A genius? I just love studying.

Chapter 113 The Problem of Euler's Legacy
July 10th, 7:30 AM
On the first floor of Zhihua Building, student volunteers from the School of Mathematics, wearing red armbands, were maintaining order.

A total of 699 contestants, representing all provinces across the country, as well as neighboring countries like Taiwan and Sing Tao, lined up and entered the examination hall one after another.

Each classroom has two proctors, and a camera at the front of the classroom rotates back and forth, like a gun barrel intimidating everyone.

Candidates must maintain a distance of at least one meter between their seats in all directions. Candidates are only allowed to bring the stationery bag provided on the day of registration; all other items are strictly prohibited from being brought into the examination room, and any violation will be treated as a violation.

"Relax, don't feel pressured, believe in your abilities, and just perform normally."

Outside the examination room, An Chengzhang gave his usual words of encouragement.

But as soon as he said it, he felt a little strange, like he was being too arrogant!

Chen Hui nodded and turned to walk towards the security checkpoint.

Upon entering the classroom, a solemn atmosphere immediately envelops you.

Even Chen Hui, a veteran of countless battles, felt slightly nervous in this atmosphere.

It's a completely different feeling from participating in the Barbary Mathematics Competition.

However, a slight sense of tension can actually stimulate potential and allow test takers to perform better, which is why many students score higher on exams than they usually do.

The CMO exam is held over two days, on the 10th and 11th. It starts at 8:00 AM each day and lasts for four and a half hours. The afternoon and evening are free time.

When I received the test paper, there were only three questions.

The format of the CMO is the same as that of the IMO: three questions per day, four and a half hours of answering time per day, for a total of six questions. The only difference is that each question in the CMO is worth 21 points, while each question in the IMO is worth 7 points.

After glancing at the three questions and confirming that there were no problems, Chen Hui carefully examined the first question.

A sports meet was held for n days (n>1), and a total of m medals were awarded. On the first day, one medal was awarded, plus 1/7 of the remaining m-1 medals. On the second day, two medals were awarded, plus 1/7 of the remaining medals, and so on. Finally, on the nth day, n medals were awarded, and no medals were left over. How many days did the sports meet last? How many medals were awarded in total?

"The Euler legacy problem?"

Upon seeing the question, Chen Hui not only arrived at the answer but also discovered the origin of the question.

The Euler's Inheritance Problem states that a wealthy man, on his deathbed, designated a specific way to distribute his inheritance to his sons. The first son would receive 100 gold coins, then 1/10 of the remaining gold coins; the second son would receive 200 gold coins, then 1/10 of the remaining gold coins, and so on, until each son received the same amount of gold coins. The question is: how many sons did the wealthy man have in total, and how many gold coins did the wealthy man inherit?

This is an interesting problem, a classic algebra problem, but it is usually more suitable for upper elementary school students to practice.

这道题解法也很多，最简单的就是设富豪遗产金币为x，所以第一个孩子得到的金币就是100+(x-100)*0.1=90+0.1x。

第二个孩子得到的金币是200+(x-(90+0.1x)-200)*0.1，而两个孩子获得的遗产相等，自然就能算出X为8100，也就能算出富豪有9个儿子。

Of course, there are many other interesting solutions to this problem, such as setting the unknown variable as the number of the rich man's sons, or using the interest of arithmetic sequences...

But the difficulty of this question is definitely no higher than that of elementary school level.

Of course, there won't be elementary school level questions on CMO, so this question has been slightly modified.

The question doesn't specify that the number of medals awarded each day is the same, but the underlying principle is the same. Anyone who has taken junior high school math will find this problem easy to solve.

Let's assume the number of medals remaining on day K is rk. Then the number of medals awarded is mk = k + 1/7(rk - k).

Then the number of medals remaining on day K+1 is r(k+1) = rk - mk = 6/7 (rk - k).

That is, rk - 7/6r(k+1) = k.

所以有r1=m，r1-7/6r2=1……r(n-1)-7/6rn=n-1，rn=n。

等式两边同时乘以(7/6)^(n-2)，然后等式两边相加之后就能逐项相消，最后得到m=1+2*6/7+……+n(7/6)^(n-1)。

再使用点小技巧，用m-7/6m就能得到-1/6m=(1+7/6+……+(7/6)^(n-1))-n(7/6)^n，右边式子的左半边部分明显是等比数列，利用公式求和，最后化简，就能得到m=36+(n-6)*(7^n)/6^(n-1)。

With one formula and two unknowns, it is obviously impossible to find the specific value.

However, the problem states that n>1, so n-6 must be less than 6^(n-1). Since 7^n and 6^(n-1) are coprime, and m and n are positive integers, m cannot have a fractional part. Therefore, n can only be equal to 6, and m can only be 36.

