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AoPS Competition Preparation
The Art of Problem Solving, Volume 1: The Basics
Grades 8 -11. The Art of Problem Solving, Volume 1, is the classic problem solving textbook used by many successful MATHCOUNTS programs, and have been an important building block for students who, like the authors, performed well enough on the American Mathematics Contest series to qualify for the Math Olympiad Summer Program which trains students for the United States International Math Olympiad team.
Volume 1 is appropriate for students just beginning in math contests. MATHCOUNTS and novice high school students particularly have found it invaluable. Although the Art of Problem Solving is widely used by students preparing for mathematics competitions, the book is not just a collection of tricks. The emphasis on learning and understanding methods rather than memorizing formulas enables students to solve large classes of problems beyond those presented in the book.
Speaking of problems, the Art of Problem Solving, Volume 1, contains over 500 examples and exercises culled from such contests as MATHCOUNTS, the Mandelbrot Competition, the AMC tests, and ARML. Full solutions (not just answers!) are available for all the problems in the solution manual.
Volume 1 is appropriate for students just beginning in math contests. MATHCOUNTS and novice high school students particularly have found it invaluable. Although the Art of Problem Solving is widely used by students preparing for mathematics competitions, the book is not just a collection of tricks. The emphasis on learning and understanding methods rather than memorizing formulas enables students to solve large classes of problems beyond those presented in the book.
Speaking of problems, the Art of Problem Solving, Volume 1, contains over 500 examples and exercises culled from such contests as MATHCOUNTS, the Mandelbrot Competition, the AMC tests, and ARML. Full solutions (not just answers!) are available for all the problems in the solution manual.
288 pages, Paperback
First published August 1, 2006
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1604 people want to read
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Displaying 1 - 19 of 19 reviews
March 12, 2018
This book is essential to anyone who actually plans to use math in their careers. I'd say the level is for an advanced early high schooler or really anyone who needs a proper refresher.
You can get good grades just by practicing and memorizing some common "scenarios". The only "challenges" school tends to give are just the same problems but with ugly numerical computations. Rarely have I been asked to truly think. Give this to your kid/cousins etc so they can learn it all the right way, the first time!
You can get good grades just by practicing and memorizing some common "scenarios". The only "challenges" school tends to give are just the same problems but with ugly numerical computations. Rarely have I been asked to truly think. Give this to your kid/cousins etc so they can learn it all the right way, the first time!
March 28, 2018
Challenging, fun math problems that really helped me improve my math skills!
November 29, 2022
As a beginers perspecive on advanced maths i found this book was very helpful and insightful as a year 10 student. It helped me understand maths from a new perspective one of Visualization and proofs. It really helps a new maths student build a strong foundation in mathematics.
September 14, 2017
Let's say it was so amusing , a book to those who want to try some competitions in math..
June 2, 2025
I'M DONE I'M DONE ASUDF90UASD8FU9S0ADF9SD8AFU LET'S GOOOOOOOOOOO
Actual review: if you want to get better at maths, you can't look down on this book. Sure, some chapters are very basic covering stuff like quadratics and coordinates, but other chapters may cover new content (e.g modular arithmetic, incenter circumcenter orthocenter), and even in the basic chapters you'll humble yourself by realising that there's a lot to unpack in the most simple concepts (e.g why probabilities are multiplied)
Of course, everyone is at a different level, but for anyone trying to built their fundamentals from the ground up this book is the perfect place to start. It provides a structured way to revisit your basics for older students (like me) and builds a comprehensive base for newer younger students. But of course, this book isn't perfect, so here are some of my criticisms+general thoughts:
1. Particularly in the modular arithmetic chapter, I felt like you didn't need to explain the basic modular rules using change of base, but a much simpler explanation could be arrived at using distributive law. Distributive law also explains the rules for -ve sign remainders e.g -3 mod 5
2. Some explanations can be very tedious sometimes (e.g angle subtended by same arc was massively overcomplicated). I think this book is best paired with a curious mind willing to explore new perspectives/approaches beyond it. When you find an explanation of a theorem you don't understand / are dissatisfied with, search it up on youtube/ask for help.
3. Be prepared to spend a lot of time on this book regardless of your skill level. I am an A level Further Maths student currently 17 years of age at the time of writing this review, yet I still managed to spend over 5 months of time reading through and completing everything AOPS1 has to offer. You may be better than me (e.g Math/STEM undergrad), you may be worse than me (primary/middle school student), but so long as you're not a prodigy/contestant with previous experience this book will seriously demand a lot of your time if you want to take the completionist route.
4. The book alone is not enough. Nobody can spend all of their time purely revisiting fundamentals, such will be very tiresome and demotivating. Get a tutor, preferably one that's IMO level/way above your skillset that can guide you through proof based questions way beyond your level once a week. I find that this is the best balance between revisiting/learning fundamental stuff that you can your own and simultaneously experiencing the growth from being pushed past your current limits.
5. Concepts require time to marinate. It's not an intelligence thing, but I think there's a limit to how fast you can progress especially with new content. Reading this book shouldn't be linear, skim through chapters that are easy and take your time with new stuff you haven't seen before.
Final remarks: overall, I've made some great progress with AOPS1. I really like how it teaches you the fundamentals of mathematics and not just the fundamentals of olympiad theorems, like sure it'll tell you about some geometry tricks but it'll also teach you e.g "why associative operations have an inverse" and "what is a transitive operator?" as well as basic proof writing and conditional logic (inverse, converse, contrapositive). One thing I'd like to reiterate though: get a tutor/mentor. You need to combine revisiting fundamentals with being pushed past your limits and honestly there's no better way to do that than with a mentor. And on that note, I'm signing off, giving this a deserved rating of 5/5. Excited to finally start reading AOPS2!
