Chapter 9, Provincial League Examination

Among the fifteen participants, the one who achieved the best results received the first prize, yet was unable to be selected for the provincial team to participate in the National High School Mathematics Competition. This time, like Qin Yuanqing and others, it was their last opportunity to participate. If they fail to be selected for the provincial team, they will miss the chance to compete nationally, let alone be chosen for the national team to compete internationally.

Zhou Hu, Shuitou No. 1 Middle School, 89 points

According to past conventions, 40% of the participants can receive the second prize, 10% of the participants can be awarded the first prize, while those who make it to the provincial team are less than 0.5%.

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The 15th question asks for the smallest positive real number k such that the inequality ab + bc + ca + k (1 / a + 1 / b + 1 / c) ≥ 9 holds for all positive real numbers a, b, and c.

The head of the mathematics group at Jinpu No. 1 Middle School gave a few instructions to everyone and then asked them to find their examination rooms, so as to avoid any confusion and panic tomorrow morning when they are looking for the exam venue.

Biology: Level 4 (20 / 100000)

Qin Yuanqing began to tackle the questions, starting with the first major section, which consisted of multiple-choice questions similar to those in the college entrance examination. Qin Yuanqing found it impossible to calculate mentally, so he continuously performed calculations on scratch paper before filling in the answers. Once he completed the multiple-choice questions, he carefully shaded the answers on the answer sheet with a 2B pencil. After verifying that everything was correct, he proceeded to the second major section. The second section comprised fill-in-the-blank questions, which appeared simple at first glance but were actually considerably more challenging than the multiple-choice questions. Sometimes, there were not just one answer but two answers required; failing to provide one answer would result in a deduction of half the points. Qin Yuanqing shaded the answers on the answer sheet with a 2B pencil and checked them for accuracy before moving on to the next section.

IQ: 145

Thus, Qin Yuanqing appeared very relaxed, eating when it was time to eat and drinking when it was time to drink, while being playfully scolded by the leader of the mathematics group for having a big heart and being carefree.

Additionally, there are two graduate students responsible for statistics, who will input the data into Excel for the convenience of final statistics and ranking

Indeed, as the level of mathematics reached Level 5, Qin Yuanqing discovered that understanding the problems of these mathematics competitions became significantly easier, as if a sense of unblocking the meridians had been achieved

Both individuals displayed expressions of surprise

"Oh, this exam paper isn't difficult at all?" After finishing question 14, Qin Yuanqing couldn't help but mutter to himself, "Could it be that I have been given a fake exam paper?"

In terms of type, there are a total of 6 multiple-choice questions, each worth 6 points, totaling 36 points. There are also 6 fill-in-the-blank questions, each worth 6 points, totaling 36 points. The third major type is the proof questions, which consist of 4 questions, each worth 20 points, totaling 80 points, with a maximum score of 152 points

As soon as the provincial competition concluded in the morning, the judges from the Provincial Mathematics Society began grading the papers in the afternoon. This time, over 1,200 high school students from across the province registered to participate, and they need to promptly grade the papers and tally the scores. Although the multiple-choice questions are graded by a computer program, the workload is still quite substantial.

Qin Yuanqing sat down, placing a signature pen, a 2B pencil, a ruler, a compass, an eraser, and a triangle ruler on the table, while the examination admission ticket and ID card were positioned in the upper right corner of the table, awaiting the distribution of the open-book exam materials

Qin Yuanqing arrived at his examination room and took a seat at his designated spot. The standards for the competition were modeled after the National College Entrance Examination, and were even stricter, as each room accommodated only twenty candidates, arranged in four rows of desks, with five desks in each row. The spacing between each row was substantial, ensuring that no one could see the answers of those seated in front, behind, or beside them. Additionally, each room was equipped with two proctors who continuously walked around the room.

After finishing at the examination venue, Qin Yuanqing returned to the school gate, where others had not yet arrived. It was not until about half an hour later that everyone gathered. The leader of the mathematics group, Zhang Jiajiji, asked everyone to check their identification cards, signature pens, 2B pencils, rulers, triangular scales, compasses, and admission tickets once again. After all, if any issues were discovered now, there would still be time to remedy them; if they were found tomorrow, there would be no time for correction.

Question 1, which is also Question 13, tests knowledge related to the intersection of a parabola and a straight line. The problem states: Given the parabola C: y = ax² (a > 0), the line y = a + 2 intersects the parabola at points A and B. Let M be the midpoint of segment AB, and a perpendicular line from M to the x-axis intersects the parabola C at point N. (1) Prove that the tangent line to the parabola C at point N is parallel to line AB. (2) Is there a real number a such that if it exists, find the value of a; if it does not exist, please explain the reason.

Li Peirong, Shuixian No. 1 Middle School, 105 points

Qin Yuanqing actually breathed a sigh of relief, as he realized that this provincial league was not as difficult as he had imagined; it was only slightly more challenging than the actual mathematics exam questions of the college entrance examination.

Both graduate students are from the Department of Mathematics, and their participation in this statistical analysis has brought them endless joy, sometimes laughing so hard that their stomachs hurt. They recorded scores of 1, which, however, are not the lowest; the lowest score is 0, and there are quite a few instances of 0.

After Qin Yuanqing filled in the name, examination admission ticket number, ID card, school name, and examination paper code on the test paper, he noticed that there were still 10 minutes left before the exam began. Not recalling any answers, he followed his usual practice and first checked the test paper, reviewing the questions, types of questions, and key knowledge points, gaining a general understanding in his mind

They will stay at a nearby hotel in the evening, a hotel without a star rating

Level 3 (100 / 10,000)

After a rigorous inspection process, apart from the participants' identification cards, examination permits, and basic examination tools, no other mobile phones or electronic devices are allowed to be brought into the competition venue

Wow, this person is amazing, having scored 140 points at Luda Shuangshi High School, currently the highest score!

