Reborn Technology Scholar

Chapter 24 First Day of Competition

In 2009, coinciding with the 50th International Mathematical Olympiad (IMO), the International Mathematical Olympiad Committee held a 50th anniversary celebration.

In this 50th anniversary celebration, many world-famous mathematicians appeared.

After the celebration, there will be a formal competition. Nearly 560 students from 105 countries and regions around the world will participate in this competition.

The entire competition lasts for one week.

Competitors will overcome mathematical problems during this week and compete for gold, silver and bronze medals in the Mathematical Olympiad. Competitors from each country come to compete in the world with the determination to win glory for their country.

On March 15th, the competition kicked off

IMO, there are six questions in total. Three questions today and three questions tomorrow. Each question is worth 7 points, and the full score is 42 points. The competition time on each competition day is 4.5 hours. You can bring any stationery and drawing tools. All electronic equipment is not allowed.

are allowed into the arena.

Because the competition lasts a long time, each contestant can bring his or her own food and drinks, and no more than three reference materials.

However, Qin Yuanqing brought some food and drink, but did not bring any other reference materials, because according to the previous situation, the reference materials were basically useless. The question maker had already considered this. If the reference materials could find a solution,

It shows that the question maker’s question writing skills are terrible.

This is just like domestic exams. Open-book exams are often much more difficult than closed-book exams.

Because the questions that contestants from their own country receive are already in their own language, there will be no language barriers for contestants when they receive the test papers.

Qin Yuanqing got the test paper. There were only three questions. The first question was the easiest. If you can't even do the first question, then you don't need to consider the next two questions.

Qin Yuanqing was very calm. The first question was the simplest, it was a sub-question, but again, it accidentally turned into a proposition.

"1,n is a positive integer, a1,a2...ak(k≥2) are different integers in {1,2,...,n}, and n|ai(ai1-

1) This is true for all i=1,2,...,k-1, prove: ak(a1-1) is not divisible by n."

Qin Yuanqing read the question three times, and secretly cursed the person who provided the question to give birth to a child without an asshole in the future. He actually set a trap so that he could answer it wrong if he was not careful.

Qin Yuanqing began to answer. He first used mathematical induction to prove that any integer i (2≤i≤k) is divisible. He concluded that when i=2, the conclusion that it can be multiplied and divided is established. Step by step

Expanding this, we finally draw the conclusion that ak(a1-1) is not divisible by n.

Then Qin Yuanqing looked at the second question.

"The centers of the circumcircle of △abc are o, p, and q on the line segments ca and ab respectively. k, l, and m are the midpoints of bp, cq, and pq respectively. The circle Г passes through k, l, and m and is tangent to pq.

.Prove: op=oq.”

Qin Yuanqing completed the examination of this question. I feel that this question is easier than the previous question. There is no trap. First I made a circle, then turned it into △abc, then made the line segments ca and ab and the two points p and q, and then

Mark the midpoints k, l, and m of bp, cq, and pq. Finally, draw a circle Г.

Then the straight line pq is tangent to the circle Г at the tangent point m, and then through the chord tangent angle theorem we get ∠qmk=∠mlk. Since points k and m are the midpoints of bp and pq respectively, km∥bq, thus we get

Out ∠qmk=∠aqp.

Therefore we get ∠mlk=∠aqp.

In the same way, ∠mkl=∠apq.

According to the equality of angles, we get △mkl∽△apo, and thus we get mk/ml=ap/aq

Because k, l, and m are the midpoints of line segments bp, cq, and pq respectively, we get km=bq/2, lm=cp/2. Put this into the above formula to get bq/cp=ap/aq. Change the formula

Convert to ap·cp=aq·bq. Through the circular power theorem, we know op2=oa2-ap·cp=oa2-aq·bq=oq2

Therefore, it is concluded that op=oq.

Qin Yuanqing did not even check, but turned the mathematical problem of drawing into images. This is what he is good at, and he has complete certainty to prove it.

Then Qin Yuanqing looked at the third question, "3, s1, s2, s3,... is a strictly increasing sequence of positive integers, and its sub-sequences ss1, ss2, ss3,... and ss11

, ss21, ss31... are all arithmetic sequences. Prove: s1, s2, s3... are an arithmetic sequence."

Looking at this question, Qin Yuanqing frowned slightly. This question was obviously much more difficult than the previous two questions. Qin Yuanqing tweaked the known conditions a little, and this question combined the arithmetic sequence and the conversion method.

Qin Yuanqing unfolded it step by step. Through the sequence and sub-sequence, they are strictly increasing positive integer sequences. Let ssk=a(k-1)d1, ssk1=b(k-1)d2(k=1,2...

...,a,b,d1,d2∈n).

Convert the problem into a function, after the sequence, use sk

Therefore a-b≤(k-1)(d2-d1)≤ad1-b. From the arbitrariness of k, we know that d2-d1=0, and we get d2=d1...

When Qin Yuanqing wrote down the conclusion of the proof, he touched his forehead and found that he was sweating. He breathed out a breath.

Then Qin Yuanqing stood up and made a gesture to hand in the paper. The invigilator walked up to him and put his examination paper into the document bag and sealed it.