Actual review: if you want to get better at maths, you can't look down on this book. Sure, some chapters are very basic covering stuff like quadratics and coordinates, but other chapters may cover new content (e.g modular arithmetic, incenter circumcenter orthocenter), and even in the basic chapters you'll humble yourself by realising that there's a lot to unpack in the most simple concepts (e.g why probabilities are multiplied)
Of course, everyone is at a different level, but for anyone trying to built their fundamentals from the ground up this book is the perfect place to start. It provides a structured way to revisit your basics for older students (like me) and builds a comprehensive base for newer younger students. But of course, this book isn't perfect, so here are some of my criticisms+general thoughts:
1. Particularly in the modular arithmetic chapter, I felt like you didn't need to explain the basic modular rules using change of base, but a much simpler explanation could be arrived at using distributive law. Distributive law also explains the rules for -ve sign remainders e.g -3 mod 5
2. Some explanations can be very tedious sometimes (e.g angle subtended by same arc was massively overcomplicated). I think this book is best paired with a curious mind willing to explore new perspectives/approaches beyond it. When you find an explanation of a theorem you don't understand / are dissatisfied with, search it up on youtube/ask for help.
3. Be prepared to spend a lot of time on this book regardless of your skill level. I am an A level Further Maths student currently 17 years of age at the time of writing this review, yet I still managed to spend over 5 months of time reading through and completing everything AOPS1 has to offer. You may be better than me (e.g Math/STEM undergrad), you may be worse than me (primary/middle school student), but so long as you're not a prodigy/contestant with previous experience this book will seriously demand a lot of your time if you want to take the completionist route.
4. The book alone is not enough. Nobody can spend all of their time purely revisiting fundamentals, such will be very tiresome and demotivating. Get a tutor, preferably one that's IMO level/way above your skillset that can guide you through proof based questions way beyond your level once a week. I find that this is the best balance between revisiting/learning fundamental stuff that you can your own and simultaneously experiencing the growth from being pushed past your current limits.
5. Concepts require time to marinate. It's not an intelligence thing, but I think there's a limit to how fast you can progress especially with new content. Reading this book shouldn't be linear, skim through chapters that are easy and take your time with new stuff you haven't seen before.
Final remarks: overall, I've made some great progress with AOPS1. I really like how it teaches you the fundamentals of mathematics and not just the fundamentals of olympiad theorems, like sure it'll tell you about some geometry tricks but it'll also teach you e.g "why associative operations have an inverse" and "what is a transitive operator?" as well as basic proof writing and conditional logic (inverse, converse, contrapositive). One thing I'd like to reiterate though: get a tutor/mentor. You need to combine revisiting fundamentals with being pushed past your limits and honestly there's no better way to do that than with a mentor. And on that note, I'm signing off, giving this a deserved rating of 5/5. Excited to finally start reading AOPS2!
June 30, 2022
I'd be harsher on the book but maybe I'm imagining things - I thought it was unnecessarily hard/not willing to explain certain points, but I did this mostly on the bus to school when I was in 7th grade in the margins so who knows - I remember this section on pigeonholes and being so confused throughout
it's funny I asked my roommate if he did this and he did and had a bad memory I think that's called collective trauma
As a whole though AoPS p good, like the forums and trading tests, (Which I gave to Ma Haha), ending up trading tests and messaging the guy who got 1st, for the win - I'm not sure if I'd recommend it to middle schoolers as a starter book though, bc even though some questions were really easy some I doubt anyone would be able to solve
it's funny I asked my roommate if he did this and he did and had a bad memory I think that's called collective trauma
As a whole though AoPS p good, like the forums and trading tests, (Which I gave to Ma Haha), ending up trading tests and messaging the guy who got 1st, for the win - I'm not sure if I'd recommend it to middle schoolers as a starter book though, bc even though some questions were really easy some I doubt anyone would be able to solve
July 4, 2023
( partial read )
I picked up this book because I missed School maths and was craving picking up a notebook and solving problems. Since I wasn’t sure I will be able to follow up completely, I picked up this easy book.
Solved questions for the topics which were interesting to me. Really enjoyed some of the questions!
Now when I feel more confident in my consistency, Onto a calculus or probability book next!
I picked up this book because I missed School maths and was craving picking up a notebook and solving problems. Since I wasn’t sure I will be able to follow up completely, I picked up this easy book.
Solved questions for the topics which were interesting to me. Really enjoyed some of the questions!
Now when I feel more confident in my consistency, Onto a calculus or probability book next!
January 13, 2020
Great book! Great method of solving equations and helping me ace a test! Thank you
January 7, 2022
The problems are good but a bit outdated. I can't believe AOPS is on goodreads. AOPS is aewsome.
January 24, 2025
Rarely will you find a book as thorough as this one.
May 18, 2025
If your middle school student yearns to level up their math understanding, maybe looking to compete in math competitions or just excited to learn at a level beyond what the normal school curriculum will demand, this series is a fantastic investment. Take it slow, maybe even do it with a friend or teacher, and wow, will you learn. All of my boys loved these books.
April 21, 2025
This book where I discover that math is not about memorizing formulas or grinding through repetition. It’s about strategic thinking, curiosity, and elegant solutions. It teaches you how to approach them, how to learn from mistakes, and how to think multiple steps ahead. Math has also other side, for me it is sort of a puzzle ( like a kid's puzzle game) which makes you wanna do it more!
February 7, 2014
best book ever
Currently reading
February 18, 2014Good Book
October 9, 2015
pretty good!
October 7, 2016
This is the book that any students who wants to try Math competitions. Problems are very challenging but fun to solve.
October 10, 2019
It’s a great book for AMC 10
Want to read
May 27, 2017656
Displaying 1 - 19 of 19 reviews