Self Study Resources
HOW TO APPROACH SELF-STUDY
Below are a selection of resources, organized vaguely by level and by topic. Working on problems will improve your problem solving in that area.
Do: Work on each problem for a reasonable time unless there's a problem you really like and want to keep working on it without time pressure. Carefully read the solutions after that time. If you did not solve the problem, try to understand why.
Is there a theorem you did not know/think of using?
Is the approach used in a proof absolute magic (rare) or have you seen a similar approach before?
What could've suggested to you that this may work?
Finally, practice writing up some of the problems. Don't abandon reading the solution just because you don't follow it immediately; sharpen your pencil and come back to it.
That being said, if you can't follow the solution despite giving it your absolute best effort, then the problem you picked is clearly substantially outside your current range as a problem solver. Move on.
Don't: Don't do serious mathematics while playing on the phone or watching TV; this is impossible. Don't read solutions without having spent the time on the problem first. Don't forget to read the solution even if you think you solved the problem. Only rarely practice problems that are substantially below your level.
Keep: Keep a list of problems that you weren’t able to solve for which you understand the solutions. The goal is to compile a list of problems you are not able to solve now, and be able to solve similar problems in 2 months.
If you are able to solve the problem 4 weeks later, remove from document. If not, there is likely a knowledge gap. Fill the knowledge gaps using Google (for small gaps) or a book below. Can’t find the right resource? Email us at admissions@mathmaddicts.org.
SELF-STUDY RESOURCES FOR SP100, SP200, and SP250
YOUNGER:
Math Circles for Elementary School Students: Berkeley 2009 and Manhattan 2011 (MSRI Mathematical Circles Library)
Books for SP200:
Middle school math skills books (in general, Art of Problem Solving books are quite good)
Other Resources:
Alcumus, a free adaptive problems database, organized by topic.
A primer on proofs, by Professor Christopher Heil.
Fun problems (with solutions) from Mandelbrot (some proofs practice in historical team play competition) & the Joe Holbrook Memorial Math Competition
Challenging online math programs:
SELF-STUDY RESOURCES FOR SP300
Books:
Introduction to middle school topics books (we would tackle in this order if you are planning on working on all four books):
Problem Solving Books:
A Decade of the Berkeley Math Circle Stankova, Zvezdelina.
Art of Problem Solving Volume 1: The Basics. If short on time, focus on the proofs chapter towards the end of the book.
Other Resources:
Alcumus, a free adaptive problems database, organized by topic.
A primer on proofs, by Professor Christopher Heil.
Fun problems (with solutions) from Mandelbrot (some proofs practice in historical team play competition) & the Joe Holbrook Memorial Math Competition
Challenging online math programs:
SELF-STUDY RESOURCES FOR GROUP SP400
Books:
A Decade of the Berkeley Math Circle Stankova, Zvezdelina.
Techniques of Problem Solving by Steven Krantz
Two primers on problem solving techniques:
Other Resources:
Primers on proofs by Professor Jenny Wilson, and by Professor Christopher Heil
Fun problems & proofs practice from Mandelbrot (some proofs practice in historical team play competition)
Practice your proofs, using USAMTS competition resources.
Challenging online math programs:
SELF-STUDY RESOURCES FOR SP500
Problem Books & Sources:
The USSR Olympiad Problem Book (Shklyarsky, Chentzov, Yaglom), mostly number theory.
Problem Solving through Problems (Loren C Larson)
Practice your proofs, using USAMTS competition resources.
These books have an introductory and challenge sections. Focus on the introductory problems.
Problem Books By Topic:
Intermediate Number Theory Text (and some more basic slides) by Prof Justin Stevens. Also has two notes on beginner and intermediate proofs.
A textbook on how to write proofs: How to Prove It: A Structured Approach
SELF-STUDY RESOURCES FOR SP600 and SP650
A reasonable time to work on a problem at this level is at least 90 minutes, but probably no more than 4 hours unless there's a problem you really like and want to keep working on it without time pressure.
Follow the do’s and don’ts in the instructions above.
Theory + Problem Books:
Number Theory: Topics in Number Theory: An Olympiad-Oriented Approach (Difficult material but it's not too hard to power through. Word of advice: understand and learn the proofs of theorems.)
Geometry: Euclidean Geometry in Mathematical Olympiads (EGMO)
Inequalities: Olympiad Inequalities by Mildorf (We suggest knowing Jensen's, AM-GM, CS, rearrangement and Schur and only try the problems that are labeled as having come from USAMO/IMO or other contests.)
Graph Theory: IMO Training 2008: Graph Theory
Another good proofs book: Proofs from the Book
Problem Books By Topic:
103 Trigonometry Problems: From the Training of the USA IMO Team
104 Number Theory Problems: From the Training of the USA IMO Team
These books have an introductory and challenge sections. Introductory problems you should either know how to solve immediately by recall or power through over the summer over the ones you haven't seen before.
Olympiad-Level Problems:
JMO-level: Try to solve every past JMO problem for least 2 hours (or until you solve it) over the summer.
USAMO-level: