The Reincarnated Science Whiz

Chapter 9: The Provincial League Exam

With the provincial competition approaching, Qin Yuanqing’s schedule suddenly became very tight. No sooner had he finished his midterms than he gathered with the others and, led by the head of the math department, boarded a bus bound for Fuzhou. The trip from Jinpu County to Fuzhou took four hours. Qin Yuanqing immediately pulled out past exam papers from the High School Mathematics League and continued studying them, engrossed in the material. Compared to college entrance exam questions, the High School Mathematics League exams were more in-depth and covered a broader range of topics, even delving into calculus. Of the fifteen students who participated, the top performer won a first prize but failed to make the provincial team to compete in the National High School Mathematics League. This was his last chance to participate, just like Qin Yuanqing and the others. If he failed to make the provincial team, he would miss out on the national competition—let alone the National Team and the chance to compete internationally. After a four-hour journey, the bus arrived at Fuzhou Experimental High School, the venue for this year’s provincial High School Mathematics League. Contestants from various cities and counties would compete here. Even if winning a First Prize didn’t secure direct admission to a top-tier university, it would still grant bonus points on the college entrance exam. According to past trends, 40% of participants receive a second prize, 10% win a first prize, and fewer than 0. 5% make it onto the provincial team. The head of the math department at Jinpu No. 1 High School gave everyone a few instructions and then directed them to locate their exam rooms, so they wouldn’t panic or lose their composure tomorrow morning if they couldn’t find them. They would be staying at a nearby hotel that night—one without a star rating. After inspecting the exam venue, Qin Yuanqing returned to the school gate. The others hadn’t come back yet; it took about half an hour for everyone to gather. Zhang Jiaji, the head of the math department, had everyone double-check their ID cards, ballpoint pens, 2B pencils, rulers, set squares, compasses, and admission tickets. After all, if any issues were discovered now, there was still time to remedy them; if they weren’t discovered until tomorrow, there would be no time to fix them. Upon checking into the hotel—where they were assigned two people per room—Qin Yuanqing didn’t chat idly with his roommate but instead took out the exam questions to study. His time was precious; he couldn’t afford to waste a single minute or second. Gradually, Qin Yuanqing began to realize that while many of the competition problems seemed to go beyond the curriculum, they actually didn’t—they were simply extensions of the material, with increased difficulty. As long as he continued to analyze them in depth, he could still solve them. Suddenly, Qin Yuanqing’s face lit up with joy when he realized that his attributes had changed: Host: Qin Yuanqing Age: 18 IQ: 145 Emotional Intelligence: 100 Subjects: Chinese: Level 4 (20/100,000) Math: Level 5 (0/1,000,000) English: Level 3 (100/10,000) Physics: Level 4 (20/100,000) Chemistry: Level 4 (20 out of 100,000) Biology: Level 4 (20/100,000) Physical Fitness: Level 1 (0/100) He never expected that his math ability would reach Level 5 at this point. While Level 5 wouldn’t make a huge difference in his high school exam scores, it could significantly boost his performance in competitions like the high school math league. Sure enough, as his Math level reached Level 5, Qin Yuanqing found that understanding these math league problems became much easier—it felt as if a blockage had been cleared. Qin Yuanqing took advantage of the remaining time that evening to work through the problems one by one. It wasn’t until midnight, when Zhang Jiaji came to check on them and reminded them it was time for bed, that Qin Yuanqing reluctantly put his book away. After taking a shower, he fell fast asleep. After a rigorous inspection process, participants were permitted to bring only their ID cards, admission tickets, and basic exam materials into the testing room; cell phones and all other electronic devices were strictly prohibited. Qin Yuanqing arrived at his exam room and took his seat. The competition’s standards were modeled after the college entrance exam—and were even stricter—as each room accommodated only twenty test-takers. The desks were arranged in four rows of five, with ample spacing between each row to ensure no one could see the answers of those sitting in front, behind, or to the sides. Additionally, each room was staffed with two proctors who constantly patrolled the room. Qin Yuanqing sat down and arranged his ballpoint pen, 2B pencil, ruler, compass, eraser, and set square on the desk, placing his admission ticket and ID card in the upper right corner of the desk as he waited for the open-book exam materials to be distributed. The public address system first announced the exam room rules,the exam duration, and important instructions for the candidates. The proctors then retrieved the exam booklets, demonstrating that they had not been opened beforehand. After having two students in the front row verify and sign off on them, the proctors removed the exam papers from inside and distributed them one by one, along with scratch paper. They reminded everyone to first fill in their names, ID numbers, admission ticket numbers, and the exam code, and not to rush into answering questions. Candidates were instructed to wait for the announcement over the PA system before beginning the exam; otherwise, they would be considered to be cheating. After Qin Yuanqing filled in his name, admission ticket number, ID number, school name, and exam code on the test booklet, he saw that there were still 10 minutes left before the exam began. Not knowing where to start, he followed his usual habit of first reviewing the test booklet. He glanced over the questions, question types, and key topics, gaining a general understanding of the material. At exactly 9:00 a. m. , the exam began on time. Qin Yuanqing began working on the questions. The first section, just like the college entrance exam, consisted of multiple-choice questions. Qin Yuanqing couldn’t