The total time to write out the answers should not exceed two minutes!
Not only Chen Hui, but many students in the classroom also showed happy smiles. It seems that the CMO is going to go easy on everyone this year.

Chen Hui didn't laugh. Although the professor from Jiangcheng University had made a promise to him, he wasn't sure if the promise would still stand if he didn't perform well at the CMO.

From the very beginning, he knew that in the end, this world was determined by his strength.

Looking at the second question,
Let A be the sum of the digits of the decimal number 4444^4444, and B be the sum of the digits of A. Find the sum of the digits of B.

The question was quite interesting. After reading it, Chen Hui's nervousness completely disappeared, and he became fully absorbed in the problem. He had done many math problems and participated in many competitions. At first, he just wanted to make money and improve his situation.

But gradually, when he saw interesting topics, he couldn't help but feel a sense of excitement.

Despite his perfect score in the final of the Acer-Arseam race, CMO and Acer-Arseam are two completely different tracks. Acer-Arseam is like Formula 1 racing, which emphasizes using the best car and the most exquisite technology to win the championship.

CMO, on the other hand, involves having competitors ride mountain bikes downhill from the mountaintop.

Winning an F1 racing championship doesn't offer much help for downhill cycling.

In this classroom, the other examinees who had just been smiling began to frown.

Upon seeing this, the two proctors standing at the podium and the back of the classroom looked up at each other and revealed "healthy" smiles.

This CMO was organized by the School of Mathematics at Yanbei University. The exam was quite large, so it naturally required the assistance of students from the School of Mathematics. The two proctors were also graduate students from the School of Mathematics.

They noticed today's questions when the papers were handed out. They thought the teacher had gone a bit too far this time, but when they thought about how they felt during their final exams and saw these kids with their brows furrowed, they felt inexplicably happy.

This is only the second question. When these little guys see the third question, they'll probably be even more "surprised".

Neither of the two graduate students had yet figured out how to prove that problem.

Thinking of this, the two laughed even more heartily.

Chen Hui frowned for a second, then relaxed.

You can't really see anything just by looking at 4444^4444, but if you write down a slightly larger number, it's easy to spot the pattern.

很显然，在十进制中，任何一个数字n与他的各位数字之和模9是同余的，例如2025%9=（2+0+2+5）%9=0，这很好证明。    只需要将由k位数字组成的n写成n=10^k·dk+……+10^1·d1+10^0·d0这种形式，学过一点二进制的同学很容易就能想到这种表达方式。

然后只需要稍微处理一下，将原式写成n=（10^k-1）dk+dk……+(10^1-1)d1+d1+d0，显然，10^k-1模9等于0，所以n模9，就等于dk+……+d1+d0，上面的结论得证。

有了上面的结论后，很容易就能得出，B的各位数字之和C与B模9同余，C又与4444^4444模9同余，4444^4444%9=(493*9+7)^4444%9=7^(3*1481+1)%9=(7^3)^1481*7%9=(9*38+1)^1481*7%9=7。

而4444^4444<(10^4)^4444=10^17776，当这个数字的每位数都是9时，它的各位数之和能够取到最大值，也就是说A≤9*17776=159984。

在小于159984的数字中，各位数之和最大的是99999，即A的各位数字之和B≤5*9=45。

Among numbers less than 45, the largest sum of digits is 39, meaning the sum of digits of B is C≤12.

Based on the initial conditions, C%9=7, so C can only be 7!
Your math level has improved from 2 (72%) to 73%.

The moment I wrote down the answer, the barrage of comments popped up again.

My proficiency has been improving faster and faster lately.

Chen Hui knew that this was largely due to his newly enhanced insight. It was somewhat similar to a game character who had obtained a powerful artifact. The character's level was low, but the stats were already overpowered. With a wave of his hand, a group of monsters would be instantly killed, and his experience points would increase rapidly, naturally leading to a rapid increase in level.

Of course, doing practice problems has always been a great way to improve proficiency, especially when it comes to problems that are a bit challenging!

Chen Huicai realized that his proficiency did not improve when he participated in the Asa competition. Now it seems that the Asa competition questions may be difficult and require profound knowledge to solve, and require in-depth study of a certain direction of mathematics, but the ingenuity may not be as good as the CMO.

Mathematics always strives for simplicity and elegance!
When Chen Hui looked at the third question, the minute hand of the clock at the back of the classroom had just crossed 90 degrees. The other students were either frowning and thinking hard or busy doing calculations on their scratch paper.

When the two proctors noticed him turning the page, they instinctively stepped forward to check on him, but due to their duties, the student at the back of the classroom stopped and allowed the other student to approach Chen Hui.

