Problem Sets
Published:
写在最前面的话
(全文：📝 聊聊大学数学做题那档子事）。
我始终认为，学习数学离不开习题 —— 数学是“算”懂的，而不是“看”懂的。
正如站在岸边再久也学不会游泳，唯有亲自演算、深入思考，才能真正掌握数学的逻辑与本质。因此学习时，切勿一味追求 “高观点” 而眼高手低，忽略了做题这一最基础也最关键的实践环节。
但“必须做题”并不等于“盲目做题”，这点必须厘清。
很多人对解题的意义有明显误解：以为题目做得越多，考试命中率就越高。
事实并非如此——我之所以强调做题，是因为优质习题能帮助我们加深对概念的理解、提升运算技巧、锻炼逻辑思维。但要特别警惕：如果忽视课本里的基本概念、定义、定理（尤其是定理的证明过程），只靠大量演算，不仅学不好数学，遇到稍微变形的题目也会立刻无从下手。
因此，我不建议一开始就盲目挑战过难的题目，也不主张陷入机械的“题海战术”，更不认可过早沉迷于“奇技淫巧”——这些做法只会让人偏离数学学习的核心。
基于此，我的建议是：先把课本上的例题和习题独立解出来。解题时，从中领会其所依据的是什么基本概念、什么定理，解完后再思考是否还能用别的方法解决，真正把这道习题搞懂。而不是又多背了一个套路、死记了一个公式。
最后请记住：做题只是途径，目的是借此打磨思维，让自己具备更清晰的逻辑和更系统的分析能力。千万不要本末倒置，把途径当成目的；况且说到“刷题”，AI在这方面早已远胜人类。
A Note Up Front
I’ve always believed that learning mathematics is inseparable from working through problems — math is something you compute to understand, not something you watch to understand.
Just as you’ll never learn to swim by standing on the shore, you can’t truly grasp the logic and essence of mathematics without doing the work yourself: solving problems, thinking deeply, and testing your understanding. Don’t get caught up in lofty “big-picture” ideas at the expense of the most basic — and most essential — practice of problem-solving.
But “doing problems” is not the same as “doing problems blindly.” That distinction matters.
Many people misunderstand the point of problem-solving: they think the more problems they do, the better their exam scores will be.
It doesn’t work that way. I emphasize problem-solving because high-quality exercises not only deepen your understanding of concepts but also sharpen your computational skills and strengthen your logical thinking. But beware: if you neglect the fundamentals in your textbook — concepts, definitions, theorems (especially their proofs) — and rely only on endless drills, you’ll not only fail to learn math well but also freeze up when faced with even slightly unfamiliar problems.
That’s why I don’t recommend rushing into overly difficult problems at the start, falling into the “problem-drilling” trap, or obsessing over clever tricks. All of these take you away from the heart of learning mathematics.
My advice: start by working through the examples and exercises in your textbook on your own. As you solve them, pay attention to which concepts or theorems each step is based on. Once you’ve finished, think about whether there’s another way to solve the same problem. Really understand it — don’t just memorize another trick or formula.
And finally, remember this: problem-solving is a means, not an end. The real goal is to refine your thinking so you develop clearer logic and a more systematic way of analyzing problems. Don’t confuse the path for the destination. And as for sheer “drilling,” AI already outperforms humans by far in that department.
关于题目的说明
本板块的题目来自我过去教学过程中的一些积累：有的摘自优秀教材，有的来自网上偶然看到的学习资源，也有来自学生提出的问题，还有一些取自不同类型的真题考试。当时觉得这些题目要么有点意思，要么颇具代表性，就顺手记录下来，稍加整理后放在这里。
特别说明：
- 这些题目不是课程作业，没有任何强制性，也与课程考试无直接关系。大家可以根据自己的学习节奏和需求，自主选择是否练习，不必有任何压力。毕竟，网上已有非常丰富的学习资源可供选择。
- 题目难度大致在我认为的“基础题—中档题”范围（针对非数学专业）：大多数是基本概念、定理的直接应用，少数可能需要对知识有更深入的理解后才能灵活运用，但绝不会涉及什么“奇技淫巧”。
- 所有题目没有答案或解析，一方面是因为我当时没有特意收集，另一方面对我个人而言，解题也不依赖答案。
- 分享这些题目的核心目的，是鼓励大家独立思考、自主探索。如果练习时遇到疑问，可以结合课程内容复习，或者和同学一起讨论。
- 题目会随缘更新。是否更新、更新节奏，主要取决于我当时是否在讲授相关课程内容，以及我的日常空闲时间。
About the Problems
The problems in this section are drawn from my experiences as a teacher over the years: some are taken from well-written textbooks, others from online resources I have encountered, some from students’ own questions, and still others from various types of past exam papers. At the time, I felt these problems were either interesting or representative, so I simply noted them down and later organized them here.
A few clarifications:
- These problems are not course assignments, carry no obligation, and have no direct connection with exams. You are free to decide whether to use them for practice, entirely based on your own pace and needs. There is no pressure—after all, plenty of study resources are already available online.
- The difficulty level is roughly what I consider basic to intermediate (for non-math majors): most are direct applications of concepts and theorems, while a few may require deeper understanding for flexible application. None involves “tricks” or obscure techniques.
- No solutions or answer keys are provided. I did not collect them at the time, and personally, I do not rely on them when solving problems.
- The main purpose of sharing these problems is to encourage independent thinking and self-exploration. If you encounter difficulties, try to revisit the course material or discuss with your peers.
- The problems will be updated from time to time—whether and when updates happen mainly depend on whether I am teaching related content and on my available spare time.
Courses
Calculus
- Functions, Limits, Continuity (函数、极限)
- Continuity (连续性)
- Derivative: Definition and Calculation (导数的定义和计算)
- Applications of Derivatives (导数的应用)
Linear Algebra
Probability & Statistics
- Events, Probability, Counting Methods (随机事件、概率、计数方法)
- Conditional Probability, Bayes’ Theorem (条件概率、贝叶斯定理)
- Random Variables and Distributions (随机变量及其分布)