Exam Room 8.

Xiao Ran was immersed in the joy of the Olympiad at this time.

His brain was very excited today, and the process of solving the problem was unusually smooth.

He met little resistance from the 10 multiple-choice questions, and quickly wiped them all out.

The fill-in-the-blank questions are not very difficult, and although there are some pitfalls, they basically do not take much time.

But just as Xiao Ran was about to continue his march, he suddenly found himself encountering a hurdle.

This is the last of 4 fill-in-the-blank questions.

[It is known that sin(x+sinx)=cos(x-cosx), where x∈[0,π], then x=().]

The stems are short and seem unremarkable, but they are a bit difficult.

At the beginning, Xiao Ran didn't take it seriously, he was in full swing, and he was going to take it down when he came up.

But three minutes later, when Xiao Ran found that he hadn't solved it yet, he suddenly realized that the little boss was coming.

In his previous life, Xiao Ran liked to compare the entire Olympiad test paper to a copy, and each type of question was a level.

And the questions are the monsters in the level, and the candidates need to fight the monsters to clear the level.

Just now, he went all the way unimpeded, and he naturally understood that those questions were low-level monsters.

And now, he finally meets the elite monster in the trigram, which he usually calls the mini-boss.

Only by defeating this mini-boss can he successfully enter the final level of the dungeon and fight to the death with the real monster.

Thinking of this, Xiao Ran calmed down his mood, and he realized that he was a little impatient.

So he didn't start writing immediately, but carefully reviewed the problem stem and the previous calculation, and looked at it for a while before starting to write again.

[The original equation is equivalent to cos(π/2-x-sin x)=cos(x-cosx)

then x-cosx=2kπ+π/2-x-sin x(k∈Z)(1)

or x-cosx=2kπ-(π/2-x-sin x)(k∈Z)(2)

From (1), we get: 2x+sinx-cosx=2kπ+π/2, and the function f(x)=2x+sinx-cosx is an increasing function at [0,π],

-1=f(0)<2kπ+π/2

By (2) calculus, self-contradiction, discarded.

Therefore x=π/4. 】

After the calculation, the miniboss fell to the ground with a bang!

Xiao Ran exhaled lightly, and after this question, he also realized that he couldn't underestimate these questions.

You must calm your mind and never be impatient, otherwise you will unknowingly fall into it and waste more precious time.

So, when he began to solve the big question, he regained his usual composure.

Hu Jianmin has been paying attention to Fang Yiyang in his school.

He hoped that Fang Yiyang would be able to achieve better results this year and win glory for Caozhou No. 1 Middle School on the stage of the province and even the whole country.

Fang Yiyang's performance today did not disappoint him, because Fang Yiyang did it quickly.

As a senior mathematics teacher who has been teaching for many years, he is naturally very interested in the test questions of this competition.

So, after a few laps, he took the time to flip through the questions this time.

He first looked at the previous questions, and finally came to a general conclusion.

That is, most of the fuss this time is decent, and only a few angles are a little tricky.

For example, the last question of the fill-in-the-blank question seems simple, but it requires a high level of basic skills for the candidate.

If you're not careful, you may end up in a loop.

Just now, Hu Jianmin went down and took a turn, and sure enough, he found that many candidates were stuck in this question.

Some racked their brains to calculate there, but Hu Jianmin took a look and knew that he was wasting his efforts in vain, because the direction was wrong.

This candidate wanted to go over there.

He is not to blame for this, this question does look like it is examining trigonometric transformations.

Hu Jianmin sighed slightly, and deliberately came to Fang Yiyang's side to take a look.

At this time, he happened to see that Fang Yiyang had just calculated this problem, and the method was right.

Seeing this, he couldn't help but breathe a sigh of relief, it seemed that with Fang Yiyang's level, this question was obviously not enough.

He is worthy of being the No. 1 seed player in No. 1 Middle School and Caozhou City.

Hu Jianmin turned around and was about to leave, but at this time, he accidentally saw Xiao Ran's test paper and suddenly felt a little surprised.

Because, Xiao Ran actually did this question at this time.

You know, he's 5 minutes behind the rest.

What surprised Hu Jianmin even more was that his method of solving the problem was not wrong.

Could it be that this Xiao Ran turned out to be a hidden master?

Hu Jianmin was full of suspicion, but he didn't stay next to Xiao Ran for long, but his heart became more curious about Xiao Ran.

He decided to continue to pay attention to Xiao Ran of Caozhou No. 2 Middle School next.

Xiao Ran didn't know that because he was fast enough to do the questions, he had already registered with the invigilator.

He continued to write non-stop.

Maybe it's because today's state is exceptionally good, but also because the first few questions this year are not too difficult, Xiao Ran broke three big questions in a row.

It's overwhelming!

However, he did not let up, because he knew that the next two big questions were the main event of the day.

Both of these questions are 15 points, and the total score is 30 points, which is a large weight.

But, surprisingly, the 18th and penultimate question doesn't seem to be that difficult.

This is a geometry problem that requires proof that the ratios of the 4 line segments present in a tetrahedron are equal.

Xiao Ran thought for a while, then tried to discuss it in two situations, but accidentally came up with it.

This even Xiao Ran himself didn't expect.

At first, he thought that he had missed something, but after a closer inspection, there seemed to be no problem.

So, Xiao Ran stopped caring about it and began to look at the last finale.

[It is known that the function f(x)=ax^4+bx^3+cx^2+dx satisfies.]

(1) a, b, c, d are all greater than 0; (2) For any x∈{-2,-1,0,1,2}, f(x) is an integer; ③f(1)=1，f(5)=70.

Try to determine whether f(x) is an integer for each integer x, and give an argument for your conclusion. 】

After reading it, Xiao Ran knew that the ultimate boss was coming.

Success or failure is a question here!

So, he cheered up the spirit of twelve points and began to calculate carefully.

Hu Jianmin suddenly felt that Fang Yiyang was a little bad.

Because Fang Yiyang has been stuck on the last question for 5 minutes, there has been no progress for a long time.

He looked at Xiao Ran next to Fang Yiyang again, who was also trying.

The other students in the exam room had not yet started to do the last question, and many were even stumped by the penultimate question.

Fang Yiyang and Xiao Ran were the fastest progressing in the entire No. 8 examination room.

Hu Jianmin looked at the last question this time, it was difficult, I am afraid that it will really open the gap between the candidates this time.

This question can also really see who is the real math master, is it Fang Yiyang, or Xiao Ran?

Although the other party's strength is full of confidence, Hu Jianmin knows that this Xiao Ran should not be underestimated.

He thought that Xiao Ran was just a bronze who came over to paddle casually, but he didn't expect that he was really not simple.

Because, Xiao Ran's progress in doing the question went hand in hand with Fang Yiyang.

This is not enough, what is more important is that the answer to the big question made by Xiao Ran is completely consistent with Fang Yiyang.

Now, both of them have done the last question together, and they have been thinking about it for a while, but neither of them has been able to answer the official answer sheet for a long time.

It seems that all have encountered some difficulties to a greater or lesser extent.

Seeing this, Hu Jianmin didn't dare to stay next to them for long, he was afraid that it would affect the state of the two of them.

So, he prepared to go back to the podium and talk about it later.

But at this moment, he suddenly saw Xiao Ran write!