"If you win in the (math) competition, you will naturally feel happy and even proud, but if you are frustrated in the competition, you don't need to be too sad or disappointed in your math ability. In order to win the competition, we need to rely on some special talents, but these talents are completely unnecessary for fruitful research work. " This is a fragment of the preface written by Andrei Andrey Kolmogorov, a great mathematician in the former Soviet Union, for an Olympic competition. For mathematics education, Koch also has many original opinions. He pointed out that the first thing of mathematics competition is to cultivate students' interest in mathematics and discover their mathematical talent. If this work is not done well in advance, a math contest will be held in the lower grades, and most people will gradually lose their ability to solve problems and even lose interest in mathematics. This is indeed a real insight! In China, Koch's fears have indeed been constantly confirmed. The reason is that the logic rigor emphasized by middle school mathematics is very different from the intelligence game of primary school competition. If the foundation is not well laid and biased training is carried out, many students will not adapt to mathematics in middle school; However, the mathematics in universities is very different from that in middle schools, which is one of the reasons why some Olympians are not suitable for mathematical research. Wiles, a great mathematician who solved Fermat's Last Theorem, was rated as "not a genius" by Gals. Gols is a Fields Medal winner and a gold medal winner of the International Maritime Organization. One of his bases is that wiles didn't win the IMO gold medal. Gals has no intention of belittling wiles. His words have two meanings. One is to show the great difference between hard science research and the Olympic Games. Secondly, he also thinks that winning medals at IMO requires mathematical talent. The training of foreign Olympic athletes is not as big as ours, so it is really smart to win medals at IMO. For example, 1990 Xiao Lafuge, one of the four perfect players of Beijing IMO, whose brother won the Fields Prize in 2002; People think that little Lafarge is more talented. He has won many awards and is likely to win the Fields Medal in the future. In contrast, the winners of Olympic Mathematics at all levels in China have achieved good results, but they have not yet reached the level of Fields Medal, which is also closely related to their learning methods during their college and postgraduate years. Wang Yuan, an academician of the China Academy of Sciences and a famous mathematician, thinks that, on the whole, the propositional level of the China Competition is relatively high, but there is still a big gap with the international level, and some problems have been overdone. The level of the proposition is reflected in whether it is enlightening and interesting. Hua also believes that it is more difficult to ask good questions than to solve them. In fact, China's first place in the world is not as certain as some people think, at least the strength of Russia and the United States can never be underestimated. In particular, doing digression is not good for becoming an excellent mathematician, which caused Qiu Chengtong's anxiety. In contrast, the proposition level of the former Soviet Union is relatively high. For example, there is a problem in the Moscow game: Alibaba tried to sneak into the cave. There is a drum at the mouth of the cave. There are four identical holes on the side of the drum, which form the four vertices of the square. A switch is installed in each hole. The switch has two states of "up" and "down". (note: the eyes can't see! If the states of the four switches are consistent, the portal can be opened. Now you are allowed to put your finger into any two holes, and you can know its state by touching the switch. You can change or not change its state at will. But every time you do this, the drum will rotate so fast that it is impossible to confirm which switches have just been touched after stopping. Proof: Alibaba needs to put his finger into the hole five times at most. It is easy to know that two operations (two holes on the side and two holes on the diagonal) will turn at least three switches to "up". If the door is not opened, this means that the fourth switch is in the "down" state. At this time, Alibaba will put his finger into the two holes on the diagonal. If he meets the down switch, he turns to "up" and enters the cave. If both switches are on, turn off one of them. In this way, it is obvious that two adjacent switches are "up" and the other two adjacent switches are "down". Then Alibaba started along the square; If the two switches are in the same state, he will change the state and enter the cave; If the two switches are in different states, he has to change their states, find the switch diagonally for the last time, and change the state inside, so as to do it five times at most. This topic is very exciting. It examines the decision-making under different information and needs your understanding and insight into the nature of the problem. Such good questions abound in the competitions of the former Soviet Union, and it should be said that it is beneficial to think about them.