Reborn Tech Scholar

Chapter 24 First Day Competition
In 2009, the 50th International Mathematical Olympiad (IMO) was held, and the International Mathematical Olympiad Committee held its 50th anniversary celebration.

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

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

The entire competition lasts a week.

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

On March 3th, the competition kicked off
IMO has a total of six questions, three questions today and three questions tomorrow, each question is 7 points, the full score is 42 points.The competition time of each competition day is 4.5 hours. Any stationery and drawing tools can be brought. All electronic devices are not allowed to be brought into the competition field.

Due to the long competition time, each contestant can bring their own food and drink into the arena, and can bring no more than three reference materials.

However, Qin Yuan brought some food and drink, and did not bring any other reference materials, because according to the previous situation, the reference materials were basically useless. The questioner had already considered these. If the reference materials can find a solution, It shows that the question maker's level of questioning is too bad.

Just like domestic exams, open-book exams are often much harder than closed-book exams.

Because the national contestants get the questions, they have already changed to the native language, so there will be no language barriers for the contestants to get the test papers.

When Qin Yuanqing got the test paper, there were only three questions, and the first question was the easiest.

Qin Yuanqing was very calm. The first question was the easiest, and it was a sub-question, but in the same way, it turned into a proposition.

"1, n is a positive integer, a1, a2ak (k ≥ 2) are distinct integers in {1, 2, ., n}, and n|ai(ai+1-1) for all i=1 , ., k-2 are all established, prove: ak (a1-1) is not divisible by n."

Qin Yuanqing read the question three times, and secretly scolded the person who provided the question to have no buttholes in the future.

Qin Yuanqing began to answer. First, he used mathematical induction to prove that any integer i (2≤i≤k) is divisible. When i=2, it is known that it can be multiplied and divided.Expand this step by step, and finally come to the conclusion that ak(a1-1) is not divisible by n.

Then Qin Yuanqing looked at the second question again.

"The center of the circumcircle of △ABC is O, P and Q are on 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. .Proof: OP=OQ.”

Qin Yuanqing completed this question, but he felt that this question was easier than the previous question, and there were no traps.First make a circle, then change it into △ABC, then make CA, AB line segment and two points P, Q, and then mark the midpoints K, L, M of BP, CQ, PQ.Finally make a circle G.

Then the straight line PQ is tangent to the circle Г, the tangent point M, and then ∠QMK=∠MLK is obtained by the chord angle theorem.Since the points K and M are the midpoints of BP and PQ, respectively, KM∥BQ, thus ∠QMK=∠AQP.

Hence ∠MLK=∠AQP.

Similarly, ∠MKL=∠APQ.

According to the equality of angles, △MKL∽△APO is obtained, so that MK/ML=AP/AQ
Because K, L, and M are the midpoints of the line segments BP, CQ, and PQ, respectively, we get KM=BQ/2, LM=CP/2, and put this into the above formula to get BQ/CP=AP/AQ, and the formula Convert to AP·CP=AQ·BQ.By the circular power theorem, we know that OP2=OA2-AP·CP=OA2-AQ·BQ=OQ2
So, it is concluded that OP=OQ.

Qin Yuanqing didn't even check, and turned the drawn mathematical problems into images. This is what he is good at, and he has complete assurance 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 SS1+1, SS2+1, SS3+ 1 is an arithmetic sequence. Prove: S1, S2, S3 is an arithmetic sequence."

Looking at this question, Qin Yuanqing frowned slightly. This question is obviously much more difficult than the previous two questions. Qin Yuanqing scoured the known conditions a little. This question combines arithmetic progression and transformation.

Qin Yuanqing expands it step by step, through the sequence and sub-sequences are strictly increasing positive integer sequence, set Ssk=a+(k-1)d1, SSk+1=b+(k-1)d2 (k=1, 2, a, b, d1, d2 ∈ N+).

将问题转为函数、数列后，以Sk<Sk+1<Sk+1及{Sn}的单调性，知对任意的正整数k，有SSk<Ssk+1≤SSk+1。即a+（k-1）d1<b+（k-1）d2≤a+kd1
因此a-b≤（k-1）（d2-d1）≤a+d1-b。由k的任意性知d2-d1=0，得到d2=d1。。。。。。

When Qin Yuanqing wrote the proof conclusion, he touched his forehead and found that he was sweating, and let out a sigh of relief.

Then Qin Yuanqing stood up and made a hand gesture.The invigilator walked up to him and put his exam papers in a file bag and sealed it.

Qin Yuanqing left the examination room easily and without any pressure.Since the answer is answered, then there is nothing wrong.

