Working Through the AoPS Precalculus Curriculum
Precalculus is the gatekeeper course for anyone trying to get into a rigorous math program. The Art Of Problem Solving book on the subject, written by Richard Rusczyk and Sandor Lehoczky, is the one most people end up using whether they plan to or not. It's dense, it expects you to actually do the work, and it does not hold your hand through any of it. The book covers functions, trigonometry, polar coordinates, conic sections, vectors, and a serious introduction to complex numbers. What makes it different from a standard textbook is that it treats every topic as a puzzle to be unpacked rather than a list of formulas to memorize. You'll spend more time deriving properties from first principles than you will plugging numbers into a calculator. I remember working through the section on logarithmic identities with a student. The book presents the change of base formula without any hand-holding, then immediately asks you to prove it using only the definition of a logarithm. Most students hit a wall there because they've spent years being told to "just use the formula." The workaround is simple: write out log_a(x) = y, convert it to exponential form, take log_b of both sides, and apply the power rule. Once you see that derivation laid out, the formula stops being magic and becomes something you can reconstruct in five minutes if you blank during a contest.
The problem sets are divided into four tiers: Problem Solving, Challenge Problems, Answer Key problems, and the deeper Olympiad-style questions at the end of each chapter. The first tier is manageable on its own. The Challenge Problems are where you learn whether you actually understood the material or just copied the examples. The final set will break you if you haven't been patient enough in the earlier chapters. One thing beginners consistently miss is that the trigonometry section is not a review. It assumes you have seen the unit circle once before and it moves immediately into sum and difference formulas, complex exponential forms, and De Moivre's theorem. If you go in thinking this is just "trig again" you will fall behind quickly. The complex number chapter alone covers enough ground to replace an entire semester of a college course that treats the topic as an afterthought. Here's a specific edge case that trips people up: the polar coordinates section includes a problem involving the intersection of a rose curve r = sin(3) and the circle r = 1/2. Standard algebra gives you sin(3) = 1/2, which yields six solutions in [0, 2). But if you graph both curves, you'll notice the rose has three petals and the circle intersects each petal twice, giving exactly six points. The trap is that some students double-count or miss the origin because it satisfies the circle equation but requires = 0, , or 2 for the rose, and the polar coordinate representation of the origin is not unique. I have my students always verify solutions by plotting, even when the algebra seems complete. It takes about three minutes on Desmos and saves you from losing points on a contest problem.
The answer key only gives answers, not solutions. This is intentional but it means you cannot self-study this book without external support unless you are genuinely comfortable working through dead ends. There is a solution manual available separately, and if you are using this on your own, I would recommend getting it. The alternative is spending two hours stuck on a single proof when thirty minutes with the manual would clear it up.
Get the Full Details
Practical advice for getting through the material
Don't rush the functions chapter. It looks easy because it reviews domain, range, composition, and inverses, but the inverse function problems near the end introduce ideas that become critical when you reach the transcendental equations section later. Skipping ahead and coming back costs more time than just sitting with the harder problems from the start. The vectors chapter introduces dot products and cross products in three dimensions. The algebraic derivations are fine. The geometric intuition is where students struggle, and the book does not spend as much time on visualization as it should. I pair this section with a few video lectures from MIT OpenCourseWare on multivariable calculus, specifically the ones covering geometric interpretations of vector operations. Thirty minutes of that clears up confusion that might otherwise take two days of wrestling with the text. When you hit the conic sections, pay attention to the parametric representations. The book uses them sparingly here but they reappear in the complex numbers chapter and in competition problems where a Cartesian approach becomes computationally unwieldy. Knowing that an ellipse can be parameterized as (a cos t, b sin t) turns a lot of messy algebra into something you can handle mentally.
The book has real limitations. It is not designed for a traditional classroom setting with a teacher guiding discussion. The exposition assumes a certain level of mathematical maturity that many students have not yet developed. It also contains occasional typographical errors, mostly in the later chapters, so if a problem seems to have no solution, check whether there is a misprint before concluding you are missing something. The errata list is posted on the AoPS website but it is incomplete. If your goal is competition math, this book will serve you well, especially for AIME-level preparation. If your goal is standard college prep, it is more than sufficient but you may find other resources more aligned with what your calculus class expects. The transition from AoPS Precalculus to a typical AP Calculus course is generally smooth, though some students find the pace jumps unexpectedly once derivatives appear. You can find the current edition through the Art Of Problem Solving store, Amazon, or Barnes and Noble. The second edition is the most recent and includes updated problem sets. Avoid used copies from before 2014 if possible because the first edition has some misprints that were corrected in later printings.
The real test of whether this book is working for you is whether you can look at a problem you've never seen before and feel confident that the tools in the chapters will eventually apply. That confidence doesn't come from reading the examples twice. It comes from working through every Challenge Problem at least once, even if you get it wrong. The struggle is where the learning happens, and the book is designed to make you struggle productively. I've seen students burn out trying to finish every problem in a single sitting. That's a bad strategy. Working through three or four problems deeply is worth more than skimming twenty. The material accumulates, and each chapter builds on the last. Gaps from rushing create compounding problems later, particularly in trigonometry and complex numbers where everything connects tightly. There is no shortcut around doing the work. The book won't do it for you, and neither will any supplement. But the payoff is genuine: students who put in the time with this material tend to handle first-year calculus with far less panic than their peers who only memorized procedures. That's the actual metric that matters at the end of the day.
