The IMO exam is the same as the exam in the CMO competition and training team that Zhao Xiancai participated in when he was in China.

The only difference is that each of the six questions in these two days is only worth seven points.

When Zhao Xiancai saw the first question on the paper he was handed out, he was a little surprised because the first question was very simple.

This question was originally in English, but when it was sent to Zhao Xiancai, it had already been translated into Chinese.

The title of the first question is as follows. Let a0
(n 1).

The question stems are very concise, and Zhao Xiancai, who has gone through so many IMO test questions, has already discovered some patterns.

In IMO test questions, generally if the question stem is very short, then the difficulty of the question is about the same, and it is usually not too difficult.

But if the question stem is longer, the difficulty of the question will usually be more difficult.

The same is true for this question. After reading the question, Zhao Xiancai had already thought of a solution to this question before he even entered the state or triggered the [Study Hard] passive.

The first solution he thought of was proof by contradiction.

That is to say, first assume that there is no n that satisfies the condition in the question, and then use this condition to deduce a result that is obviously contradictory to the known conditions in the question. Then we can prove that there is such a positive integer n, and then prove it again.

Proof by contradiction is also a method often used in sequence proof questions.

After thinking of this method, Zhao Xiancai immediately started writing.

After he finished writing, he had entered a state of "immersive learning", and the effect of his passive skill [Study Well] had also been triggered.

Therefore, after he finished writing the first method, he immediately thought of the second method, which was to use the construction method to first construct a new sequence.

Then use the proof sequence to prove the content of the question. This method seems to be simpler.

After Zhao Xiancai went through the general process of this method in his mind, he wrote the second solution on the test paper.

After finishing the first question, Zhao Xiancai focused on the second question.

This second question was also quite simple. After solving it quickly, Zhao Xiancai looked at today's third question.

Today's third question was a bit more difficult than the first two questions. When Zhao Xiancai read the question, he guessed that someone in his team must not get full marks for this question.

After spending about ten minutes to solve the third question, Zhao Xiancai spent some more time writing out the second solution to the second question and the third question.

When he finished all this and looked at the time, he realized that there was still more than an hour and nearly two hours until the exam was over.

"I didn't expect that I didn't spend as much time on the three questions on the first day of this year's imo as I did during the training team exam. But maybe it's because I've improved during this time?"

Zhao Xiancai put down his pen, looked around at the candidates who were mostly frowning and meditating, and thought to himself.

It's a pity that IMO exams cannot be submitted in advance. In the following time, Zhao Xiancai, who had nothing to do, did not find a third solution to these questions. Instead, he ate the snacks provided by the organizer and emptied his mind.

When I took the CMO exam and the training team exam before, the organizer also provided small snacks. Those small ringtones were some common domestic chocolates, biscuits, bread, etc., as well as drinks.

When we were at the National Fair, coconut brand mango juice drinks were distributed.

He had heard of the Coconut Brand coconut juice, but he had never seen the Coconut Brand mango juice drink before, either on TV or in real life.

After all, this exam lasts four and a half hours, so candidates will inevitably be hungry.

The candidates around him had obviously noticed Zhao Xiancai. Seeing Zhao Xiancai eating snacks leisurely and looking extremely relaxed, these candidates from all over the world also complained in their hearts.

"Did this person finish writing so quickly? It seems like he is another old man."

"Why didn't he check it after he finished writing? Are you so sure that he did everything right?"

"Asians are perverts in the field of mathematics competitions."

"Isn't this the outrageous Chinese player we saw during the opening ceremony?"

…

However, Zhao Xiancai didn't know what these people were complaining about him in their hearts. As long as he didn't violate the on-site examination discipline, the invigilators wouldn't say anything about him.

After the exam was over, when everyone went to the place where the organizer provided lunch, Huang Yiming asked everyone: "Do you think today's three questions are difficult?"

"It's okay. I feel like it's not as difficult as when we took the exam in the training team or when we were training with the national team. It's quite simple."

"I don't think it's that difficult either."

"I think it's a little difficult, but at least I've finished writing it, and there should be no problems with my proof process."

"I also feel that it is a bit difficult. I probably won't get full marks for the last question. How many points can I get?"

"My problem is also with the last question."

This last sentence was said by Huang Yiming himself, and the previous sentence said that the person who failed to get full marks on the last question was Chen Lantian, who had always been "in the same boat" with him.

Qi Ruilin was silent now, and they all turned their attention to Qi Ruilin.

"I also feel that the last question is quite difficult, but I have finished it, but I don't know if it is correct or not," Qi Ruilin said.

"If that's right, just ask Zhao Xiancai and Gao Peicai. If they say it's right, then there must be no problem. Zhao Xiancai, how did you two prove the last question?"

Chen Lantian looked at Zhao Xiancai and Gao Peicai and asked them.

"Draw a few auxiliary lines, and then prove two lemmas..."

The last question was a geometric proof question. Zhao Xiancai gave a rough outline of the two proof methods he wrote on the test paper.

This chapter is not over yet, please click on the next page to continue reading! "Wucao, do you dare to write two solutions to this kind of game? You are not afraid of any mistakes in one solution. Even if you solve the whole team with another solution, they will

I won’t give you full marks either?”

After hearing what Zhao Xiancai said, Chen Lantian and the others' first reaction was not that Zhao Xiancai used these two methods, but that he actually used two problem-solving methods in this situation.

"You didn't use two methods to solve the other two questions, right?"

Chen Lantian asked again.

"Yes, the main reason is that the first two questions are too simple. I used two methods to write the first question from the beginning, but that's it. When I used the first method to finish the second question, I only used a total of

It took about thirty minutes.

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It was too early when I finished writing the first time. I didn’t have an hour, and I couldn’t hand in the paper in advance, so I wrote it the second time.

Even so, it was still more than an hour when I finally stopped writing, and I ate snacks in the examination room for an hour, which made me not very hungry now." Zhao Xiancai explained.

After hearing Zhao Xiancai's explanation, Chen Lantian and Huang Yiming looked at Gao Peicai again and said to him: "How did you prove the third question?"

"I used the same method as Zhao Xiancai." Gao Peicai said.

After that, Zhou Wen and Qi Ruilin also talked about their proof methods for the third question.

Zhou Wen's method is the same as Zhao Xiancai's first method, but Qi Ruilin's method is different from Zhao Xiancai's.

From the previous training at Modu Middle School, everyone knew that Qi Ruilin was actually inferior to Zhao Xiancai and Gao Peicai, and his strength was about the same as Zhou Wen.

But the difference between him and Zhou Wen is that his ideas are always so different and novel, and his problem-solving methods are always different from what Zhao Xiancai and others thought of.

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