When Qin Yuanqing left the examination room, he realized that he was the first to hand in the papers. None of the members of the Huaxia Olympic Math Team had handed in the papers, and none of the Olympic math teams from other countries had yet handed in the papers.

"How did you feel on the first day of the exam?" The deputy team leader asked quickly when he saw Qin Yuanqing.

"It's normal, it's very easy!" Qin Yuanqing waved his hand dashingly: "It's not difficult for the training camp yet, don't worry, you can't run with 42 points!"

Hearing this, the deputy team leader breathed a sigh of relief. In this Huaxia Olympic math team, Qin Yuanqing is the ace in existence and the ballast stone. Since Qin Yuanqing said so, it means that this year is not difficult.

"It's the first question. I don't know which country gave the question. I set a trap, and if I'm not careful, I'll make a mistake. It's too immoral, and I'm playing tricks with us high school students!" Qin Yuanqing said in a snarl.

Then Qin Yuanqing saw not far away, a tall white man looked over, his eyes were not good, the deputy leader quickly covered Qin Yuanqing's mouth with his hand, and whispered: "I heard that the first question is from Australia. , that person is the deputy team leader of the Australian Olympic Math Team!"

Qin Yuanqing was speechless.

He is talking ill of people behind his back, and he is heard by others, which is too degrading.

But when he heard it was Australia, he immediately felt that Australia was simply wicked. Before his rebirth, Australia didn't know which tendon was wrong, and he was arguing with Huaxia, which led to a lot of scolding on the Internet.Now, the other two questions are very normal, especially the last question is very good, but the first question is playing with your heart, which is really abnormal in Australia.

Qin Yuanqing couldn't understand, Australia is so brainless, why so many Chinese immigrated to Australia, and as a result, their heads were also affected. For example, Liang Mouyan, who was famous in early 2020, had an arrogant and unreasonable attitude, and even shouted for help, claiming that Someone harassed her, if it weren't for the video, it would be really unreasonable.After being deported, she even asked the Chinese people to apologize to her and reimburse her for the air ticket, which made her head rust.

About half an hour later, an Indian walked out of the examination room. Qin Yuanqing asked curiously, "Deputy leader, India's mathematics is very strong?"

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

"Ramanujan is very powerful, and the Ramanujan conjecture series is relatively powerful." Qin Yuanqing nodded slightly.

As for the Russian players who came out later, in the Soviet era, mathematics was very strong, and many great mathematicians were born. Mathematics is also very strong, for example, Grigory Perelman is a fierce man who cracked the Poincaré conjecture, because he proved the Poincaré conjecture, and he did not know the thousands of mathematical conjectures associated with it. Theorem can be said to promote the historical process of geometry and topology by oneself.

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

When the members of the Huaxia Olympic Games team arrived, everyone returned to the hotel together. No one would check the answer, which would only interfere with the next day's game.

Qin Yuanqing returned to the home court and used the computer to connect to the Internet to find the world's mathematical powers. The United States, Europe, Russia, and Eastern Ying all belong to the world's mathematical powers, and more than one Philippine Prize winner has been born. The ranking of mathematics majors in universities, or the Institute of Mathematics, mathematics professional magazines, etc., are undisputedly the world's first mathematics power.

And Huaxia, although it has often won IMO gold medals in this decade, it is not a math powerhouse, at most a math powerhouse.

Qin Yuanqing also thought of what he had seen and heard outside a few days ago. Ordinary people in Europe and the United States have very bad computing skills, but their education is to cultivate children's interest. Mathematics is a subject that pays attention to talent and logic. It is very demanding and has no mathematical talent. , lack of logical thinking, and will not enter the gate of mathematics at all, and those who are interested, because they dare to be interested, often have a strong self-learning ability, and what they are interested in is often twice the result with half the effort.

Similarly, the development of mathematical thinking is also very important.Huaxia is a country with a large population, and it is also popularizing nine-year compulsory education. The pursuit of fairness and justice has led to the need for a large number of teachers. For Huaxia, the first is to satisfy the quantity first, and the last is the quality.This leads to the cramming-style teaching method in the education process. The test scores of the students trained in this way are almost the same, but thinking is a problem.

At the university stage, there is a gap between the mathematics ability of Huaxia college students and foreign countries. Foreign mathematical geniuses can get a good mathematical thinking training from an early age.

Foreign education is elite education, domestic education is civilian education, and the gap in education system has led to different results.

And the domestic education system has led to the training of millions of engineers in the country, with high-quality and cheap engineer labor, so that around 2018, 'engineer bonus' has become a new hot word.

　The collection and recommendation tickets are so miserable, and they are on the street again!
　　


(End of this chapter)