Advanced mathematical problem-solving guide with detailed, step-by-step solutions and analysis, aimed at building graduate-level problem-solving skills for math competitions, AI/ML, data science, and theoretical work. This guide exists to fill a gap that most standard mathematics books and solution manuals leave open.
Typical problem books give you three things: the statement, a solution, and perhaps a short remark. They almost never tell you what is really happening in your head when you solve the problem well:
- How you recognize the relevant ideas,
- Which “triggers” in the problem tell you which tools to try,
- How the structure of this problem belongs to a wider family of problems.
My guide is intentionally built around this missing layer. It does not just present solutions; it emphasizes meta-recognition and transfer of strategy.
For each problem, I highlight the key moments in the solution: the point where a hidden symmetry becomes visible, where a clever substitution suggests itself, or where a standard theorem suddenly becomes relevant.
Instead of silently jumping from step A to step B, the guide explicitly asks:
- What pattern are we recognizing here?
- Why was this lemma or technique the natural thing to try?
- How could we have predicted this step from the problem’s structure?
This makes the book not just a collection of problems, but a training manual in mathematical thinking. You are not only learning what solves this problem, but what kind of mind-move solves an entire class of problems.
The level and style of the material are maybe even more needed for:
- students preparing for mathematical competitions (national Olympiads, undergraduate/graduate contests), and
- those moving toward theoretical mathematics (analysis, algebra, etc.), where proof-writing, abstraction, and generalization are central, as most of theorems take decades even for most of geniuses to discover, therefore you have to master generalizing and implying right groundwork in right directions to push forward.
Solutions are written in a graduate-level style: rigorous, detailed, and explicit about the underlying ideas. The goal is not to impress you with brevity, but to show the complete logical and conceptual path, so that you can reuse these ideas in unfamiliar settings.
This guide is not a random collection of tricks. It is the result of years of personal work, feedback, and refinement, including guidance and commentary from PhD-level mathematicians and professors.
Over time, I systematically identified:
- which skills each type of problem actually trains,
- which habits distinguish strong problem solvers from average ones, and
- how to make those skills visible and trainable instead of leaving them as “intuition” or “talent”.
Each problem is therefore chosen and annotated with a purpose: what this problem is really teaching you.
If you are reading this, you are probably someone who genuinely enjoys mathematics. The guide is written with that reader in mind: a motivated learner who wants to go beyond classroom explanations and standard exam prep.
The explanations are structured so that you can:
- work through the problems without needing a live instructor,
- reconstruct the reasoning step by step, and
- see how to generalize the method to other contexts.
In other words, it is meant to function as a kind of personal mentor in book form.
Modern general-purpose LLMs are useful tools, but they have clear limitations for this specific kind of deep training:
- They often produce correct-looking solutions without clearly identifying the critical conceptual steps.
- They usually cannot reliably detect what you, personally, are misunderstanding, or which skill in your toolkit is missing.
- Their answers tend to be optimized for being helpful in general, not for building a systematic, rigorous problem-solving framework that you can reuse and refine.
This guide is designed precisely for that missing role: to give you mathematically rigorous, fully worked-out reasoning together with a clear map of:
- what idea appeared,
- why it appeared,
- how to recognize when it should appear again in new problems.
In short, compared to other math books, this guide is important because it does not just show you how to solve these problems; it trains you to understand why these methods work, when to use them, and how to generalize them. If you are serious about becoming a stronger problem solver—for competitions or for theoretical research—this is exactly the kind of resource you need.
The material included in this collection—most of it, in fact—already existed by the end of 2023. I then worked to make it more rigorous, polished, and general. This file was mainly transformed into LaTeX format using online tools and some help from large language models.