calculate the answers in his head, so he kept working them out on scrap paper before filling in the answers. Once he had finished the multiple-choice section, he filled in the answers on the answer sheet with a 2B pencil. After double-checking to ensure everything was correct, he moved on to the second section. The second section consisted of fill-in-the-blank questions. These questions seemed simple at first glance but were actually much harder than the multiple-choice ones. Sometimes there were not just one but two correct answers, and omitting even one would result in a half-point deduction. Moreover, no one could be certain whether their answers were correct or not, and under these circumstances, there was simply no time to review them a second time. In terms of question types, there were 6 multiple-choice questions, each worth 6 points, for a total of 36 points. There were also 6 fill-in-the-blank questions, each worth 6 points, for a total of 36 points. The third section consisted of 4 problem-solving and proof questions, each worth 20 points, for a total of 80 points, bringing the maximum possible score to 152 points. The first problem in the open-ended section—Question 13—tests knowledge related to the intersection of a parabola and a line. The problem is as follows: Given the parabola C: y = ax² (a > 0) and the line y = a + 2, which intersects the parabola at points A and B, let M be the midpoint of segment AB. Draw a perpendicular from M to the x-axis, which intersects the parabola C at point N. (1) Prove that the tangent line to the parabola C at point N is parallel to AB. (2) Does a real number a exist such that If so, find the value of a; if not, explain why. Qin Yuanqing set up the parabola and the line as a system of equations on the exam paper. Since they are points of intersection, y is equal, leading to ax² − x − 2 = 0. Then, let A(x₁, y₁) and B(x₂, y₂); use these to solve for the values, thereby deriving a new equation. Finally, use the slope k of the tangent line to parabola C at point N to prove the parallelism. For sub-question (2) and similar problems, one first assumes the existence of a real number a such that the equation holds, then uses the given conditions to deduce the value of a. If a can be calculated, it exists; if there is no solution, it does not exist. “Huh, this exam isn’t that hard after all?” Qin Yuanqing muttered to himself after finishing Question 14: “Could it be that I’m working on a fake exam?” Although he was puzzled—it didn’t seem like a particularly difficult exam after all—it was considerably easier than the problems on the National High School Mathematics League. While there was a fair amount of calculation involved, the difficulty certainly hadn’t skyrocketed. Question 15 asked 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. Qin Yuanqing immediately thought of 1. Setting a = b = c = 1, he found that k ≥ 2. He then substituted k = 2 into the original equation to form a new equation, which he used to prove that the inequality held true. Qin Yuanqing moved on to Question 16, the final problem—a proof question that combined necessary and sufficient conditions with inequalities, making it the most challenging. Qin Yuanqing furrowed his brow slightly and broke the problem down step by step, writing out a lengthy proof. Just as he finished writing the conclusion, the bell rang. The proctor announced the end of the exam; every student stood up, and the proctor collected the answer sheets one by one. As soon as the exam ended and they stepped out of the exam room, some students began complaining: “How could it be this hard? Is this even fair?” Qin Yuanqing, on the other hand, actually breathed a sigh of relief. He found that this provincial competition wasn’t as difficult as he’d imagined—it was only slightly harder than the actual college entrance exam math questions. So Qin Yuanqing seemed quite relaxed—eating and drinking as he pleased—prompting the head of the math department to tease him, calling him “thick-skinned” and “carefree.” As soon as the morning session of the provincial competition ended, the judges from the Provincial Mathematical Society began grading the exams that afternoon. This time, more than 1,200 high school students from across the province had registered to compete, so they needed to grade the exams and tally the scores as quickly as possible. Although the multiple-choice questions were graded by computer programs, the workload was still enormous. In addition, two graduate students were in charge of data entry, inputting the results into Excel to facilitate the final tally and ranking. “Lin Feng, Lushi Island Shuangshi High School, 110 points.” “Zhou Hu, Shuitou No. 1 High School, 89 points.” “Wang Lin, Provincial Experimental High School, 118 points.” “Li Peirong, Shuixian No. 1 High School, 105 points.” “Tang Liang, Jianyang No. 1 High School, 8 points... Wow, only 8 points? I could score more than that with my eyes closed.” “Look, there’s even someone here who got 4 points—they wrote a solution for each question and got 1 point for each.” The two graduate students, both from the Department of Mathematics, found endless amusement in compiling these statistics; sometimes they laughed until their stomachs hurt. They found scores as low as 1 point, but that wasn’t the lowest—the lowest was 0 points, and there were quite a few of those. The two were speechless. How lazy could someone be—too lazy to even write out a solution? Or maybe their handwriting was just too sloppy; the word “solution” looked like some kind of mysterious symbol. “Wow, this person is amazing! From Lushi Island Shuangshi High School, they scored 140 points—that’s the highest score so far!” “No, look—this one’s even more impressive. Qin Yuanqing from Jinpu No. 1 High School scored 152 points! The first perfect-score exam actually came from Jinpu County No. 1 High School in Shuixian City!” Both of them looked astonished. Jinpu No. 1 High School was the best school in Jinpu County and was considered a key high school even within Shuixian City. However, on a provincial scale, it seemed rather insignificant and had no reputation at all. After all, it wasn’t a provincial key high school; everyone only knew about those provincial key high schools—who would even know about a high school in an ordinary county?