This question seems quite novel, but it is actually not difficult.

Yi Cheng replied:

First of all, we know from the title:

Suppose the landlord is set C

Then the number of cards of C is 10, which can be written as the set C{C1, C2...... C10}

The set of A is 8, and the same is A{A1, A2...... A8}

……

Then both C and A have a straight:

You can set at least C1+1=C2 and C2+1=C3......

Similarly, A1+1=A2, A2+1=A3......

B says he has only one pair and B has no straight.

B1=B2 can be set, and there is no difference between 5 consecutive numbers that is 1 for each other.

The elements in the other set come from two sets of numbers from 1 to 13, and they are mutually exclusive.

That is, if the 1 of spades appears in A, it must not appear in B and C.

……

Yi Cheng wrote it down all the way and found that this question was a manual job.

The difficulty of this question is not the previous part, but the later game.

Yi Cheng finished the first half.

Then continue to do the splitting and tidying:

A can be split into two sets: a straight set and a non-straight set,

B splits into pair sets and single-card sets,

C splits into straight and non-straight sets,

C plays first.

Then there will be two cases: the set C straight is greater or smaller than the set A straight......

Then you can roughly get several models:

……

Yi Cheng shook his head as he did the question.

It can be dealt with with the Nash equilibrium that the werewolf killed yesterday, or it can be done with the stupidest exhaustive method.

In other words, this question is destined to not open up the score difference.

The order of magnitude is not large, and others will definitely be able to do it in 2 hours through exhaustive measures.

Ay.

It's uncomfortable, it's uncomfortable.

Yi Cheng sighed in his heart.

Finally, according to different card types, the corresponding probability models are sorted out and discussed separately.

Yi Cheng's question is over.

ok。

21 points to hand.

However, this problem was a lot of calculations, and it wasted almost an hour of his time.

……

Yi Cheng moved on to the third question.

[At the birthday party, there is a group of friends, as the birthday star, you have to cut the cake for them, and the cake must ensure that each piece is cut with the same volume and cream, so that no children will be unhappy.

s is a convex set in the xy plane.

Convex set: In the real number R (or on the complex number C) vector space, the set S is called the convex set, if the points in the line connecting any two points in S are in the set S.

For Euclidean spaces, intuitively, the convex set is convex. In one-dimensional space, a convex set is a single point or an unbroken line (including straight lines, rays, line segments); 2. The convex set in three-dimensional space is an intuitively convex figure. 】

The convex set is specifically explained in the title.

The cake is a distinct convex set that can be seen with the naked eye.

Yi Cheng has no doubts about this.

He continued to examine the question—

[Assuming the height of the cake is h, h& amp; amp; amp; amp; amp; amp; amp; amp; amp; amp; amp; gt; 0, defined as a set of points C={(x,y,z)|(x,y,z∈S, and 0 is less than or equal to z, less than or equal to h)}

Then C is a cake with a height of h based on S.

The height of the cake is consistent, assuming that the other surfaces except the bottom surface are evenly creamed.

Then, if the area of this k set is figured out, and the perimeter length of the original S is also the same, then it is called a perfect division of k of S.

If all of its dividing lines are segments from a single point, the division is said to be a star-shaped perfect division.

Test Proof:

Any planar convex set has a perfect division of 3 stars. 】

, a cut cake, why are you Rory talking so much?

Yi Cheng expressed doubts about the language proficiency of the questioner.

He is already at the level of lv2 literary learning, plus the champion of the 8 Chinese Poetry Conference, he is now qualified to complain.

To put it simply, for example, if a circle is divided into a rice character and becomes 6 equal divisions, then this rice character division is called a 6-star perfect division.

Now all it needs to prove is that no matter what shape, as long as it is a convex set, it can be perfectly divided into 3 stars.

Icheng began to argue on the draft.

But half an hour into the work, he suddenly found -

Your sister's question looks simple, but it's actually very difficult.

Why?

Because before this problem can be proved, 7 lemma need to be proved in a row.

This is only 5 houses short of the assassination of Athena.

Yi Cheng thought to himself, even if you are 7 gourd babies, Lao Tzu will beat you to death.

The big baby is:

It is proved that for the convex set S, there is a 3 equal length division of an edge: S1, S2, and S3, and the area enclosed by S1, S2, and S3 is less than 1/3 of the area of S.

Second baby:

It is proved that for the convex set S, S1, S2 and S3 are an equal division of the sides of S, then the area enclosed by S1, S2 and S3 is at most only one not less than 1/3 of the area of S

……

Qiwa: If the side of S is divided into three consecutive segments of the same length, and the area enclosed by these three segments does not exceed 1/3 of the area of S, then the proposition proves that there is a perfect division of 3 stars in any plane convex set.

Yi Cheng only felt hot and sweaty.

Along the way, he sacrificed all kinds of magic weapons, and finally surrendered the 7 gourd babies one by one.

In the end, grandpa came to deliver it.

He successfully accepted his grandfather.

When he looked up, it was half past eleven.

I didn't expect this last question to be so time-consuming.

It is worthy of the title of the Chinese Mathematics Olympiad.

3 questions, it's like spending most of my life.

On the scroll, Yi Cheng was densely filled with words, including scratch paper.

When he finishes the question, he presses his hand slightly on his temples to clear his mind.

After two minutes, start checking the answers.

……

"Huh, Mr. Meng, I'm sorry for you."

After the exam, Mr. Meng came to pick up everyone.

Before returning to the hotel, several players were tested and cried.

Wu Qingqing said with a handful of snot and tears, "I have failed your respectful teachings, and I am not worthy of going to Qingbei." ”

"Me too, I'm a garbage ......" Jiang Wenbo cried with a sad face, "I don't even have a firm grasp of the first question, and the second question is gone in half ......the time."

"You know, your biggest problem is that you don't think right, and you only think about picking up girls." Xia Wenyu poked his head with his finger.

Jiang Wenbo immediately blushed and didn't dare to look at Li Anruo.

"It's all Li Anruo's fault, he took us to play werewolf killing the day before yesterday and broke his brain."

"Hey, you can't blame her." Yi Cheng stood up and responded, "You must know that today's landlord must also use the Nash balance. ”

"Nash...... What the hell is that? ”

The crowd looked frustrated.

Lao Meng made a certain judgment from everyone's expressions and the conversation just now.

Yi Cheng, this young man's qualifications are very high, this time the question will definitely not be difficult for him, seeing that he just said the truth, he should have answered it all.

In addition, Yan Ziqi didn't say a word just now, her face was a little dignified, I'm afraid it's a little troublesome.

Among the others, the youngest Fang Mingyu was silent, not knowing what to think, maybe he had a certain certainty.

……

Wu Qingqing cried and shouted, and Jiang Wenbo had to fight, these two people are estimated to only get some process points, and it is estimated that there is no hope of getting the first country.

Not to mention entering the national training team to compete in next year's IMO.

"Teacher Meng, you're right, I'm only worthy of a second-rate like Jiangcheng University." Wu Qingqing wiped away the tears on her face, her eyes showed a fierce light, "I'll change my mind after I go back." ”

Belch......

"No, no, no." Lao Meng stretched out his hand and patted the back of Wu Qingqing's head, "Let's talk about it later." ”