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==Introduction==
The best way to prepare for math contests is to '''do lots of practice problems''' and learn the material necessary to solve these problems. There are also many books and online handouts and lectures you can use to improve your problem-solving skills.  Depending on your current abilities, you may want to start out with different practice problems, different books, and different areas of the forums. You should also try to strengthen the areas you are not as good at. This guide is intended to help you get started.

==Overview==
If you don't feel like going too deep and want a straightforward answer, here it is:

===Beginner===
To score well on the low-level competitions (like [[Mathcounts]] and [[AMC 8]]), read the following AoPS books and take their AOPS Academy/Online classes simultaneously in this order:
*Prealgebra
*Introduction to Algebra
*Introduction to Geometry
*Introduction to Number Theory
*Introduction to Counting & Probability
*Volume 1
Use AoPS's [[Alcumus]] tool constantly to practice all of these topics. When you answer problems incorrectly or take a long time to answer one, reread the part in the book that corresponds with that question. The more you do this, the better your skills will get.

===Advanced===
To score well on the high level competitions (like [[AMC 10]] and [[AIME]]), read the following AOPS books and take their AOPS Academy/Online classes simultaneously in this order:
*Intermediate Algebra
*Intermediate Counting and Probability
*Volume 2
*Precalculus
*Calculus
Use AoPS's [[Alcumus]] tool constantly to practice all of these topics. When you answer problems incorrectly or take a long time to answer one, reread the section in the book that corresponds with that question. The more you do this, the better your skills will get.

==Books==
===Art of Problem Solving Books=== 
'''AoPS books''' are excellent resources to help prepare for math contests. They cover a broad range of topics, mostly focusing on [[Algebra]], [[Geometry]], [[Number theory]], and [[Combinatorics]].
*Art of Problem Solving Introduction series and Volume 1 - [[Mathcounts]], [[AMC 8]], [[AMC 10]]
*Art of Problem Solving Intermediate series and Volume 2 - [[AMC 12]], [[AIME]], [[USAMO]], [[MOP]]
AoPS books break down specific areas of mathematics. These books are on 3 levels: Introductory, Intermediate, and Advanced.  The Advanced series, as well as part of the Intermediate series, has not yet been published.  These books are indexed [{{SERVER}}/store here].  Excerpts are provided, as well as pretests and posttests to see if the books are on the right level for you. A very important note is that the prealgebra series will cover topics from algebra, number theory, geometry, and counting & probability, but justs skims through the important parts. Theoretically, with extensive practice and going over the entire book multiple times, you could score well on the basic level competitions like Mathcounts or AMC 8.

*Prealgebra - [[Mathcounts]], [[MOEMS]] 
*Introduction to Algebra - [[Mathcounts]], [[AMC 8]]
*Introduction to Number Theory - [[Mathcounts]], [[AMC 8]], [[AMC 10]], [[AMC 12]], [[AIME]]
*Introduction to Geometry - [[Mathcounts]], [[AMC 8]], [[AMC 10]], [[AMC 12]], [[AIME]], [[HMMT]]
*Introduction to Counting & Probability - [[Mathcounts]], [[AMC 8]], [[AMC 10]], [[AMC 12]], [[AIME]]
*Intermediate Algebra - [[AMC 10]], [[AMC 12]], [[AIME]], [[USAMO]], [[HMMT]]
*Intermediate Counting & Probability - [[AMC 12]], [[AIME]], [[HMMT]], [[USAMO]]
*Precalculus - [[AMC 12]], [[AIME]], [[USAMO]]
*Calculus - [[HMMT]], [[Putnam]]

===Other Books===
Here are a few more books good for contests:
*[//www.amazon.com/The-Art-Craft-Problem-Solving/dp/0471135712 Art and Craft of Problem Solving] by Paul Zeitz
*[//www.amazon.com/Problem-Solving-Strategies-Problem-Books-Mathematics/dp/0387982191/ref=pd_bxgy_b_text_b Problem-Solving Strategies] by Arthur Engel
*[//www.amazon.com/Geometry-Revisited-New-Mathematical-Library/dp/0883856190 Geometry Revisited] by H.S.M. Coxeter & Samuel L. Greitzer
*[//www.amazon.com/102-Combinatorial-Problems-Titu-Andreescu/dp/0817643176 102 Combinatorial Problems] by Titu Andreescu & Zuming Feng
*[//www.amazon.com/103-Trigonometry-Problems-Training-Team/dp/0817643346/ref=pd_sim_b_1 103 Trigonometry Problems] by Titu Andreescu & Zuming Feng
*[//www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=pd_bxgy_b_text_b 104 Number Theory Problems] by Titu Andreescu & Zuming Feng
*[//www.maa.org/press/ebooks/euclidean-geometry-in-mathematical-olympiads Euclidean Geometry in Math Olympiads] by Evan Chen
*Ritvik Rustagi's [//www.tmasacademy.com/ace-the-amc10-12-free-book ACE The AMC 10 and 12 book] is a great resource to use for AMC 10 and AMC 12. The book has over 200 pages and contains 250+ problems with detailed solutions.
*Free [//www.omegalearn.org/mastering-amc8 Mastering AMC 8 book] is a good way to learn and review the topics on the AMC 8
*Free [//www.omegalearn.org/mastering-amc1012 Mastering AMC 10/12 book] is a good way to learn and review the topics on the AMC 10/12
Other books can also be found online.

