The Art of Problem Solving, affectionately known as “AoPS” for short, is a math program offering books and courses covering high-school mathematics from an advanced and challenging perspective. The heart of AoPS is a two-volume set of books, mentioned before in “Resources for Math Learning (Intro to Math)”:
The Art of Problem Solving, Volume 1: the Basics by Sandor Lehoczky and Richard Rusczyk (solutions)
The Art of Problem Solving, Volume 2: and Beyond by Richard Rusczyk and Sandor Lehoczky (solutions)
Although AoPS is aimed at high-schoolers, I decided to include it as part of my math learning project because it covers a remarkable breadth of subjects and served as a great way to review the elementary curriculum without getting bored. Here are just some of the topics included in Volume 1:
Algebra, including exponents and logarithms, complex numbers, and quadratic equations; analysis, including functions, sequences and series, and set theory; introductory probability and statistics; a little elementary number theory; beginning combinatorics; an intro to proof; and an enormous amount of geometry, a sorely neglected topic in the elementary curriculum.
And in Volume 2:
A ton more geometry, including Ptolemy’s theorem, Ceva’s theorem, Menelaus’s theorem, and Stewart’s theorem; more depth on probability, combinatorics, and complex numbers; a surprising amount of number theory, including linear and quadratic congruences, Fermat’s little theorem, Euler’s \( \phi \) function, and an entire chapter on Diophantine equations; and even a chapter on graph theory.
In addition to covering surprisingly advanced subjects (Diophantine equations in high school?!), AoPS can be quite challenging even on subjects that seem fairly easy. Some examples from my own posts include “Solution to a Contest Math Problem”, “Another Contest Math Problem”, and “An AoPS Problem for Half Tau Day 2025”.
As indicated by those post titles, AoPS very much belongs to the “contest math” genre, which is rigorous and challenging but also places a premium on speed and cleverness. This can be an advantage, of course, but it’s important to remember that progress in the mathematical sciences often requires more tenacity and focus than speed and cleverness. As mathematician Botong Wang noted of his collaborator June Huh:1
“I have this math competition experience, that as a mathematician you have to be clever, you have to be fast,” he said. “But June is the opposite… If you talk to him for five minutes about some calculus problem, you’d think this guy wouldn’t pass a qualifying exam. He’s very slow.” So slow, in fact, that at first Wang thought they were wasting a lot of time on easy problems they already understood. But then he realized that Huh was learning even seemingly simple concepts in a much deeper way—and in precisely the way that would later prove useful.
So useful, in fact, that Huh won the Fields Medal in 2022.2
Despite these minor limitations, I believe that contest math can have a place in any math learning project, whether high school, college, grad school, or independent study. And there’s no doubt that AoPS is a great way to go.