After last year's incident, Asai was already very popular this year. Chen Hui followed Jiang Sheng's path last year, which pushed Asai's popularity to another level. Therefore, many mathematics students have at least heard of Chen Hui.

The two proctors noticed Chen Hui as he entered the examination room. They were naturally curious about the level of this young man who had caused such a stir online.

The proctor standing on the podium politely nodded to the students, indicating that he appreciated their gesture.

The students at the back of the classroom responded with smiles.

However, the proctor on the podium did not stride towards Chen Hui, as that would seem too deliberate and might lead to complaints about disrupting the exam.

So he slowly walked down from the podium, pretending to be inspecting, and tried to make as little noise as possible as he slowly approached Chen Hui's location.

[3. In space, there are n spherical planets with equal radii. If a certain area on the surface of one planet cannot be seen by any of the other planets, we call this area "hidden." Prove that the sum of the areas of all these "hidden corners" equals the surface area of ​​one planet.]

When n=3, the conclusion is obvious, as shown in the figure below.
Three points can define a plane, so this problem can be transformed into a two-dimensional plane for analysis. The yellow part is the area that cannot be seen from other planets. Obviously, the sum of the central angles of these three regions is 360 degrees, which is consistent with the conclusion.

But when extended to n planets, the situation becomes complicated.

Chen Hui didn't stop writing; he started drawing circles on the answer sheet.

Proof: As shown in the diagram above, if we choose any direction to locate the North Pole, then all n planets will have a North Pole in that direction. Obviously, only the North Pole of the northernmost planet cannot be seen by other planets, while the North Pole of other planets can be seen by planets further north.

Now, let's introduce a reference planet. We iterate through all the points on this planet, and each time we select one point as the North Pole. In this direction, only the point on the northernmost planet is a hidden point. Therefore, each point on this reference planet corresponds to a hidden point, and there are no other hidden points in this direction.

Therefore, after traversing this reference planet, the sum of all hidden points is the surface area of ​​this planet.

Q.E.D.

This question is not a conventional math problem. In Chen Hui's view, it tests students' spatial imagination ability, which is the foundation for learning geometry and topology, such as quotient spaces and closed surfaces in topology.

Unfortunately, Chen Hui was quite good at topology, so he saw through the essence of the problem at a glance.

Even now, the proctor on the podium has not yet come to Chen Hui.

Seeing Chen Hui raise his hand, he was overjoyed. It was like a pillow being delivered to someone who was sleepy. Without any hesitation, he took three steps at a time and came to Chen Hui's side. "Excuse me, do you need any help?"

"Hello, I'd like to hand in my paper!"

Chen Hui, having tidied up his stationery, replied.

He couldn't study in the exam room, and the exam was four and a half hours long. He didn't want to waste so much precious time here.

"???"

Guan Yifan looked completely bewildered, craning his neck like a crayfish to look at Chen Hui's answer sheet.

All three questions are filled out!
But how long has it been?

Guan Yifan subconsciously looked up at the clock at the back of the classroom: 8:17!
The test-setting team estimated that a problem that would take math geniuses across the country four and a half hours to complete, you finished in seventeen minutes.
He had just seen Chen Hui flip through the answer sheet two minutes ago, which meant that this guy had answered the third question in just two minutes?
What happened?
Is this magic?!

He could understand solving the first problem in two minutes, but that was the third one!

He still had no clue after reading the question.

He was a graduate student in the Department of Mathematics at Yanbei University, and was also a CMO gold medalist back then.

Anyone who can get into Yanbei University is a genius.

Chen Hui had no idea what the proctor was thinking. After picking up his stationery bag, he got up and walked out of the classroom.

The other students also looked over and couldn't help but shake their heads inwardly. This guy had only been taking the exam for a little over ten minutes and he already wanted to go to the bathroom?
Although four and a half hours seems like a long time, if you focus on answering the questions, the four and a half hours will fly by. Therefore, they all cleared out their inventory before entering the examination room and even deliberately controlled their diet and drank less water in order to maintain a good state and save more time.

Those who can come to CMO are geniuses among geniuses. If they don't work hard on these details, how can they surpass other players?

This guy has absolutely no respect for the CMO!
It's no wonder they thought that way. After all, the exam had only been going on for a little over ten minutes. They couldn't possibly assume that this guy had handed in his paper, could they?
"Hey, student, you can only hand in your paper 30 minutes after the exam starts!"

It wasn't until Chen Hui walked out of the classroom that the stunned proctor reacted and shouted at Chen Hui's retreating figure.

"???"

The students, who had been looking down to continue working on their questions, looked up again, their eyes blank as they stared at the proctor and the empty classroom doorway.

(End of this chapter)