Chapter 216 Final Question

Mingcheng No. 1 Middle School, Song Ranran.

The little girl who got perfect scores in the preliminary test of the league.

As the president of the provincial mathematics association, he was very impressed by her.

As early as the first exam, although he was not present to invigilate the exam, Song Ranran's name was mentioned verbally several times by the teachers during the subsequent marking session.

The handwriting is beautiful, the problem-solving process is perfect, the answers are all correct, and the answer sheet is like a work of art.

I just don’t know if my performance in this re-examination will be equally impressive...

The president expressed his expectation for this.

Song Ranran, whose attention was completely attracted by the topic, did not realize that there was someone standing behind her. She quickly deduced the problem-solving ideas in her mind. The tip of the black pen rubbed the manuscript paper and made a rustling sound. Lines of mathematical formulas were written out. After discovering

After realizing that the direction of her thinking was correct, her eyes brightened slightly and she immediately jumped to the next step.

The invigilator looked at her for a while, then nodded his jaw in a slight arc.

But then, the satisfaction in his eyes slowly faded away, and he felt a little uncomfortable.

Song Ranran's progress in solving the problem is not slow, even very fast. Occasionally, her thoughts drift, but she can quickly turn back after the experiment——

At least, this is somewhat beyond the president’s expectations.

These four questions for this year's exam were carefully selected and prepared for these students.

When he solved the problem, he wasted a lot of brain cells. When it was the students' turn to solve the problem, they had to be "beaten".

The chief invigilator stared at Song Ranran's answer sheet with an unpredictable expression for a while. However, he couldn't stand still and walked back and forth again, secretly judging the performance of other students.

Fortunately, not all candidates are as successful as Song Ranran.

Occasionally, a few students scratched their heads and scratched their heads in a reserved manner, which the president of the Mathematics Association found quite interesting.

During the exam, the invigilator should not disturb the students too much.

The president also exercised caution and observed, but never made a sound. He also kept a certain distance from the students. After going down and walking around twice, he returned to the stage and sat down again.

With about an hour left before the end of the exam, Song Ranran finished the last question.

Generally speaking, the difficulty of the questions that can appear at the end of the paper is by no means ordinary.

Of course, sometimes the solution to the final question may not be that difficult, but the requirements for thinking are definitely very high.

Candidates need to use limited time to find a perfect idea in the examination room. It is undeniable that the difficulty is doubled.

Song Ranran stretched out her hand to rub her slightly aching temples, adjusted her mentality, and lowered her head to read the questions.

This final question continues the minimalist style of the paper, with the question being just two lines long.

This is a proof question.

Assume that positive integers a and b satisfy (a+b) divided by (ab+1) equal to k (k belongs to N), and verify that K is the square of a certain positive integer.

There were not many words in the title, but the girl's eyes swept over it over and over again, trying to find some inspiration.

Is there any theorem related to it?

Have you encountered similar problems before?

Or...backwards?

…

My thoughts gradually subsided, as if everything had become quiet.

Song Ranran closed her eyes and simulated several ideas, then opened her eyes and tried to verify them on paper.

Before I knew it, nearly half an hour had passed.

There is not much time left for the exam.

Finally, Song Ranran's lips curled up slightly.

After trying again and again, she finally found a solution that was relatively uncomplicated and perfectly proven.

The most practical one is undoubtedly proof by contradiction.

Assuming that k is not the square of a positive integer, then a is not equal to b. Consider the indefinite equation a-kab+(b-k)=0.

If a=b, then (2-k)a=k, it follows that k=1, which is inconsistent with the assumption.

…

In the last three minutes before the end of the exam, Song Ranran stopped writing.

The sun shone in from the window, and the brilliance enveloped the tips of her black hair, making the girl look like a pearl.

President of the Mathematics Association: The problem I spent so many brain cells on must let me see the effect~

Candidate: What a bad old man.

In terms of questions, Zhaozhao checked some of the final questions of several historical Mathematical Olympiad competitions, such as the one mentioned in the article. Those who are interested can try it on their own. Zhaozhao will not describe the problem-solving process in detail here.

In fact, sometimes the solution to such competition questions may not be that difficult, but if you have to use limited time to find the perfect idea in the examination room, it is undeniable that the difficulty is doubled. Whether it is an examination or a competition, your mental quality must be stable and don't panic.
Chapter completed!