# Difference between revisions of "How should I prepare?"

Mysmartmouth (talk | contribs) (New page: A common question that regularly comes up on the forum is "How can I prepare for MathCounts/AMC/ARML/AIME/USAMO?" This page is intended to answer these questions.) |
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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 the problems. There are also many books and online handouts/lectures you can use to improve your problem-solving skills. Depending on your current abilities, you will want to start out with different practice problems, different books, and in different areas of the forums. You should also try to strengthen in the areas you are not as good at. This guide is intended to help you get started. | ||

+ | |||

+ | == Books == | ||

+ | |||

+ | The '''Art of Problem Solving books''' are an excellent resource to help prepare for math contests. They cover a broad range of topics, from algebra to geometry to number theory to combinatorics and much much more. | ||

+ | |||

+ | * Art of Problem Solving Volume 1 - [[Mathcounts]], [[AMC 8]], [[AMC 10]] | ||

+ | * Art of Problem Solving Volume 2 - [[AMC 12]], [[AIME]], [[USAMO]], [[MOP]] | ||

+ | |||

+ | The '''AoPS textbooks''' 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 [https://artofproblemsolving.com/store/recommendations.php#state/240 here]. Excerpts are provided, as well as pretests and posttests to see if the books are on the right level for you. '''Alcumus''' is a good resource even if you do not own any of the AoPS books. | ||

+ | |||

+ | * 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]] | ||

+ | |||

+ | Here are a few more '''books good for preparation for higher level contests''' such as AMC 12, AIME, and USAMO (though some can be found online): | ||

+ | |||

+ | * [http://www.amazon.com/The-Art-Craft-Problem-Solving/dp/0471135712 Art and Craft of Problem Solving] by Paul Zeitz | ||

+ | |||

+ | * [http://www.amazon.com/Problem-Solving-Strategies-Problem-Books-Mathematics/dp/0387982191/ref=pd_bxgy_b_text_b Problem-Solving Strategies] by Arthur Engel | ||

+ | |||

+ | * [http://www.amazon.com/Geometry-Revisited-New-Mathematical-Library/dp/0883856190 Geometry Revisited] by H.S.M. Coxeter & Samuel L. Greitzer | ||

+ | |||

+ | * [http://www.amazon.com/102-Combinatorial-Problems-Titu-Andreescu/dp/0817643176 102 Combinatorial Problems] by Titu Andreescu & Zuming Feng | ||

+ | |||

+ | * [http://www.amazon.com/103-Trigonometry-Problems-Training-Team/dp/0817643346/ref=pd_sim_b_1 103 Trigonometry Problems] by Titu Andreescu & Zuming Feng | ||

+ | |||

+ | * [http://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 | ||

+ | |||

+ | * [http://www.maa.org/press/ebooks/euclidean-geometry-in-mathematical-olympiads Euclidean Geometry in Math Olympiads] by Evan Chen | ||

+ | |||

+ | |||

+ | ===AMC 8:=== | ||

+ | |||

+ | Algebra: Introduction to Algebra | ||

+ | |||

+ | Geometry: Introduction to Geometry | ||

+ | |||

+ | Combinatorics: Introduction to Counting & Probability | ||

+ | |||

+ | General: Math competitions-Middle School | ||

+ | |||

+ | -Note:(Not much Number Theory is asked on the AMC 8). | ||

+ | |||

+ | |||

+ | ===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: === | ||

+ | |||

+ | Algebra: Intermediate Algebra | ||

+ | |||

+ | Geometry: Introduction to Geometry | ||

+ | |||

+ | Combinatorics: Intermediate Counting & Probability, Introduction to Counting and Probability | ||

+ | |||

+ | Number Theory: Introduction to Number Theory | ||

+ | |||

+ | General: Volume 2 | ||

+ | |||

+ | Extra: Precalculus | ||

+ | |||

+ | == Practice Problems == | ||

+ | Old practice problems (with solutions) sorted by contest and year are available on the Wiki and the Resources section. Many practice problems are also available on the forums. | ||

