Chapter 39: Start of the Competition
The nine players participating in the training camp nodded seriously. This was a problem they had to avoid. They had to remember that their opponents were top-ranked high school students in the country, all of whom were geniuses. They had to sprint for better results and not be careless at all!
They were all top students in mathematics. They suddenly realized that Lao Zhang had only named nine people, and the remaining one was the god. They couldn't help but glance at Wu Tong secretly, wanting to see how Lao Zhang would explain to them how to save this trump card?
"Don't watch it. As for Wu Tong, I don't have much to say. Just keep performing normally. The national competition is not your end. I'll give you a small goal. Get a place in the national team first!"
She got full marks in the league, could easily solve IMO classic problems, and even studied number theory in university textbooks. What else can he say? Looking at this girl's mentality of taking on-the-spot performance as a matter of course, he didn't even need to remind her to play normally and keep a good attitude!
He even had a little extravagant hope that this girl could join the national team and win a full score gold medal in IMO. Then their Zhongyuan Province would be very famous!
"Does Teacher Zhang have anything else to arrange?"
"Everyone has practiced and prepared everything they need. There is no need to add to the icing on the cake by doing more questions today. As long as you maintain a good attitude, make sure to get enough rest, and face the competition with plenty of energy, it will be fine! Don't get overly excited before the exam and lose sleep, as it will affect your performance and the gain will not outweigh the loss!" Teacher Zhang, the housekeeper of the provincial team, gave some additional instructions on the details.
On the 14th, the Chinese High School Mathematical Olympiad officially kicked off.
"I'm here to wish all the students good luck in getting good results!" With the passionate encouragement from the two team leaders, Lao Zhang and Xiao Zhang, all the contestants from the Zhongyuan Provincial Team, like hundreds of contestants from all over the country, walked into the examination room with high morale.
At 7:40, all candidates were in their seats.
Eight o'clock.
The test papers have been distributed and the 24th China High School Mathematical Olympiad has officially begun.
CMO completely simulates the format of the International Mathematical Olympiad (IMO). The exam is divided into two days, with one test per day, each test lasting 4.5 hours, and a total of three questions. The only difference is the score. The total score of each question in CMO is 21 points, six questions in two days, and a total of 126 points, which is three times that of IMO. This is to meet the cognitive habits of Chinese people. IMO is 7 points per question, with a total score of 42 points.
Four and a half hours is a long time for a normal exam, but it is still tight for the candidates who are taking the CMO. Although there are only three questions in each CMO, these three questions are not easy to answer. For many candidates, even if they are given twice as much time, it is not too much.
Especially this year is an odd-numbered year, according to the rules, the questions are always harder than those in even-numbered years. In addition, the Chinese team has won great victories in the Olympic Games in the past two years, and has overcome many difficulties. Compared with the outstanding results of last year's IMO, which won five gold medals and one silver medal with a total score of 217 points and won the first place in the team, the difficulty of this year's questions can be imagined if we want to select talents to enter the international arena and continue to shine and achieve better results.
Many candidates become confused as soon as they read the questions on the test paper and start to rack their brains to think hard. Only a very small number of them start to write down the answers.
Wu Tong got the test paper, checked the information as usual, and took a quick look. It was still the usual CMO question setting method. The first question was a geometric proof. There were two small questions in total, and it was obvious that it was a given proof question with a lot of work.
She looked at the difficulty of the questions and found that they were not as difficult as the classic IMO questions, but they were also more difficult than the national competition questions. The second question was already as difficult as the finale question in the national competition.
However, such difficulty is still not a hurdle for Wu Tong.
Wu Tong read the question carefully, and a brainstorm started in her mind. Her sharp thoughts ran like a flash of lightning, sparking a spark of inspiration. Her thoughts expanded, and the direction emerged like a stream flowing down the river. She began to deduce on the draft paper.
Let Q and R be the midpoints of OB and OC respectively.
Connect EQ, MQ...
Therefore, △EQM=MRF, so EM=FM,
Similarly, EN=FN,
So EM·FN=EN·FM
The first question was solved, and the second question continued smoothly. This question was more complicated. Wu Tong wrote a whole page about the proof process, and the final proof result was negative.
As long as the analysis is clear, doing proof questions is actually much simpler than other calculation questions. There is no need for tedious calculations, and it is very refreshing to deduce step by step. Wu Tong actually likes doing proof questions very much.
After organizing the proof process, Wu Tong copied it onto the test paper and finished the first question.
The second question was about prime numbers. The question was really simple, just one sentence, but the problem was very broad, asking all pairs of prime numbers (p, q)... The difficulty of this question increased sharply. Wu Tong carefully deduced on the draft paper and quickly found the direction.
If 2|pq, let p=2, then 2q=·····
According to Fermat's little theorem, we get...
Similarly, K<1, a contradiction! That is, there is no (p, q) that meets the requirements.
In summary, all the prime number pairs (p, q) that meet the requirements are (2, 3) (3, 2) (2, 5) (5, 2) (5, 5) (5, 313) and (313, 5).
After solving the second question again, Wu Tong moved on to the last big question.
This problem could not stop Wu Tong's progress. The spark of inspiration burst out completely, and Wu Tong quickly came up with an ingenious solution. He first proved a lemma, then used the lemma to derive the conditions for its satisfaction. Finally, he solved the problem in two steps: the number of convex m-gons whose vertices belong to P and have exactly two acute angles. His answer was quite brilliant.
For Wu Tong, the key point to note in this question is the use of mathematical symbols, which must be written neatly and clearly.
The third question was completed, which also meant that Wu Tong had completed all three questions on the first day of the exam. Wu Tong looked at the time and saw that just over an hour had passed. Now Wu Tong had a clear judgment on her derivation results. As long as she did not read the question wrong, the result could not be wrong.
It took Wu Tong more than ten minutes to check carefully, which was equivalent to speeding up the process to verify it again in her mind. After confirming that everything was correct, she raised her hand to hand in the paper. At this time, the exam time had just passed an hour and a half.
In the examination room, many students were still struggling with the first question, those who finished quickly were still working on the second question, and almost no one had finished the third question.
These top students from various schools can be said to be the best in the country ranked within the top 200 to 300. At this moment, I deeply feel that they are like the poor students in school. Their brains are blank and their brains are almost dried up because of the difficult questions.
At this time, Wu Tong's actions were like thunder to the candidates who noticed her actions.
How much time has passed since everyone handed in their papers?
Did they waste too much time, or did some people just get burned out and give up on the exam?
The invigilator was also shocked. It was only half an hour? Was this candidate sure he wanted to hand in his paper and end the exam? However, when he came over and saw Wu Tong's neatly written paper, his doubts turned into silence.
Kids nowadays are really amazing! If they were asked to take this year's test, which is unprecedentedly difficult, they might not be able to finish the test so quickly.
With such test questions, there were actually candidates who were able to complete and hand in the answer sheet three hours in advance. He must be another top candidate who can reserve a place in the national team!
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