Qin Yuanqing left the examination room relaxed and stress-free. Since he had answered the question, there was nothing wrong.

When Qin Yuanqing left the examination room, he realized that he was the first to hand in his paper. None of the members of the Chinese Mathematical Olympiad team had yet to hand in their paper, and none of the other countries' Mathematical Olympiad teams had yet to hand in their paper.

"How do you feel about the first day of competition?" the deputy team leader asked quickly when he saw Qin Yuanqing.

"It's just so so, it's very easy!" Qin Yuanqing waved his hand coolly: "It's not as difficult as the training test. Don't worry, you can't run with 42 points!"

The deputy team leader breathed a sigh of relief when he heard this. In this Chinese Mathematical Olympiad team, Qin Yuanqing is the ace and the ballast. Since Qin Yuanqing said this, it means that this year's difficulty will not be great.

"Even for the first question, I don't know which country came up with it. It's a trap. If you're not careful, you'll get it wrong. It's so unethical. You're playing tricks on us high school students!" Qin Yuanqing complained.

Then Qin Yuanqing saw a tall white man not far away with a bad look in his eyes. The deputy leader quickly covered Qin Yuanqing's mouth with his hand and whispered: "I heard that the first question was asked by Australia.

, that person is the deputy leader of the Australian Mathematical Olympiad team!"

Qin Yuanqing was speechless.

I was saying bad things about people behind their backs, and others heard me. This was too damaging to my character.

But when he heard it was Australia, he immediately felt that Australia was simply evil. Before he was reborn, Australia didn't know what was wrong with it, and it was arguing with China, which caused a lot of scolding on the Internet. Now, the other two questions are normal, especially

The last question was answered very well, but the first question was played too hard. This Australian is really out of his mind.

Qin Yuanqing couldn't understand why so many Chinese people immigrated to Australia because Australia was so stupid. As a result, their brains were also affected. For example, Liang Mouyan, who became famous in early 2020, had an arrogant and unreasonable attitude, and even shouted for help, claiming

Someone was harassing her, but if it weren't for the video, it would be hard to explain why. After being deported, she even asked the Chinese people to apologize to her and reimburse her for the air tickets. It just made her brain rusty.

About half an hour later, an Indian came out of the examination room. Qin Yuanqing asked curiously: "Deputy leader, Ah San is very good at mathematics?"

The deputy team leader said: "That's natural. Indians' mathematics level is also ranked among the best in the world. The Ramanujan Award, second only to the Philippine Prize, is named after the Indian mathematician Ramanujan."

"Ramanujan is a very powerful person, and Ramanujan's conjecture series is relatively powerful." Qin Yuanqing nodded slightly.

As for the Russian players who came out next, mathematics was very strong in the Soviet era, and many great mathematicians were born. Sergey, Driefeld, etc. all won the Philippine Prize. Russia, which inherited most of the Soviet heritage,

Mathematics is naturally very strong. For example, Grigory Perelman was the one who cracked the Poincaré conjecture. Because he proved the Poincaré conjecture, he did not know that thousands of related mathematical conjectures became

Theorem can be said to have single-handedly advanced the historical process of geometry and topology.

Even though Perelman is a weirdo who doesn't like to be interviewed or appear in public, there is no doubt that Perelman is definitely one of the greatest mathematicians in the world today.

After the members of the Chinese Mathematical Olympiad team arrived, everyone returned to the hotel together. No one would check the answer, as it would only interfere with the next day's competition.

Qin Yuanqing returned to his home court and used the computer to connect to the Internet to search for the world's mathematical powers. The United States, Europe, Russia, and Japan are all world's mathematical powers. They have produced more than one Philippine Prize winner, and the United States is the most powerful country in mathematics.

Rankings of university mathematics majors, mathematics institutes, mathematics professional magazines, etc. are all undisputed as the world's number one mathematics power.

As for China, although it has often won IMO gold medals in the past ten years, it is not considered a strong country in mathematics. At most, it can be regarded as a big country in mathematics.

Qin Yuanqing thought about what he had seen and heard outside a few days ago. Ordinary people in Europe and America have very poor computing skills, but their education is designed to cultivate children's interests. Mathematics is a subject that emphasizes talent and logic. It is very demanding and there is no mathematical talent.

, lack of logical thinking, will not enter the door of mathematics at all, and those who are interested, because they dare to be interested, often have strong self-learning ability, and often get twice the result with twice the result.

Similarly, the development of mathematical thinking is also very important. China is a country with a large population and has universalized nine-year compulsory education. It pursues fairness and justice, which leads to the need for a large number of teachers. For China, first of all

Satisfying quantity, and finally quality. This leads to the cramming teaching method in the education process. The students trained in this way have almost no test scores, but their thinking is a problem.

At the university level, there is a gap between the mathematical abilities of Chinese college students and foreign students. Foreign mathematical geniuses can get good mathematical thinking training from an early age, and with accumulated experience, they can show superb abilities at the university level.

Elite education is provided abroad, while civilian education is provided domestically. The gap in the education system leads to different results.

The domestic education system has led to the cultivation of millions of engineers in the country, a high-quality and cheap engineering labor force, so that around 2018, "engineer dividend" became a new hot word.