I Touching Day 2.

London time, daylight saving time, nine in the morning.

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

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

As for Fang, the beautiful little red man glanced at Fang, then raised his middle finger. The meaning was very obvious. Boy, if you have the ability, you can hand me in advance today!

Fang stared at the other party again, looking at his burly body, golden show, and deep eyes. He suddenly smiled, showing a smile happily, and his snow-white teeth were exposed, which made the beautiful little red man stunned. This Chinese team player was crazy?

He seemed to have overlooked one thing: from beginning to end, he should not regard Captain America's players as his opponent...

The beauty is indeed very strong in practice, which is understandable, but in terms of basic ability, the national team dares to be the first, but who else dares to challenge?

This is not arrogant, nor is it arrogant, it is self-confidence, it is the foundation! This is something left over from Chinese culture for five thousand years, and what has always been preserved in its bones is not comparable to others.

Chinese people have always stood on the shoulders of giants.

In recent years, although the US team won the first place in the I-Tun event, looking at their faces, if you put aside the national flag, you can see familiar faces, which are yellow-skinned and black-headed Chinese.

Real beauties don't need to worry at all, nor do they need to treat them as opponents.

Just ignore it!

Fang soon started to do the questions and ignored the American players, which made the American players stunned. I am like this, are you still so calm?

I Touching Day 2.

The first question.

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

The above, n is the power.

When he saw this, Fang smiled.

In recent years, there has been basically no number theory topics without addition in the I-Tun event. Faced with this problem with only multiplication, something popped up in Fang's mind.

Number of prime factors!

When using the number of prime factors, Lejeune theorem will inevitably be used.

Prime numbers are not unfamiliar to the square. This is the most basic thing, but it is also the most complex thing. So far, many mathematicians have been trapped in prime numbers.

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

For example, the Riemann conjecture, the Fibonacci sequence, and even the Goldbach conjecture are all difficult problems caused by prime numbers and have not been solved yet. Until now, there are still a large number of mathematicians working in this direction, hoping to break these conjectures.

Fang started to learn mathematics seriously and had the most contact with prime numbers. He was not a great mathematician, and he didn't even need to solve the world-wide mathematical problems. His troubles were just to solve the problem in front of him.

But since the professor has such a question, there are naturally answers.

However, he has already measured such a question, and even faced this question, he didn't care at all.

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

He can handle it easily!

After he listed two lines of formulas, he soon found out where the main thing to think about in this question is.

When p2 and 3 are on both sides of the equation.

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

V3(k!)＞k/3-1

k/3-1＜n/4

n/4>k/3-11/3(m(n-1))/2-1

Get -3/2＜n＜4,

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

Substitute n into the formula.

Fang came up with two solutions.

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

"Done!"

"Including this question, I have already won the full score of four questions, which has already won the silver medal. Of course, if the contestants of this class are not good, it is not a big problem to get the gold medal with such a score, but my goal is not the case at all. I want to get the full score of the I-Touch event individual competition to fill my grand slam in mathematics. All of them are full scores, so that my youth has no regrets, and my grades become legends and famous in history! No one has surpassed me!"

Fang Nei's heart was full of ambitions and spirit, and began to focus on the second question.

topic:

Given the integer n2.n(n1) football players with different heights stand in a row, the team coach hopes to remove n(n-1) from these players, so that the remaining 2n players in this row meet 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 two players with the third and fourth heights.

...

(n)There are no other players among them.

Proof: This can always be done.

Fang started to do it and solved this question in 53 minutes. The conclusion was valid and it could be done.

The time spent on the two questions is shorter than that of the day. It cannot be said that these two questions are relatively simple. It is just that for the party, these two questions are what he is good at, so it is easy to get the scores of both of them without any effort.

In less than two hours, I just focused my attention on the third question.

This question is related to the venue of this year's I-Touch event. The professor who set the questions really means to be quick, but it is obviously not just a reason for the trick. It is obviously a charm for it to be selected.

The coins of the Bath Bank bank are minted with h to avoid injury, and T is minted on the other side. Harry has n such coins and arranges these coins in a row from left to right. He repeatedly performs the following operations: If there are exactly k (>o) coins h face up, wait until he flips the k-th coin from left to right: If all coins are T face up, stop the operation.

For example: when n3 and the initial state is ThT, the operation process is ThThthThTTTTTT, and it is stopped after three operations are performed in total.

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