Yesterday afternoon, the closing ceremony of the final (CMO) of the 2023 National Mathematics Olympiad for Middle School Students was held in Wuhan No. 3 Middle School in Wuhan, Hubei Province, and the results of the competition were officially announced. A total of 678 students participated in the competition, including 229 gold medals and 302 silver medals, as in previous years, this year's national training team is still composed of the top 60 in the gold medal list, of which 60 students who enter the national training team can obtain the qualification of Tsinghua University and Peking University. 
Considering that many people are not familiar with the CMO process, I will briefly introduce here that the competition route is divided into three steps:The first step is to participate in the National High School Mathematics Joint Competition, abbreviated as Gaolian. It is usually held on the second Sunday of September. For example, this year's Gaolian was held on the morning of September 10, divided into one test and two tests, with a 20-minute break in between. There are a total of 11 questions in the first test, including 8 fill-in-the-blank questions and 3 answer questions, and the scope of the test is the in-class knowledge of high school, and the test time is 80 minutesThere are a total of 4 questions in the second test, the scope is plane geometry, algebra, number theory, and combination, of which plane geometry is expanded on the basis of junior high school, and the exam time is 2 hours and 50 minutes. The high-level competition is usually judged by each province.
First, second and third prizes.
The second step is to participate in the National Mathematics Winter for Middle School StudentsCamp, also known as the National Mathematics Olympiad for Middle School Students, is abbreviated as CMO. Outstanding players from the high-level competition are recommended by all provinces, autonomous regions and municipalities directly under the central government to participate in the CMO competition, except for last year, which was slightly late due to the impact of masks, and in recent years, this event has been held in November. The competition lasts for two days, with 3 questions per day, and the exam time is 4 hours and 30 minutes per day. Three awards will be awarded according to the results of the competition: gold, silver and bronze.
The third step is to participate in the International Mathematical Olympiad, abbreviated as IMO. In the second step of the competition, the top 60 players will be selected for the national training team, and then 6 players will be selected to form a national team to participate in the IMO competition and compete with masters from all over the world. The schedule is the same as that of the CMO, the competition time is two days, there are 3 questions per day, and the daily exam time is still 4 hours and 30 minutes. Three awards will be awarded according to the results of the competition: gold, silver and bronze. After the results of the 2023 CMO competition were announcedI compared this year's results with those of the past few years, and I only had one feeling: the competition is getting more and more volatile, and the competition is becoming more and more intense. 
Performance 1: More and more students are in the first year of high school among the winners. Of the 60-member roster of this year's training team, 17 students are still in their first year of high schoolIn the 2022 60-person training list, 16 students are from the first year of high schoolIn the 2021 training team's 60-member roster, 10 students are from the first year of high school. It can be seen that among the students who have entered the training team, there are more and more gods from the first year of high school. In addition, there are many CMO gold and silver medals that also come from the first year of high school, and these are also cow babies. Many of them will still participate in the competition next year to compete for the guaranteed quota of the training team. 
You know, if you want to win a high award in the CMO competition, you must first win the first prize of the provincial high union competition, and the results of the first test of the high union must be very good, basically close to the full score, and the scope of the first test of the high union is similar to the college entrance examination, but the difficulty is higher than the college entrance examination. These cow babies who can enter the training team in the first year of high school have a high probability of completing high school mathematics in the first year of junior high school or even elementary school and entering the Xi stage of mathematics competition. Judging from the age of the winners, the road to competition will only get more and more rolling. If you want to take the competition route, you must highlight an early word, and you can only plan early, learn in advance, and brush the questions in elementary school, the sooner the better.
Performance 2: The level of the winners is getting higher and higher. The number of CMO gold and silver medals is not fixed, and is based on the performance of the participants. The higher the total score of the players, the greater the number of gold and silver medals. This year's CMO produced 229 gold medals and 302 silver medalsThe 2022 CMO produced 201 gold medals and 258 silver medalsThe 2021 CMO produced 183 gold medals and 219 silver medals. You can see itThe number of gold and silver medals is increasing, which means that the level of the winners is getting higher and higher. But in the past few years, the way out of the competition has been much narrower than in the past:Only the 60-member roster who entered the national training team can be eligible for escort;Only those who have won the gold and silver medals of CMO can be admitted with reduced points in the Strong Foundation Program. When the level of the contestants generally improves, the competition route will definitely become more and more rolled.
Performance 3: The concentration of winners is getting higher and higher. From the perspective of provinces, among the 60 members of the national training team this year, Zhejiang accounted for 11, Shanghai also accounted for 11, Hubei accounted for 10, Beijing accounted for 6, Hunan accounted for 5, and Guangdong accounted for 5. In these six provinces and cities alone, 48 members of the training team have been produced, accounting for 80% of the total number. In stark contrast, Tianjin, Hebei, Jiangxi, Anhui and other provinces and cities did not even have a single member of the training team. From the perspective of schools, among the 60 members of the national training team, 9 are from Shanghai Middle School, 6 are from Renmin University High School, 5 are from Wuchang Experimental High School, 3 are from Shenzhen Middle School, and 3 are from Yali Middle School. You can see itIf you want to win in the competition route, you can only hope to compete with strong schools in strong provincesOtherwise, there is a high probability that it will not work. Because the competition route not only requires students to be highly talented, but also needs targeted individual coaching from good coaches. (ps: Who said that talented cow babies don't make up classes?)）In a word: the road of competition is getting narrower and narrower.
You've seen it all, like it and watch a math problem before you go. Now let's move on to the math time, today's topic is a number theory problem, the topic is from a mathematics competition in Australia, and the knowledge used to solve the problem does not exceed the second grade of junior high school. The question is only for competition students, students who do not plan to participate in the competition should ignore this question.
Food for Thought (4 and a half stars difficulty):For the real number a, [a] is used to denote the largest integer that does not exceed a. Verification: For any positive integer m greater than 4, there is a positive integer n, satisfying [(m-1)!/m]=n(m-1)。Welcome to write your answer in the comment area, if no one gives the correct detailed answer, tomorrow I will give the answer in the same name gzh.