I Just Want to Be a Quiet Top Student

　　 Xu Da and Zhu Yuanzhang played a game of chess. On the premise of defeating Zhu Yuanzhang, Xu Da used chess pieces to put the word "Hurray" on the chessboard. What is the probability of this incident? And explain why.

　　 The Chinese Mathematical Society asked questions and challenged the candidates’ bottom line.

　　 MMP must understand the rules of Go to do a math problem, just ask if you are upset, this kind of problem may be solved by Chinese students, and the crooked fruit will cry after reading it.

　　 This is the National Championship, the most difficult high school mathematics arena in China.

　　 If there is a player who doesn’t know anything about Go, it’s a sad reminder. You don’t have any special skills. Are you embarrassed to participate in the math national finals?

　　 Shen Qi can play Go, Chess, Flying Chess and Animal Chess. Regardless of chess skills, he knows the rules.

　　 And Shen Qi knows that many other national finalists understand various chess rules and play very well. For example, the group of players from the Northern Hubei Province Mathematical Competition team can play blind chess with their eyes closed.

　　 "Look at the essence through the phenomenon. The essence of this question has nothing to do with Zhu Yuanzhang and Xu Da. It is just a math problem. Except for the last sentence, which asks about probability and this game record, the allusions in the front are all pretexts."

　　 Shen Qi quickly thought of Fermat and Pascal’s theory on the distribution of gambling money. In a sense, playing chess is also a kind of gambling, and people under the bridge have long lived by this.

　　 Since it is gambling, it is necessary to apply relevant professional knowledge of probability theory and number theory.

　　甭 Regardless of how the word "Long live" was vacated, it is just a probability event, a trick for people who understand mathematics. If Zhu Yuanzhang understands mathematics, he will immediately cure Xu Da's crime and reward Mochou Lake with a woolen thread.

Fermat and Pascal’s joint gambling distribution theory and the subsequent derived related theories are the theoretical basis for the world’s major casinos to make no loss. The probability of the word "Hurray" is calculated, and two kings and four are calculated. Second, the theoretical principles of the remaining one-handed straight are similar.

　　 Shen Qi wrote: Let the sun be p and the white be q, if p is the probability of a single event, then q is the probability of the event not appearing.

Then, the probability of the event occurring at least m times in n trials is equal to the n-th power expansion of (p+q), from the n-th item of p to the m-th item including p multiplied by q (nm) The sum of the items up to the second item.

　　……

　　 Based on this theory, Shen Qi quickly calculated the probability of the word "Hurray", which was only 0.2% in 10,000, and elaborated on the reason.

　　It is not difficult to calculate probability. You can become the king of gambling if you have mastered the above mathematical principles. The difficult thing is to walk out of the casino alive with sound limbs.

　　 Shen Qi inferred that it is true that Zhu Yuanzhang and Xu Da played chess, but Xu Da's victory over Mochou Lake with the word "Long live" is most likely a legend.

　　 From a mathematical point of view, "Hurray" can only appear once when you play 50,000 games. A game can be as short as tens of minutes and as long as a few hours. It is more than three or five games in a day.

　　 Zhu Yuanzhang and Xu Da don’t do anything else every day, they just play chess. It takes 27 years to see the "Hurray" presented by black and white.

　　 Zhu Yuanzhang is the founding emperor, he doesn't need to deal with state affairs?

　　 Of course, it is also possible that the "Hurray" event appears randomly for the first time in 50,000 times.

　　 So this is a legend, not to be taken seriously.

"This first question looks very painful at first, but after finishing it, it doesn't hurt, and there is even a little jittering pleasure. This question is actually quite interesting. Oh, I haven't been to Mochou Lake. Okay. I want to see it." Shen Qi ate a Snickers and celebrated his success in cracking the title of the national finals.

　　 Without stopping, Shen Qi enters the solution to the second question, which is a plane analytic geometry problem.

　　 For Shen Qi of Mathematics Level 5, it is not difficult to analyze the affine transformation of two-dimensional homogeneous coordinates with determinants. It is nothing more than seeking a set of invariants for rotation, translation and reflection.

　　 the single-ellipse geometry corresponds to the subgroup of the projective transformation. This seems to be a natural axiom, but don’t be confused by its appearance, otherwise you will go astray.

　　The most sane number contestant only needs to go straight to Huanglong and find the absolute shape of the imaginary ellipse on the plane. The second question is to send points.

　　 Shen Qi uses an economical and practical way to find the virtual ellipse. The Klein continuous transformation written in the university textbook is too complicated, and it is completely troublesome for himself.

　　 Poincaré, known as the last "all-round scholar" in the world, is obviously more flexible. Shen Qi likes to use Poincaré's many views and conclusions.

　　From provincial competitions to national competitions, Shen Qi used Poincaré's theory to solve problems more than once. Poincaré is a generalist in mathematics and a master in physics, astronomy, philosophy and other fields.

　　 In the plane coordinate system, there are many ways to obtain the absolute form through a curve. Poincaré's degeneration method is simply a magic weapon for the competition system. Shen Qi uses this kind of degeneration method, which is simple and rude. It's very easy to use, just like tailor-made for math competitions.

　　 Two hours later, Shen Qi solved two questions. He drank a sip of Dongpeng special drink and beaten chicken blood, accompanied by bear biscuits, snickers and wife cakes to replenish his energy.

　　 Don’t think that mathematics is purely mental work, and it’s also physically exhausting. Just sit there and keep writing and writing, and ask you if you’re tired after writing for two hours.

　　 60 national finalists are distributed in 7 classrooms. Shen Qi and his team have 10 players from 10 different provinces and cities.

　　 There are as many as 11 invigilators, one-on-one marking the remaining one to fight guerrillas.

　　 After all, it is the national finals. The gold medal in the national competition is very heavy. After the big wave ~www.readwn.com~, only six people can win the gold medal. These six lucky winners will surely become the sweet pastry of the famous universities.

　　The invigilator responsible for guarding Shen Qi walked behind Shen Qi and hummed softly, "Is the classmate full?"

　　 "I'm full, and then eat the spicy bar graph for auspicious, prosperous." Shen Qi left the spicy bar with compasses in his right hand, eating while drawing, and drew a concentric circle.

　　 "Classmate, why did you bring so many snacks into the room?" The invigilator frowned and asked in a low voice.

　　 Shen Qi is also strange: "Hey, aren't you allowed to bring food? 4.5 hours, teacher, it's almost the same as the ancient imperial examinations. It's a protracted battle. If you don't eat, you will be hungry."

"Eat less, eat too much and you will get sleepy." The invigilator kindly reminded him, and glanced at Shen Qi's exam papers. The Southern Guangdong Provincial Mathematical Competition Team, Shen Qi, okay, this little boy, he finished two questions in just two hours. , He also enjoys eating spicy strips, he is very likely to be a dark horse!

The invigilator who was marking Shen Qi was a staff member of the Chinese Mathematical Society. He was a full-time mathematics researcher himself. He paced to the door of the classroom and asked a younger guy: "National Jue No. 56 contestant, Shen Qi, he pre-examined. How many points?"

The young man had an IPAD in his hand. He swiped out the information and glanced at it: "Shen Qi, the national pre-qualification is 21 points, full marks. It is not easy for the Nanyue team players who are in danger to advance to the national finals to get full marks. Shen Qi is the only national pre-qualification score of their team ."

"Well, it's an individual talent. The purpose of our Mathematical Society is to drop talents in any style." The older invigilator nodded and said to the young man: "Xiaowen, go and buy me the spicy strips, which Shen Qi brought with you. That same style. It's past 11 o'clock, hungry."

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