with provocative eyes. Before they came to participate in the competition, their teacher had told them that China had produced a very powerful newcomer this year, and that was this guy named Chen Hui.

They had also done the real test questions for Huaxia's CMO this year, which were indeed very difficult, but they were able to get full marks. They didn't think they were any worse than that newcomer from Huaxia!
I heard he submitted his CMO paper in just a few minutes on the first day, so let's see who's the fastest this time at the IMO!

Chen Hui remained completely unmoved, even wanting to laugh a little. Of course, not in a mocking way; he didn't take their provocation to heart. Just as an adult wouldn't hold a grudge against the offensive words of a three- or four-year-old, Chen Hui only sensed a youthful spirit in them, which he found rather endearing.

After improving his math skills to level 3, he felt that he had at least the strength of a math graduate student. Participating in the IMO would be a bit like bullying someone weaker than him. If it weren't for the free attribute points, he wouldn't have participated in the competition.

Soon, the test papers were distributed.

The exam process for IMO and CMO is similar, both consisting of two days of exams, with three questions each day, lasting four hours, from 9:30 am to 1:30 pm.

After quickly scanning the three questions, Chen Hui already had a good grasp of the concepts.

1. There are 21 girls and 21 boys participating in a math competition.
a. Each participant answered a maximum of 6 questions correctly.
b. For any pair of boys and girls, there is at least one question that they both answered correctly.

Prove that there exists a problem in which at least three girls and at least three boys can solve it correctly at the same time.

As expected of the first question, it was probably designed to save face for the participants and prevent them from failing to score. The question was very gentle, and Chen Hui immediately had a solution.

To prove this problem, we only need to use a concept that everyone learned in elementary school: the pigeonhole principle, or the drawer principle.

The principle can be simply summarized as follows: if there are ten pigeons and you want to put them in nine cages, then there must be two pigeons in one cage.

This theorem may seem simple, but it can often solve many complex problems, especially those concerning existence. It is often a sharp weapon.

The current question is no exception.

Since we are using the pigeonhole principle to solve this problem, we first need to create a 21x21 table. Each row and column represents a boy and a girl, respectively. The square in the middle represents any question that the boy and girl can answer correctly at the same time. According to the problem statement, for any pair of boys and girls, there is at least one question that they both answered correctly.

Suppose that if at least three boys answer the question correctly, fill in an M in the cell; if at least three girls answer it correctly, fill in an F. In other words, if the question that both boy number 3 and girl number 4 answer correctly is Q1, then the cell at coordinate (3, 4) represents question Q1.

If three boys answer Q1 correctly, then fill in an M in that box. If exactly five girls answer correctly, then fill in an F.

Therefore, the proof of this problem becomes proving that at least one cell in this table contains both M and F.

We assume that such a situation does not exist, but according to the problem statement, each participant answered at most 6 questions correctly, and for any pair of boys and girls, there is at least one question that they both answered correctly. Therefore, we can construct such a minimum case.

If a boy answers only one question correctly, then the question he answered correctly must be the one corresponding to the square. This means that 21 girls answered that question correctly, so the square in the row where the boy is located will be filled with F.

To minimize the number of F's, the only way is for the boy to answer 6 questions correctly, with only two girls answering five of them correctly, and then 11 girls answering the remaining question correctly. Therefore, there will only be 11 F's.

Therefore, each row containing boys will have at least 11 F's, and similarly, each column containing girls will have at least 11 M's.

那么这样所产生的M和F的个数就为21X11X2，但是格子总数只有21X21，根据鸽笼原理，至少存在21个格子同时被填上了F和M。

In just two minutes, Chen Hui completed the proof of the first problem. He believed that even Rui Rui, who was still in elementary school, could solve this problem quickly. He thought there was no one who couldn't do it!

(End of this chapter)