Become a master from now on

Chapter 188 The Last Question (Ask for Monthly Ticket Subscription!)
The second day of the IMO event.

London time, daylight saving time, nine o'clock in the morning.

A group of contestants got the test papers and began to answer.

The atmosphere today is obviously more dignified than yesterday's. Everyone's expressions are very grim. The contestants who got good results yesterday must get better results today.

As for Fang Chao, the beautiful little red guy glanced at Fang Chao, and then put up a middle finger, which meant clearly, boy, if you have the ability, you can hand in the paper earlier today!
Fang Chao stared at the other party again, seeing his burly body, blond hair, and deep eyes, he suddenly smiled, showing a very happy smile, the white teeth were exposed, which made the pretty little red man I was taken aback for a moment, is this Chinese team player crazy?

He seems to have overlooked one thing. From the beginning to the end, he should not regard the US team players as his opponents...

The Yankees are really strong in practice, which is understandable, but in terms of basic ability level, the national team dares to be the first, who dares to challenge other countries?
This is not arrogance, nor arrogance, this is self-confidence, this is the foundation!This is something left over from 5000 years of Chinese culture, something that has always been preserved in the bones, and it is beyond the reach of others.

Chinese people have always stood on the shoulders of giants.

In recent years, although the US team has won first place in IMO competitions, look at their faces. If you put aside the national flag, you can see familiar faces. They are yellow skinned. , black-haired Chinese.

Real Yankees don't need to worry about them at all, and don't need to treat them as opponents.

Just ignore it!

Fang Chao soon started to do the questions, ignoring the American players, which made the American players stunned, I am already like this, are you still so calm?

The second day of the IMO event.

first question.

求所有正整数对（k，n），满足，k！=（2n-1）（2n-2）（2n-4）……（2n-2（n-1））
[Above, n is the power].

Fang Chao smiled when he saw this.

In recent years, there have been basically no number theory problems without addition in IMO competitions. Facing this kind of problem with only multiplication, something popped up in Fang Chao's mind.

The number of prime factors!
In the case of using the number of prime factors, Legendre's theorem must be used.

Prime numbers are no strangers to Fang Chao. This is the most basic thing, but it is also the most complicated thing. So far, how many mathematicians have been trapped in prime numbers.

You say it is easy, it is also easy, you say it is difficult, it is really difficult.

For example, Riemann's conjecture, Fibonacci sequence, and even Goldbach's conjecture, they are all problems caused by prime numbers, so far no one has broken through, and until now, there are still a large number of mathematicians working in this direction. Work hard, hoping to break through these conjectures.

Fang Chao has been in contact with prime numbers since he started studying mathematics in a serious way. He is not a great mathematician, and he doesn't even need to solve world-wide mathematical problems. His trouble is just to solve the problem in front of him. .

But since the professor asked such a question, there is naturally an answer to it.

But Fang Chao had already weighed in on such a question, and even faced this question, he didn't take it to heart at all.

The first question on the second day is not a big problem.

He can handle it easily!
After he listed the two lines of formulas, he soon discovered where the main thing to think about was this question.

The situation on both sides of the equation when p=2.

After half an hour, Fang Chao wrote down the last few steps of this question.

V3(k!) ≥ [k/3] > k/3-1
k/3-1<n/4
n/4＞k/3-1=≥1/3（m（n-1））/2-1
Obtain -3/2＜n＜4,

That is, n can only take three numbers of 1, 2, and 3.

Substitute n into the formula.

Fang Chao came up with two solutions.

(k, n) = (1, 1) or (3, 2).

"It's done!"

"Counting this question, I have already scored full marks for four questions, which is already the score line for the silver medal. Of course, if this year's contestants are not good, it is not a big problem to win the gold medal with this score. , but my goal is not the case at all. I want to get the full score in the individual competition of the IMO event, so as to fill my grand slam in the mathematics competition. His achievements have become legends and his name will go down in history! No one can surpass him!"

Fang Chao's heart was high-spirited and high-spirited, and he began to focus on the second question.

topic:

Given an integer N≥2. N(N+1) football players with different heights stand in a row, and the team coach wishes to remove N(N-1) players from these players, making the remaining The 2N players satisfy the following N conditions.

(1) There is no other player between the two tallest players among them.

(2) There is no other player between the third and fourth tallest players among them.

……

(N) There are no other players between the two shortest players among them.

Proof: This can always be done.

Fang Chao started to work, and solved this problem after 53 minutes. His conclusion was valid and it could be done.

The time spent on the two questions was shorter than the time spent on the first day. It cannot be said that the two questions are relatively simple, but for Fang Chao, it just so happens that these two questions are what he is good at, so it does not take much time. With little effort, it is easy to win the scores of the two courses.

In less than two hours, Fang Chao fixed his eyes on the third question.

This question is related to the venue of this year's IMO competition, and the professor who proposed the question really meant to be tricky, but it is obviously not just a coincidence that it can be selected as one of the questions. That's where the charm lies.

The coins issued by the Bank of Bath have an H on the side to avoid damage and a T on the other side. Harry has n such coins and arranges them in a row from left to right. He repeatedly performs the following operations: If exactly There are k (>0) coins with H faces up, wait for him to flip the kth coin from left to right: if all coins are T faces up, stop the operation.

For example: when n=3 and the initial state is THT, the operation process is THT→HHT→HTT→TTT, and stops after a total of three operations.

(a) Proof: For each initial state, Harry always stops after a finite number of operations.

(b) For each initial state C, record L(C) as the number of operations when Harry starts from the initial state C to when he stops the operation. For example, L(THT)=3, L(TTT)=0, find the iterations of C. The average value of L(C) obtained for all 2n possible initial states.

(End of this chapter)