On November 14th, the first test of the mathematics national final officially began.

There are only three questions in total, and the time limit is four and a half hours to complete.

In the national final, where all the top students are in the country, the gold content of each question is not high.

It can even be said that from the beginning of the national finals and even the World Olympiad, it is not at the same level as the previous provincial competitions.

The classrooms were quiet.

Su Mu gently flicked open the test paper and flattened the scratch paper.

The first question is the most valuable question.

Let a,b,c,d,e≥-1 satisfy a+b+c+d+1=5, and find the maximum and minimum values of S=(a+b)(b+c)(c+d)(d+e)(e+a).

The topic is very short and short, and there are even only a few characters, but Su Mu suddenly felt a burst of pressure.

In ordinary exams, very short questions are likely to be scored questions, but in the Olympiad, especially in the Olympiad National Mathematics, the shorter the questions mean that less information can be obtained, and the more difficult it is!!

S=（a+b）（b+c）（c+d）（d+e）（e+a）

Su Mu looked at the topic, and for the first time felt the familiarity of being a scumbag, he didn't know how to write.

It is impossible to expand, and when it comes to the quintessential equation, even if it is expanded, it is not particularly useful, there must be other methods.

Seeing this first question, most of the other candidates in the examination room gasped, some had begun to break out in a cold sweat, some looked at the second question calmly, and some people may have had a flash of inspiration and started writing directly, but the next moment, the aura in their eyes suddenly dimmed a lot.

Su Mu carefully observed the first two conditions of the question, and the concept of the average principle flashed in his mind.

Find the maximum value first, obviously S is a positive value when taking the maximum value, because a+b+c+d+1=5, from the principle of average, we can know that there is at least one number greater than or equal to 1 in abcde, and each symbol appears twice, therefore, a+b, b+c, c+d, d+e, e+a, at least two non-negative values will appear.

If S is positive, then there may be 0 or 2 negative numbers in these five numbers.

If there are no negative numbers, there is a mean inequality to know that S=(a+b)(b+c)(c+d)(d+e)(e+a)≤ [(a+b+b+c+c+c+d+d+e+e+a)5]^5=32

If there is a negative number....

Su Mu paused, if there is a negative number....

If there is a negative number...

Su Mu felt that there should be no problem with his thinking, but if there was a negative number in this case, he really didn't know how to solve it.

Sure enough, it was still too difficult to participate in the mathematics national final at the level of Level 6 mathematics.

After thinking about it for more than ten minutes, Su Mu didn't figure out what to do if there was a negative number.

Doing math problems, the most uncomfortable time is stuck in this kind of place, although there are only three problems in the whole try, although it is said that there are four and a half hours to think about Su Mu.

However, in the world of mathematics.

I can't think of that, it's very normal to be stuck for several days!

Su Mu put down the pen in his hand and took a deep breath.

Sure enough, the national competition is the national competition, and even a student of his level is not at a disadvantage.

In fact, he can now go and look at the next two questions first, ease his thinking, and then do the first question.

But Su Mu inexplicably has an obsessive-compulsive disorder, so he has to make this question first.

If there is a negative number....

Damn it.

Isn't it normal to have negative numbers.

Things went back to that cycle, and Su Mu constructed seven or eight equations to explain if there were negative numbers, but none of them could play a substantial role.

Although he can now directly use skill points to improve mathematics to level seven or eight, Su Mu is still a little unwilling to admit defeat.

If everything can only be solved by the system, then what is the point of him practicing so many Olympiad problems every day?

Is it true that everything is undecided?? every day

Su Mu picked up the pen again, he didn't believe that he couldn't find a solution.

Although he knows that he himself is a bit of a bull now.

But it's just such a simple formula as ABCDE merging into one, and he doesn't believe in this evil!!

Another twenty minutes passed.

Su Mu set up a whole page of equations, nearly twenty kinds of special assignments.

Still didn't achieve good results, but Su Mu caught a line between the shadows, as long as this line is organized, it will definitely be able to make it perfect!!

If there is a negative number, then it will be discussed in several cases, assuming that the values of the negative terms are x, and y, and the positive terms are P, Q, and R, respectively, and x, y, are greater than or equal to -2, and P+Q+R≤14.

If the two terms P, Q, and R are not adjacent, because the sum of the five numbers is 5 and any one of them is greater than or equal to -1, then their sum is less than or equal to 6, and if there are two non-connected items in PQR, the sum of the two must be less than or equal to 6.

After persisting for nearly an hour, Su Mu's back was already soaked with sweat, and he finally found a slightly more feasible equation discussion group!!

Moreover, according to Su Mu's intuitive understanding of mathematics, he still thinks that there is no problem with his thinking!!

As long as you persist, you will definitely be able to write this question with your own strength!!

In the examination room, there were sounds of students drinking water one after another, and everyone answered the questions seriously.

The invigilators did their part and paid tribute to these struggling students.

“！”

Suddenly, a thought flashed through Su Mu's mind, and he hurriedly wrote it down.

"Suppose P≤Q≤R, then P+Q≤6, so PQR≤PQ(14-P-Q)≤ [(P+Q)2]^2(14-P-Q).

"Write f(x)=x^(14-x), then the derivative is f'(x)=28x-3x^2."

Maybe it's because he held back for more than an hour, and when the ideas came out, the pen in Su Mu's hand was out of control, and he completely fell into a kind of pleasure of solving the problem!!

Yes, that's it!

A glint flashed in Su Mu's eyes!

"When x≤6, f'(x)≥0, so [(P+Q)2]^2(14-P-Q)≤72"

"Solid xyPQR≤288, and when abcde is 4, -1, -1, -1, 4, etc. The maximum value is 288. ”

Su Mu's whole person was excited, and he made the first question and the first question in an hour and twenty minutes!!

He would love to put an exclamation point behind the last 288, but then he thought "! " can represent factorial, so in the end only a full stop is written as the end.

It's tough.

Why is math so hard?

Su Mu was sad and happy, sad because the topic was really difficult, and the joy was that this topic was finally overcome by Su Mu.

This great sense of accomplishment.

It's even stronger than he gets a skill point!!

If the maximum value is solved, the idea of the minimum value is much smoother.

Obviously, when S is the minimum value, the S value is negative, and if S is negative, then....

....

When the solid xyPQR is ≥-512, abcde can be taken when -1, -1, -1, -1, and 9 are taken respectively.

According to the title, the maximum value of S is 288 and the minimum value is -512.

Question 1.

Perfect Finish!!

.....