The nine players participating in the training camp nodded earnestly. This was a problem they needed to avoid. They had to keep in mind that their opponents were among the top-ranked high school students in the country, each of them a true genius. They had to strive for better results, and no time was allowed to slip!
They were all top students in mathematics. They suddenly realized that Lao Zhang had only called out the names of 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 the confidence of saving this trump card?
"Don't look at it anymore. As for Wu Tong, I don't have much to say. Just keep performing normally. The national competition isn't your end. I'll give you a small goal: grab a spot on the national team first!"
She'd scored full marks in the league, easily mastered classic IMO puzzles, and even covered number theory in university textbooks—what else could he say? And seeing how this girl took impromptu performance so lightly, he didn't even need to remind her to just play her best and maintain a positive attitude!
He even had a little hope that this girl could join the national team and win a perfect gold medal in the IMO. Then their Central Plains Province would be so famous!
"Is there anything else that Teacher Zhang wants to arrange?"
"Everyone has practiced and prepared everything they need. There is no need to do more exercises today. Just keep a good attitude, get enough rest, and face the competition with plenty of energy! Don't get overly excited before the exam and lose sleep. It will affect your performance and the result will not be worth it!" Teacher Zhang, the steward of the provincial team, gave some instructions on the details.
On the 14th, the Chinese High School Mathematical Olympiad officially kicked off.
"I'm here to wish all my classmates good results!" With the passionate encouragement of the two team leaders, Lao Zhang and Xiao Zhang, all the contestants of 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.
The CMO closely mimics the International Mathematical Olympiad (IMO) format, with one test per day over two days, each session lasting 4.5 hours and containing three questions. The only difference is the score. The CMO has a total score of 21 points per question, with six questions over two days, for a total of 126 points, three times the IMO score. This is to accommodate Chinese students' cognitive habits. The IMO has a score of 7 points per question, for a total of 42 points.
Four and a half hours is quite long for a typical exam, but it's still tight for those taking the CMO. While the CMO exam only has three questions per session, those three are not easy to answer. For many candidates, even double the time would be more than enough.
This year is especially so, as it's an odd-numbered year, and according to the rules, the questions are inherently more difficult than those in even-numbered years. Furthermore, the Chinese team has triumphed at the IMO over the past two years, overcoming numerous obstacles. Last year, they won five gold medals, one silver medal, and a combined total of 217 points, securing first place in the team competition. The difficulty of this year's questions is foreseeable, as they aim to identify talented individuals to compete internationally and continue their success.
Many candidates become confused as soon as they read the questions on the test paper and start to rack their brains in thought. Only a very small number of them start to write and deduce.
Wu Tong received the test paper, checked the information he had filled in as usual, and then took a quick glance. It was still the usual CMO questioning method. The first question was a geometric proof. It had two small questions, and it was obvious that it was a given proof question with a lot of work to do.
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, her mind ablaze with activity. Her sharp thoughts raced like a fleeting stream, sparking a spark of inspiration. Her thoughts flowed, her direction emerging like flowing water, and 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 the first question was done.
The second question was about prime numbers. The question was really simple, just one sentence, but the problem was very broad, asking for 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, 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 completely exploded, and Wu Tong quickly came up with an ingenious solution. He first proved a lemma, then used the lemma to derive the conditions for 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 internal angles. His answer was quite brilliant.
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