Checking into the hotel, two people per room, Qin Yuanqing did not engage in casual conversation with another classmate, but instead focused on studying the problems at hand. His time was very precious, and every minute and every second could not be wasted

Due to the participation in the provincial league, Qin Yuanqing's schedule suddenly became tight

After a long journey of 4 hours, the bus arrived at Rongcheng Experimental High School, which is the venue for this year's provincial high school mathematics competition. Participants from various cities and counties will compete here, and even if they cannot obtain the qualification for guaranteed admission to key universities by winning first prize, they can still earn extra points during the college entrance examination.

Qin Yuanqing continued to work on the 16th problem, which is also the last question. This is a proof problem that combines sufficient and necessary conditions with inequalities, and it is the most difficult one.

Wang Lin, Provincial Experimental Middle School, 118 points

As Qin Yuanqing took advantage of the remaining time in the evening, he reviewed the questions one by one. It was not until Zhang Jiajia came to check the room at 12 o'clock, reminding them that it was time to sleep, that Qin Yuanqing reluctantly put away the books. After taking a shower, he fell asleep immediately.

Mathematics: Level 5 (0 / 1,000,000)

Having just completed the midterm examination, we immediately gathered with others and, under the leadership of the mathematics group leader, boarded a bus to Rongcheng

Emotional intelligence: 100

Vaguely, Qin Yuanqing began to understand that although many competition problems seemed to exceed the syllabus, in reality, they did not; they merely extended the concepts and increased the difficulty. As long as one continues to delve deeper and analyze, they can still be solved.

Subject

As soon as the exam ended and I had just walked out of the examination hall, I heard students complaining about how difficult it was, questioning how one could possibly survive under such circumstances

Qin Yuanqing furrowed his brows slightly, methodically breaking down the problem step by step. He wrote a lengthy proof, and just as he finished writing the conclusion, the bell rang. The invigilator announced the end of the exam, and each student who had completed the test stood up, while the invigilator collected the papers one by one.

For point (2), problems of this nature first assume the existence of a real number a such that the equation holds, and then deduce the value of a based on the known conditions. If the value of a can be calculated, it exists; if it cannot be solved, it does not exist

Chemistry: Level 4 (20 / 100000)

Chinese Language: Level 4 (20 / 100000)

Lin Feng, Luda Shuangshi High School, 110 points

Suddenly, Qin Yuanqing's expression brightened with joy, for he discovered that his attributes had changed

Jinpu No. 1 Middle School, regarded as the best school in Jinpu County, is also considered a key high school in Shuixian City. However, on a provincial scale, it appears somewhat insignificant, lacking any notable reputation. After all, it is not classified as a provincial key middle school, and people are only aware of those provincial key middle schools; who would know about an ordinary county school

Qin Yuanqing listed the parabola and the straight line as a system of equations on the test paper. Since they intersect, it indicates that y is equal, leading to the equation ax² - x - 2 = 0. Then, let A (x₁, y₁) and B (x₂, y₂), and proceed to evaluate this, transforming it into a new equation. Using the slope k of the tangent line to parabola C at point N, we aim to prove the conclusion of parallelism.

It takes 4 hours to travel from Jinpu County to Rongcheng

The two are also at a loss for words, so lazy that they can't even be bothered to write a solution. Perhaps the handwriting is too sloppy, as if the character for 'solution' is drawn like a ghostly symbol.

Physical fitness: Level 1 (0 / 100)

Qin Yuanqing immediately thought of 1, letting a = b = c = 1, which leads to k ≥ 2. Then, substituting k = 2 into the original equation forms a new equation, thereby proving the inequality holds.

Tang Liang, Jianyang No. 1 Middle School, 8 points ... Wow, only 8 points, I could score more than 8 points even with my eyes closed

Age: 18 years old

Although I do not understand, it seems that this exam paper is not particularly difficult, being considerably simpler than the questions from the National High School Mathematics Competition. While there is a certain amount of calculation involved, there are no questions that present an overwhelming level of difficulty

The broadcast first announced the examination rules, the duration of the exam, and the points for candidates' attention. After that, the invigilator took out the examination papers, indicating that the papers had not been opened previously. After verifying and signing with two students in the front row, the invigilator then distributed the exam papers and draft paper one by one. The invigilator reminded everyone to fill in their names, identification numbers, admission tickets, and paper codes first, and not to rush to answer the questions. Only after the broadcast notification could they begin answering; otherwise, it would be considered cheating. The invigilator then distributed the exam papers and draft paper one by one, reminding everyone to fill in their names, identification numbers, admission tickets, and paper codes first, and not to rush to answer the questions. Only after the broadcast notification could they begin answering; otherwise, it would be considered cheating

He did not expect that his mathematical ability would reach level 5 at this time. While level 5 would not significantly change his scores in high school exams, it could add a considerable number of points in competitions at the high school league level

No, look, this is truly impressive, Qin Yuanqing, Jinpu No. 1 High School, 152 points! The first perfect score exam paper actually comes from Jinpu County No. 1 High School in Shuixian City!

Moreover, no one knows whether the answers they have filled in are correct or not, and in such circumstances, there is simply no time to do a second round.

The examination will commence promptly at 9:00

You see, there is also one that scores 4 points here, with each question earning 1 point for a solution

Physics: Level 4 (20 / 100000)

Host: Qin Yuanqing

Qin Yuanqing directly took out past examination questions from the high school mathematics competition for further study, engrossed in them. Compared to the college entrance examination questions, the problems in the high school mathematics competition are more in-depth and cover a broader range, even involving advanced mathematics.