Lan Jie and Yang Kewei are both science madmen.

One of the attributes of science madness is to be truth-seeking and pragmatic, the second is to like to invite people to dinner, and the third is to hoe the strong and help the weak, and care for the disadvantaged.

Yang Kewei said that his language was cracked, and Lan Jie also said that his language was cracked, but Yang Kewei was cracked to a deeper extent, so Lan Jie invited Yang Kewei to lunch.

In the afternoon math test, this math paper was generally unremarkable, but there was one calculation problem with high difficulty.

It is known that the box contains 100 red balls and 100 blue balls, and the balls of both colors contain 1,3,3^2,...... , one for each of the balls of 3^99, and the sum of the scalar numbers of the balls of the two colors is denoted as S. For a given positive integer n, if a number of balls can be taken out of the box so that the sum of their scalar numbers is n, it is called an "n-scheme" for taking the balls. Remember the number of different "n-schemes" as f(n), and try to find the value of f(1)+f(2)+......+f(S).

Is it familiar or not?

This method of asking questions has a strong personal style, and Lan Jie can also make judgments by thinking with his toes, and the red and blue balls are most likely from Mr. Han.

Mr. Han is a kind person, and he will inevitably explain the relevant knowledge in class before he asks the question.

Permutations and combinations, combination counting, combination identities, probabilities and a large number of special technical contents, Mr. Han has focused on explaining.

However, even if you listen carefully to all of Mr. Han's lessons and don't miss even a second, you won't be able to solve this red and blue ball problem.

"So this question may not have been given by Mr. Han?"

'So the teachers in the math teaching group are all keen on the red and blue balls?'

Teacher Kang, who teaches mathematics in classes 8 and 9, has just left his job, so is Teacher Kang developed? Or is he bankrupt? If he is bankrupt, Teacher Kang should work hard to repay his debts, and he can't lose his job if he loses anything. To sum up, Mr. Kang must have made a fortune!'

Lan Jie thought deeply about this math problem, and he thought about it for ten minutes, but he couldn't put pen to paper.

Fortunately, it only took Lan Jie half an hour to complete all the previous problems, and he had plenty of time to solve the red and blue ball problems.

Brush!

Yang Kewei suddenly raised his hand.

The afternoon math exam was still invigilated by Mr. Lu Zhenhua, and Mr. Lu walked to Yang Kewei's side, and he asked in a low voice: "Yang Kewei, are you done?"

Yang Kewei didn't take the initiative to influence the other classmates, he didn't make any sound, he nodded slightly with great quality, signaling him to hand in the paper.

The real male god is not only good-looking, tall, and high in test scores, but also has high personal qualities.

Yang Kewei handed in the papers and left, and Teacher Lu praised from the bottom of his heart: "Kewei, what an excellent student." ”

'Yang Kewei handed in forty minutes, do you dare to believe it?'

'Believe it or not, I do. ’

Feeling pressured, Ranger focused and used all his strength to solve the math problem.

Time passed minute by minute, and Lan Jie, who had no distractions and concentrated thinking, suddenly woke up!

What do you think Ajie, the surface phenomenon of the bells and whistles that go around this question is a trap, and the key to solving the problem is the ternary!

Second sister, you're so naughty!

Applejack, you're so clever!

Ranger Rapid Operation, for the set A={0,1,...... , S }, which can be expressed as a ternary form m=a0+3a1+3^2a2+......+3^99a99 with a hundred digits, where ai∈{0,1,2},i=0,1,2,...... ，99。

Based on the logic of the ternary, Lanjie quickly calculated the answer, f(1)+f(2)+......+f(S)=2^200-1.

Push back and check again, yes, it's 200th power of 2 minus one.

'Luckily, I'm a good one. ’

Rangier let out a long breath, and he took a few more minutes to check the question in front of him, checking that it was correct, and holding his left hand high.

Teacher Lu Zhenhua walked up to Lan Jie, and he asked, "Lan Jie, you also handed in the paper in advance?"

Lan Jie nodded lightly and walked away.

Mr. Lu carefully reviewed Lan Jie's math test papers, and after the review, Mr. Lu couldn't help but sigh: "Students with good Chinese scores are really not bad in mathematics. ”

Lan Jie, who left the examination room, wanted to talk to Yang Kewei about his life ideals, but Yang Kewei had long since disappeared.

After the math exam, Lan Jie can go back to the second and third class classrooms to study on his own, or he can go home, or play for a while and then go home.

The exam will continue tomorrow, so Lan Jie chooses to go straight home.

Passing by the large playground, Lan Jie found that the international class in his sophomore year of high school was in physical education class.

The liberal arts class and science class in the second year of high school are all exams on these two days, and they are both math exams this afternoon.

The teaching arrangement of the international class is different from that of the liberal arts and science classes, and there are no mid-term exams for the international class.

On the track, the girls of the international class are running 100 meters in a group of 8.

Lan Jie, who was passing by, stopped to watch, and he saw eight girls starting and setting off, the girls were heroic, there were royal Luo, single and double ponytails, black straight race high, running, running, chasing, chasing, eight girls, each showing their powers.

The fiercest is the girl in lane 8, she runs really fast.

Just over 50 meters in the run, the girls in lane 8 are more than two meters ahead of the second place.

This woman is Zhang Yutong, her hair has not been cut for a long time, it is long, and she has a single ponytail.

Running to the last 20 meters, Zhang Yutong beat all the girls with an absolute crushing advantage.

In the sprint stage, Zhang Yutong's rubber band in the ponytail loosened and fell, and the wind was quite strong at this time, and it must have exceeded the wind speed by more than 2 meters.

Hula!

Zhang Yutong's long hair swayed wildly, she was like a little madman running wildly in the wind, and she was about eight meters ahead of the second place when she crossed the line.

"Zhang Yutong habitually hangs and beats!"

Lan Jie has known Zhang Yutong for more than three semesters, and now he has to re-examine Zhang Yutong.

Zhang Yutong pushes the shot put very well, and the sprint is also very strong.

Compared with most girls of the same age, Zhang Yutong's upper and lower limb strength are at a high level.

It can be seen with the naked eye that although Zhang Yutong runs fast, she has not received professional training, and she purely relies on brute force to beat the rest of the girls.

"Zhang Yutong, full marks, you can move freely. "The PE teacher praised it.

"Oh buy Gadis, Yutong, you're running so fast!"

"Yutong, your legs are so long, sure enough, girls with long legs are fast!"

Several girls in the same group were impressed, and one girl took off the rubber band on her wrist and handed it to Zhang Yutong, asking Zhang Yutong to tie her hair.

"Yuxin, thank you. Zhang Yutong took the rubber band to express her gratitude, she tied her hair, and said very modestly: "Oh, I just run casually, and besides, my legs are not long, compared with eight, I am a short-legged girl." ”

Yu Xin didn't react at once: "What kind of stalk is eight?"

Yutong explained: "Eight don't even have heads, it's all legs. ”

Yuxin laughed wildly: "Poof, you're funny!"

Zhang Yutong, who was talking and laughing, suddenly found that Eight was coming, and Eight was staring at her not far away.