I really just want to be a scholar

Chapter 13 Difficult Math Papers
Qin Ke couldn't help sighing that the school committee is really a typical hard-working academic master, and the hard work he put in is far beyond human comprehension.

Thanks to Tuo Ningqingyun's Chinese notebook "Key Points for Writing Breaking Questions", he even thought of an excellent entry point for writing. Although his writing skills are limited, he has a good idea. He may get more than 60 points for a 45-point composition.

After a rough calculation of the scores, I am confident that I can do the right questions, and with the composition, it is conservatively estimated that the score can exceed 110.

This surprised Qin Ke, who had never scored more than 100 in Chinese, and for the first time saw the hope of entering the top [-] in the grade.

In the afternoon, mathematics was the subject that Qin Ke was least worried about.

Among the invigilators in the examination room happened to be Zheng Jianzhou, a math teacher. Qin Ke nodded slightly to Old Zheng, then bowed his head and started to write the test paper.

Lao Zheng looked around the examination room and observed the reactions of each student. Seeing that most of the students showed embarrassment when they saw the first question, they shook their heads in disappointment.

This time, he and three other math teachers worked out the papers together. In the end, he made the final decision, removed some simple questions, kept only the moderately difficult questions, and added three extra difficult questions totaling 15 points. The purpose of this question is naturally to screen out students who can participate in the Mathematical Olympiad.

——This is also Qin Ke's fault, making Lao Zheng mistakenly believe that gifted students are likely to be buried among ordinary students who usually don't notice, so he tried his best to raise the difficulty of this math test.

Ordinary students were unlucky. When they got the test paper, they saw that the first multiple-choice question read:
“若自然数n使得作竖式加法n＋(n＋1)＋(n＋2)均不产生进位现象，则称n为“开心数”。例如：32是“开心数”，因为32＋33＋34不产生进位现象；23不是“开心数”，因为23＋24＋25产生进位现象，那么，小于100的“开心数”的个数为()
A． 9 B． 10 C． 11 D． 12"

They were dumbfounded on the spot.

According to the usual "agreement becomes custom", isn't the first multiple-choice question all simple theorems and laws, which can be done easily so that students can continue to complete the test paper with confidence?
But what is going on with this topic now?

I didn't see the idea of ​​​​solving the problem at all, okay!
The students with poorer math scores sat on the wax directly, and they looked at the second multiple choice question with dull eyes:
"In a WeChat group, there are five people A, B, C, D, and E playing the game of grabbing red envelopes. There are currently 4 red envelopes, and each person can grab one at most, and all the red envelopes have been snatched. Among the 4 red envelopes, 2 are 5. Yuan, one 1 yuan, one 8 yuan (the same amount in the red envelope is regarded as the same red envelope), then there is a situation where both A and B grab the red envelope ()
A． 18 kinds of B． 24 types

C． 36 kinds D． 48"

Students: "..."

They almost flipped the table, did you make a mistake!Do you want to let the scumbags get a point or two?This is to force them into the abyss of zero!

The students are also frowning, the second question is so difficult, I feel that the chance of passing this time is very slim.

There is nothing to love between the scumbags and the poor students there, and the students with better math scores are also secretly shocked. The first two multiple-choice questions seem aggressive and unfriendly, and the latter questions are afraid that it will be more difficult.

The other invigilator was Xu Shichao, a new high school math teacher. After looking around at the hard-pressed expressions of the students, he also looked down at the exam paper himself, and couldn't help but mutter in his heart.

The questions in the math test paper this time are so murderous. Is this the rhythm that will stump most students?
He walked half a circle around the examination room, and found that nearly 15 minutes had passed, and most of the students were still doing the third multiple-choice question, and many students went through the examination paper from the beginning to the end and then fell asleep on the table , apparently abandoned.

There was even a female student who secretly wiped her tears while doing the questions, which was so hard to cry.

The degree of difficulty of the second-year high school math test is indeed obviously high...

Xu Shichao doubted whether there were any students who could complete the paper within the specified time for the math test of the second year of high school, let alone the three additional questions that followed.

Seeing the gloomy face of the senior Zheng, Xu Shichao didn't touch the bad luck, sighed secretly, turned around and was about to go back to the podium, but he saw something unexpectedly out of the corner of his eye.

He froze for a moment, then turned his head to take a closer look, only to see that a male student was already doing the third part of the answer questions, and it was the last question!
Xu Shichao thought he was dazzled, so he took a closer look, and sure enough, the male student wrote very quickly, and he finished the last question with a brush, and opened the test paper to do the comprehensive big question in the fourth part.

When he flipped through the test paper, Xu Shichao clearly saw that the student had finished all the multiple choice questions and fill in the blank questions.

So fast?It's only about 15 minutes since the exam starts now, right?

Could it be that they don't know how to write casually?

Xu Shichao couldn't help but stop, the student was already looking at the comprehensive question.

"The known function f(x)=xln x+ax+1, a∈R.
(1) When x>0, if the inequality f(x)≥0 about x holds true, find the value range of a;
(2)当n∈N时，证明：n/(2n＋4)＜(ln 2)^2＋(ln 3/2)^2＋…＋[ln (n＋1)/n]^2＜n/(n＋1)。”

After reading the question, Xu Shichao secretly complained again. This question is quite difficult, and the monotonicity of the function is used. The second question also involves the number sequence knowledge points in the second semester of the second year of high school. As the finale comprehensive question of the mid-term exam in the first semester of the second year of high school, it is too outrageous!

It is estimated that apart from the most famous academic masters, such as Ning Qingyun, who can solve this question, other students may just stare and give up with empty papers.

Xu Shichao was shaking his head secretly, but saw that the male student started to write the problem-solving process.

“解：(1)由f(x)≥0，得xln x＋ax＋1≥0(x＞0)，

That is, -a≤ln x+1/x is always established, that is, -a≤[(ln x+1/x)]min
……

∴－a≤1, that is, a≥－1,

∴ The value range of a is [-1, +∞)"

Xu Shichao's eyes widened. This student doesn't seem to think at all, does he?The next second after reading the question, write the answer directly?

And... this answer seems to be correct!

Under Xu Shichao's shocked gaze, the male student's pen tip moved quickly, brushing, and the answering process of more than [-] lines was completed in less than a minute and a half. The male student looked at the three additional questions.

Xu Shichao couldn't help but also looked at these three additional questions together, the more he looked at them, the more speechless they became. These three questions are really quite difficult, all of which are questions for the municipal-level Mathematical Olympiad preliminary competition, and the last question stumped him even more.

——Xu Shichao also participated in the Mathematical Olympiad when he was in high school, but he stopped at the provincial competition, but now he is asked to do the third question, and he has to spend a lot of hard work asking himself.

(End of this chapter)