===AMC 8===
The following books are recommended for AMC 8:
*Algebra: Introduction to Algebra
*Geometry: Introduction to Geometry 
*Combinatorics: Introduction to Counting & Probability
*General: Competition Math for Middle School
*Number Theory: Introduction to Number Theory (Note: There is not much Number Theory on the AMC 8.)

===AMC 10===
The following books are recommended for AMC 10:
*Algebra: Intermediate Algebra
*Geometry: Introduction to Geometry 
*Combinatorics: Introduction to Counting & Probability
*Number Theory: Introduction to Number Theory
*General: Volume 1

===AMC 12===
The following books are recommended for AMC 12:
*Algebra:  Intermediate Algebra
*Geometry: Introduction to Geometry 
*Combinatorics: Intermediate Counting & Probability (review Introduction to Counting and Probability if needed)
*Number Theory: Introduction to Number Theory
*General: Volume 2
*Extra: Precalculus

==Practice Problems==
Old practice problems and solutions sorted by contest and year are available on the AoPS Wiki and the forums. Here are some old contest archives that may be useful for practicing with:

*[[AMC 8 Problems and Solutions]] - [[AMC 8]] is a national contest for grades 8 and younger.
*[[AMC 10 Problems and Solutions]] - [[AMC 10]] is a national contest for grades 10 and younger.
*[[AMC 12 Problems and Solutions]] - [[AMC 12]] is a national contest for grades 12 and younger.
*[[AIME Problems and Solutions]] - The [[American Invitational Mathematics Examination]] is a contest administered to those who qualify with a high score on the AMC 10 or AMC 12.
*[[USAMO Problems and Solutions]] - The [[United States of America Mathematical Olympiad]] is a proof-based contest which is qualified for through a combination of AMC and AIME scores.
*[//www.hmmt.co/archive/problems HMMT] is a nice contest on a hard AIME level run by Harvard and MIT.
*[[List of mathematics competitions]].
*[[AoPS Past Contests|User-Created Contests]].

==Strategies to qualify for the AIME== 
There are certain strategies in preparing for the AMC 10/12- especially qualification for the AIME.

The AIME cutoff has ranged 84-95 on AMC 12.since 2020 when the qualification was loosened to "5% of scorers". In order to get a score in the range, a simple way is to answer 13 questions right (check your work carefully!) and leave the rest blank, which earns a score of 96. In the past, since the 2020  cutoffs have been getting slightly harder each year, and new generations of competitors don't always match the new level. This means, since the first 10 questions are solvable in half the test time by most people who prepare, they are 60 easy points. Throughout questions #10-#20, answering 3-5 shall be enough to qualify. 

Beware, though, that the AIME question #1 is harder than AMC question #10, so this strategy presumes that you *could* solve more than 15 AMC problems, but you are choosing to reduce your time/difficulty pressure and increase your confidence, to guarantee a qualifying score but not get your highest possible score. 

Qualification for the USAMO, however, is much harder. Only 260-270 people qualify every year. USAMO qualifiers need a good combination of AMC & AIME scores. The average score on the AMC 12 for a USAMO qualifier is around 114-132. There are simple ways to do this but it takes a lot of work. Answering the first 15 right, and then getting 5 out of the 10 left would usually qualify.

The AIME cutoff on the AMC 10 have ranged throughout (94.5-104) in recent years. The top 2.5% of scorers qualify. The AMC 10 does test less topics than the AMC 12 but many questions go into much more depth. Cutoffs on the AMC 10 are higher since the testing only tests topics up to Geometry. AIME ranges from Algebra to precalculus, which means only very elite scorers make it. Though the qualifying scores are high, there is indeed a good strategy. Since you get 1.5 points for each question blank, it’s good just to do what you know. Answering 15 questions right and leaving the rest blank would earn a score of 105 while answering 20 right and leaving the rest blank would earn a score of 127.5. Since contests are getting harder as said earlier, 15-18 right should be enough. 

Qualifications for the USAJMO is similar to that for the USAMO except it uses AMC 10 scores.

==Top 10 most Difficult National Math Competitions in The USA==<!--Is this part necessary?-->

Can someone add HMMT?

10. MATHCOUNTS - Pre-Algebra, Geometry, Number Theory, Combinatorics, Logic

9. AMC 10 - Intermediate Algebra, Geometry, Number Theory, Combinatorics

8. AMC 12 - Intermediate Algebra, Geometry, Number Theory, Combinatorics, Pre-Calculus

7. ARML - Advanced Algebra, Geometry, Number Theory, Combinatorics

6. AIME - Advanced Algebra, Advanced Geometry, Number Theory, Combinatorics, Pre-Calculus

5. USAMTS - Advanced Algebra, Advanced Geometry, Number Theory, Combinatorics, Pre-Calculus

4. USAJMO - Advanced Algebra, Advanced Geometry, Advanced Number Theory, Combinatorics

3. USAMO - Advanced Algebra, Very Advanced Geometry, Advanced Number Theory, Combinatorics, Advanced Pre-Calculus

1 (tie). IMO - Very Advanced Algebra, Very Advanced Geometry, Very Advanced Number Theory, Advanced Combinatorics, Advanced Pre-Calculus

1 (tie). PUTNAM - Advanced Algebra, Geometry, Number Theory, Advanced Combinatorics, Advanced Calculus

==Forums==
The forums are one of the best ways to find problems to solve, get help with problems you cannot solve, and collaborate with people of all levels and abilities.  The forum is divided into many subforums for problems of different difficulties.

*The [{{SERVER}}/community/c3 Middle School] forum is for MathCounts and AMC 8-level problems.
*The [//{{SERVER}}/community/c4 High School] forum is a good place to find AMC10/12-level and AIME-level problems.
*The [{{SERVER}}/community/c6 Olympiad] forum is a forum for problems at the olympiad level.
*The [{{SERVER}}/community/c68 LaTeX and Asymptote] forum is a place to get help with <math>\text{\LaTeX}</math>, which is what you use to type things like <math>2^3</math> on the forums. It's also for Asymptote, which is what we use to make diagrams like this one:
<asy>
draw((0,0)--(2,0)--(0,2)--cycle);
label("A",(0,0),SW);
label("B",(2,0),SE);
label("C",(0,2),NW);
</asy>

==Cheat Sheets==
Many great reference guides are available for free on the internet.
* [//www.tmasacademy.com/amcseminar Ritvik Rustagi's 53-page long handout] has all the formulas for the AMC 10 and AMC 12. The handout was made alongside a 4 hour long review seminar for AMC 10 and AMC 12. It's a great way to review and learn new topics for the AMC 10 and AMC 12.
*[//www.dropbox.com/s/icds9u5yo0xypyf/AIMElist.pdf BOGTRO's list of theory relevant to the AIME.]
*Sohil Rathi's [//www.omegalearn.org/thebookofformulas The Book of Mathematical Formulas & Strategies] is a great way to review all formulas on math contests like AMCs, AIME, MATHCOUNTS, etc
*Coach Monk's [//monks.scranton.edu/files/courses/ProblemSolving/MathCountsPlaybookBW.pdf MathCounts Playbook] is a good place to start for MathCounts-level material.
*Coach Monk's [//monks.scranton.edu/files/courses/ProblemSolving/HighSchoolPlaybook.pdf High School Playbook] goes a little more in depth, and is useful for all levels of high school mathematics.
*The Mandelbrot Competition maintains a nice list of topics you need to know for high school math competitions called [http://www.mandelbrot.org/resources/forms/indivtopics.pdf All of Math in 3 Pages].
*[http://web.archive.org/web/20060518133620/http://staff.imsa.edu/math/journal/volume4/articles/NoahSheets.pdf The Noah Sheets]
*The Popular Elementary Math Competitions [//medium.com/@edustar/popular-elementary-school-math-competitions-do-you-know-them-7298dc1d3108 Contests to build Math Interests]

==Classes==
*Free AMC 8 Fundamentals Class: https://www.omegalearn.org/amc8-fundamentals
*Free AMC 8 Advanced/Mathcounts Class: https://www.omegalearn.org/amc8-fundamentals
*Free AMC 10/12 Class: https://www.omegalearn.org/amc10-12
*Free AMC 8/10 Class: https://www.youtube.com/channel/UC-Nt9Uo03VSo2QTNIzsE_cA (Some special seminars occasionally with Olympiad Winners)
*If you are serious about improving your problem-solving skills, AoPS offers several online classes, available [{{SERVER}}/school here].
*[{{SERVER}}/woot WOOT] is an online class offered by AoPS for olympiad training. It has one of the best peer groups in the country, and is a great way to prepare for the USAMO.

==Summer Camps==
Summer programs are also a great way to improve problem-solving skills.  Some of these include:

*[https://rossprogram.org/ Ross Mathematics Program]
*[http://www.promys.org/ PROMYS] 
*[http://www.mathcamp.org/ MathCamp]
*[http://www.mathpath.org/ MathPath]
*[https://www.awesomemath.org/ AwesomeMath]
*[http://ideamath.org IdeaMath]

== See Also ==
* [http://artofproblemsolving.com/wiki/index.php?title=Resources_for_mathematics_competitions Resources for Math Competitions]
* [https://sites.google.com/site/wwphssmathclub/resources More Resources]
* [https://sites.google.com/site/wwpnorthmathclub1/contest-prep Useful Handouts]
* [http://www.andrew.cmu.edu/user/daltizio/mathstuff.html More Useful Handouts]

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