Day 2 of the IMO event.

London time, daylight saving time, 9 a.m.

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

Today's atmosphere is obviously more dignified than yesterday's, everyone's expressions are very hideous, and the players who got good results yesterday must get better results today.

As for Fang Chao's side, Mei Xiaochi glanced at Fang Chao, and then raised a middle finger, the meaning is obvious, boy, if you have the ability, you can hand in the paper early for Lao Tzu today!

Fang Chao stared at the other party again, looking at his burly body, golden hair, and deep eyes, he suddenly smiled, very happily showing a smile, the snow-white teeth were exposed, making the beautiful little red guy stunned, this Chinese team player is crazy

He seems to have overlooked one thing, from start to finish, he shouldn't treat the American players as his opponents......

The United States is indeed very strong in practice, which is understandable, but in terms of basic ability level, the national team dares to call the first, and who dares to call other countries the first

This is not arrogance, nor arrogance, this is self-confidence, this is the essence! This is something left over from 5,000 years of Chinese culture, and what has always been retained in the bones is not comparable to others.

Chinese people have always stood on the shoulders of giants.

In recent years, although the American team has won the first place in the IMO competition, look at their faces, if you put aside the national flag, then you can see a familiar face, which is a yellow-skinned, black-haired Chinese.

Real beauties don't need to worry at all, and they don't need to be treated as opponents.

Just ignore it!

Fang Chao soon began to do the question, ignoring the American player, which made the American player stunned, I am like this, you are still so calm

Day 2 of the IMO event.

The first question.

Find all pairs of positive integers (k,n), satisfy, k! =（2n-1）（2n-2）（2n-4）…… （2n-2（n-1））

[Above, n is the power].

When he saw this, Fang Chao smiled.

In recent years, there has been basically no number theory problem without addition on the IMO event, and in the face of this kind of problem that only multiplication, something popped up in Fang Chao's mind.

Number of prime factors!

In the case of using the number of prime factors, then Legendre's theorem will inevitably be used.

Prime numbers are no stranger to Fang Chao, they are the most basic things, but they are also the most complex things, and so far, how many mathematicians have been trapped in prime numbers.

You say it's easy, it's simple, you say it's hard, it's really hard.

For example, the Riemann conjecture, the Fibonacci sequence, and even the Goldbach conjecture are all problems caused by prime numbers, and no one has solved them so far, and there are still a large number of mathematicians working in this direction to solve these conjectures.

Fang Chao has been learning mathematics since he started to learn mathematics seriously, and the most contact is prime numbers, he is not a great mathematician, he doesn't even need to solve the world's mathematical problems, his troubles are just to solve the problem in front of him.

But since such a question is taught, then there is naturally an answer.

And Fang Chao had already measured it in the face of such a question, and even in the face of this question, he didn't pay attention to it at all.

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

He can do it easily!

After he laid out the two-line formula, he quickly discovered where the main thing to think about was this problem.

When p=2,3 is on both sides of the equation.

Half an hour later, 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

-3/2

For example, if n=3 and the initial state is tht, the operation process is ththhthttttt, and the operation stops after a total of three operations.

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

(b) For each initial state C, denote l as the number of operations Harry had from the beginning of state C to the time he stopped the operation, e.g. l(tht)=3, l(ttt)=0, and find the average of l(c) obtained when c took all possible initial states to the power of 2n.