(Thanks to the two thousand rewards of the husky called the lion at home!)

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The first question is a complete scoring question.

This question may still have a certain degree of difficulty for those players from weak countries.

But for the students in the first few rows, they are all members from major competition powers, so there is naturally no pressure to do it.

The difference is nothing more than the speed of completion.

The less time you spend completing the first question, the more time you can spend on the next two questions.

Question 1: [Find all positive integer pairs m,n≥3 so that there are infinitely many positive integers a, so that (a^m+a-1)/(a^n+a^2-1) is an integer.] 】

Bi Qi touched his chin and muttered for a few seconds, and then he had an idea in his mind.

Holding the pen, the tip of the pen slurped the formulas on the scratch paper, while muttering a long list of things that ordinary people couldn't understand at all.

"First of all, one thing that can be determined is m≥n, then the next thing you need to construct is to construct two functions. ”

"f(x)=(x^m+x-1),g(x)=(x^n+x^2-1), let f(x)=r(x)g(x)+s(x), r(x) and s(x) should all belong to integer polynomials. ”

Then, give it a Pei Shu theorem, which gives the greatest common factor for the existence of r(x) and s(x). ”

“…… Here, let's go straight to the infinite descending method! to reduce the power of the equation. Recycle...... find m=5, n=3, then we only need to prove that for any integer a, (a^5+a-1)/(a^3+a^2-1) are all integers!"

In ten minutes, Biqi completed the first step of transformation.

That is, the values of m and n in the question stem are determined, and the problem is transformed into an inequality proof question with only the difficulty of the ordinary college entrance examination.

"It's incredibly easy!"

Twenty minutes into the exam, Bi Qi looked at the first question on the scratch paper that had been proven by himself, and smiled easily.

IMO's title is not as scary as he imagined!

Thinking like this, Bizzie's gaze fell on the second algebra problem.

Although I already know the Lagrange multiplier, it is the key to solving this problem.

However, the specific derivation process still needs to be carefully thought and sorted out.

…………

Inside the lounge.

Every move of each candidate in the venue was watched by the team leaders from all over the world.

At present, most of the candidates are still working on the first question.

Therefore, except for a few candidates, the expressions on the faces of most of the candidates are quite normal, and there is no such expression of bitterness and hatred.

Apparently the progress is relatively smooth.

This made many coaches breathe a sigh of relief.

Fortunately, it didn't appear, and their team members were stuck in the first question of the sky collapse picture.

"Hey, someone has already finished the first question?" said the Canadian team leader.

"Where?" said the Canadian team leader, which attracted the attention of many people.

After all, it has only been more than 20 minutes, and although the difficulty of the first question is not too high, it is still a little too fast to complete the answer at this point in time.

"Here, look!" the Canadian leader gestured to a figure on the surveillance screen.

And that figure is none other than Bi Qi, a classmate of the Huaguo team.

From the monitoring screen, it can be vaguely seen that the answer sheet corresponding to the first question on Bi Qi's desktop has been filled with dense formulas.

At this time, Bi Qi was facing the second question, sketching on the test paper.

One of the team leaders spoke, "This ...... It seems to be a member of the Huaguo team. ”

Team Hua Guo ......

Several people turned their heads in unison and looked at Gu Lu, who was leaning on the sofa and playing with his mobile phone with Erlang's legs.

Although Park Changqu, who was sitting next to Gu Lu, had been acting very calm, he had actually kept his attention on the monitoring screen.

The conversation of the team leaders just now, Park Changqu naturally heard everything.

When he learned that a member of the Huaguo team had completed the first question to answer, Park Changqu's whole heart couldn't help but pwitch.

Just when Park Changqu's heart was about to beat to his throat, the Canadian team leader screamed again.

"Someone has completed the first question again, this time ...... The members of the Korean team!"

"Hiss——! not just one Korean player, but two Korean players have completed it all!"

Whew-

Big rocks fall to the ground.

After enjoying everyone's envious gaze, Park Changqu glanced at Gu Lu, whose expression was still indifferent, and said with a smile, "Team Leader Gu, it seems that our Korean team is still slightly better?"

"460 again?" Gu Lu shrugged, put the phone with the gray screen on the desktop, and turned his head to look at Pu Changqu, "By the way, what did you just say?"

Park Changqu: "......"

In the next 10 minutes, candidates completed the first question.

However, the 12 students who have completed the first question so far, all of whom are in the top two rows, belong to the top six countries in the last IMO group rankings.

None of the students in the back row completed the answer.

The strength of the competition power is undoubtedly demonstrated at this moment!

"Heh, what's the use of being so fast, and there's no guarantee that the answer will be correct. It's better to do it slower and more steady. The leader of a small country spoke in a sour tone.

…………

Sitting next to Biqi, the player named Kim An-tae is the second member of the Korean team to complete the first question after Biqi, as the Canadian team leader said.

After completing the first question, Kim An-tae looked at the time.

Twenty-five minutes, much faster than he expected.

He subconsciously wanted to see the progress of the others, and the Korean teammate on the left hand side was still doing the math on the scratch paper, which didn't surprise him.

After all, whether in the training match or in the knockout round, he has always been in the first and second place among the six members of the Korean team.

And the teammate next to him often hovers on the verge of elimination.

Turning his head to look to the right, he looked at Bizzie.

Biqi's greeting of "Savadika" before the exam left a deep impression on Kim An-tae.

He quickly glanced at Bi Qi's desktop, and then did not calm down for a moment.

Question 2!

This Chinese is doing the second question!

Since he was sitting next to Bizzie, he saw it much more clearly than the group leaders in the lounge could see on the surveillance screen.

Judging from the outlined symbols on the test paper, this Chinese native has been doing the second question for a while.

I...... It's not the fastest!

Jin Antai forcibly suppressed the panic in his heart, and held back his gaze from looking at Bi Qi's side.

He lagged behind, but not too much.

With his strength, it will not be a problem to easily catch it back.

But the premise is, don't panic!

Kim An-tae remembered the admonition that Team Leader Park Changqu had told him last night, took a deep breath, stabilized his mind, and looked at the second question on the test paper.

…………

Time passed little by little.

Whether it is the exam venue downstairs or the team leader lounge upstairs, the tense atmosphere is gradually permeating.

It wasn't until the hour came that there were players in the back row who completed the first question one after another.

The coaches of several small countries breathed a sigh of relief.

The time difference of half an hour is barely within the acceptable range of them.

Next, it's up to them to see if they can solve the second question with three and a half hours left.

As for the third question......

The leaders of those small countries never thought that there would be players in their team who could do it.

If you can get the first two questions right, it's all burning high incense!

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PS: Before ten o'clock, stabilize the top 300 recommended votes, and there will be more!!