The second question is a game question, but this game requires more brainpower than ordinary games. It requires you to prove that there are 2010^2010^2010 gold coins in the last box.
It was an interesting question. Wu Tong wrote down the solution process with great interest. He used the stem condition to set the operation condition, and used the directional condition to calculate the number of operations... Finally, after a finite number of times, he found that there were no gold coins in other boxes, but there were 2010^2010^2010 gold coins in the last sixth box.
The third question is another difficult watershed question, a deep number series problem, which puts together the usual IMO question-setting strategy, with geometry, algebra, and number theory categories all competing to appear.
Thinking freely, Wu Tong wrote down the complex reasoning for the final question in one fluent breath, without pausing. Finally, he wrote, "To sum up, the conclusion stands." He checked his pen, raised his hand, and handed in his paper. Within the hour, Wu Tong left the exam room.
There was bright natural light outside, and after an hour, the temperature was not much different from before she entered the examination room.
The IMO competition was a complete success. My secondary preparations for going to university early were also a complete success.
When Wu Tong came out, as they stepped into the final examination room, Professor Cheng and Professor Li, who had finally been released from confinement, were basking in the warm sunshine by the window and sipping fragrant coffee.
Both of them have similar study abroad experiences and can be considered as the new generation of youth. Their acceptance of coffee is much more sophisticated than that of Wu Tong, a country bumpkin who stepped out of the country for the first time!
"Congratulations, Wu Tong! You've broken the fastest IMO test-taking record! Are you confident you'll get full marks?" Professor Li smiled and saluted Wu Tong from a distance. "Come, sit over here. I ordered a small dessert and pistachio yogurt ice cream for you. They've just been served!"
Wu Tong's timing was still incredibly precise. He knew it wouldn't take more than an hour. Professor Li, who was there to translate the questions, didn't need Wu Tong to write them out by hand today.
"The score depends on the judgement of the organizing committee!" Wu Tong smiled easily. She could still guarantee the correctness of the result, but the specific score would depend on whether the organizing committee would find fault with it.
"Hahaha... Don't worry about that. As long as the process is fine and the result is correct, no matter how reckless they are, we in China are not afraid of anyone. Teachers are no pushovers!" Li Yishen was full of vigor and arrogance. They, the team leaders, all had the power to mark the papers, and no one could blacken their students' grades for no reason.
Wu Tong’s achievements are all there for us to see, and the organizing committee believes in them and doesn’t dare to do too many small moves!
While eating desserts and exchanging ideas, from the initial discussion on the last three questions of the IMO to further extended exploration in the fields of geometry, number theory, and algebra, Wu Tong's effortless connection, rigorous logic, powerful and clever thinking, and in-depth analysis and solutions to problems made the two professors praise Wu Tong's knowledge reserves.
You know, the two of them can be said to be the backbone of Peking University. They have been studying mathematics for more than 20 years. I dare not say that they are particularly outstanding in mathematics, but I think their knowledge is solid and worthy of their titles and students.
But when I look at the underage child Wu Tong communicating harmoniously with them again, I can't help but feel that the new generation is pushing the old generation out, and the old generation dies on the beach.
When K=1, for all natural numbers k, there are infinitely many prime number pairs (p, p + 2k).
Staring at the ice cream in front of her, which was dotted with pistachio crumbs and had a unique beauty, Wu Tong suddenly fell into deep thought. The conditions for the twin prime conjecture popped up in her mind. The ice cream in front of her was no longer ordinary ice cream, but a series of wonderful formulas with unique mathematical language.
The right Lambda function?
Viongradov's three prime number theorem?
The infinite extension of topology...
Suddenly, Wu Tong had a flash of enlightenment. Instinctively, she pulled out a pen and paper from her backpack and began to deduce on a piece of scratch paper... She was completely lost in her own world. A series of mathematical formulas coalesced under her pen, and in her mind, her thoughts expanded endlessly, stirring up an endless mathematical storm.
p1, p2, ..., pn, let N = p1 × p2 × ... × pn. If N + 1 is a prime number, then N + 1 must be greater than p1, p2, ..., pn
···5*7*11*13···p*3*5*7*11*11*17`````(p-2)/p····
The wind blew in gently from the window, but at this moment, Li Yisheng and Professor Cheng no longer had the leisurely and comfortable mood they had just now.
"This is..." Professor Cheng recalled what he had heard before and looked at Li Yishen in disbelief.
"Haha..." Li Yisheng shrugged and spread his hands. This state was very familiar to him. As you can imagine, we witnessed the child's inspiration burst out again.
Did we discuss anything special just now? Were we in the same dimension just now?
How come we are still the same, without any inspiration from the white dove, but the kid opposite us is writing eloquently and launching an attack on the twin prime conjecture right in front of us?
The small round table for resting was not big. Even without moving, the two of them could see what Wu Tong had started to write. The familiar formula appeared, and they both thought of the twin prime conjecture that Wu Tong had been studying.
All I can say is that our brains are vastly different from those of geniuses. To a genius, the traces of wind, the contours of rolling clouds, even the patterns on ice cream... in their eyes, they are all extensions of mathematics!
The two teachers dared not utter a single sound, performing a silent pantomime. Li Yisheng smiled bitterly, unable to conceal the excitement in his eyes. He was anxious about his own brain, but also excited. This child belonged to China, was a student of their Peking University! He could even shamelessly say that he had taught her!
Li Yisheng and Professor Cheng perked up, keeping an eye on their surroundings, preventing anything from disrupting Wu Tong's inspiration and interrupting this moment of potential miracles. They also carefully helped Wu Tong organize and preserve the draft papers she had placed on the table. These would become treasures of the caliber of cultural relics in the future!
Wu Tong continued to deduce this until the end of the exam, then she raised her head to stretch her shoulders and neck, and met two pairs of eyes that were looking at her eagerly and passionately.
"Teacher Li, Professor Cheng, what are you doing?" She was almost scared.
"Wu Tong, have you...proven this?" Li Yisheng and Professor Cheng asked in a low, cryptic voice. This wasn't China; no matter how cautious they were, it couldn't be excessive.
Wu Tong shook his head. "We haven't even finished yet. How can it be that simple?"
Oh, that's good. Li Yishen and Professor Cheng breathed a sigh of relief. The progress at this stage was not bad. Every new step was a gain. They just breathed a sigh of relief and were about to comfort Wu Tong, telling her not to worry, as her breakthrough and progress were already very fast.
Wu Tong paused and then continued, "Fortunately, the general direction has been decided. I should be able to send myself a certificate as a gift before my birthday!"
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