+ | |||

+ | Clevermath provides weekly problems. (Currently, Clevermath is not available.) | ||

+ | |||

+ | Here are some old contest archives that may be useful for practicing with: | ||

+ | |||

+ | American Mathematics Competitions: | ||

+ | * [[AMC 8 Problems and Solutions]] - [http://www.math.ksu.edu/main/handbook/ProblemSets AMC 8] is a national contest for grades 8 and younger. | ||

+ | |||

+ | * [[AMC 10 Problems and Solutions]] - [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43 AMC 10] is a national contest for grades 10 and younger. | ||

+ | |||

+ | * [[AMC 12 Problems and Solutions]] - [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44 AMC 12] is a national contest for grades 12 and younger. | ||

+ | |||

+ | American Invitational Mathematical Examination: | ||

+ | * [[AIME Problems and Solutions]] - [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45 AIME] is a contest administered to those who qualify with a high score on the AMC 10/12. | ||

+ | |||

+ | United States of America Mathematical Olympiad: | ||

+ | * [[USAMO Problems and Solutions]] - [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO] is a proof-based contest which must be qualified for through a combination of AMC & AIME scores. | ||

+ | |||

+ | Harvard-MIT Mathematics Tournament: | ||

+ | * [https://www.hmmt.co/archive/problems/ HMMT] is a nice contest on a hard AIME level. | ||

+ | |||

+ | * [http://www.artofproblemsolving.com/Forum/resources.php More Contests]. | ||

+ | |||

+ | * [http://www.artofproblemsolving.com/wiki/index.php?title=AoPS_Past_Contests User-Created Contests]. | ||

+ | |||

+ | |||

+ | |||

+ | |||

+ | |||

+ | There are certain strategies in preparing for the AMC 10/12- especially qualification for the AIME. | ||

+ | |||

+ | |||

+ | |||

+ | The AIME cutoff on the AMC 12 have ranged throughout (84-96) in the recent years. The top 5% of scorers qualify. In order to get a score in the range, a simple way is to answer 13 questions right and leave the rest blank which earns a score of 96. In the past, cutoffs have been around 100.5 but it’s very rare as contests are getting slightly harder year by year. This means, since the first 10 questions should be solved by nearly anyone, they are 60 “free” points. Throughout questions 10-20, answering 3-5 shall be enough. | ||

+ | |||

+ | |||

+ | |||

+ | |||

+ | 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 (96-120) 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 they use AMC 10 scores. | ||

+ | |||

+ | |||

+ | |||

+ | |||

+ | |||

+ | |||

+ | Top 10 most Difficult math Competitions(National) in The USA: | ||

+ | |||

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

+ | |||

+ | |||

+ | 9. AMC 10 - Intermediate Algebra, Geometry, Advanced 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 | ||

+ | |||

+ | |||

+ | 2. IMO - Very Advanced Algebra, Very Advanced Geometry, Very Advanced Number Theory, Advanced Combinatorics, Advanced Pre-Calculus | ||

+ | |||

+ | |||

+ | 1. PUTNAM - Advanced Algebra, Geometry, Number Theory, Advanced Combinatorics, *Extremely 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 [https://artofproblemsolving.com/community/c3_middle_school_math Middle School] forum is for MathCounts and AMC 8/10-level problems. | ||

+ | |||

+ | * The [https://artofproblemsolving.com/community/c4_high_school_math High School] forum is a good place to find AMC10/12-level and AIME-level problems. | ||

+ | |||

+ | * The [https://artofproblemsolving.com/community/c6_high_school_olympiads Olympiad] forum is a forum for problems at the olympiad level. | ||

+ | |||

+ | * The [http://www.artofproblemsolving.com/community/c68_latex_and_asymptote 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: <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. | ||

+ | * ExploreMath's 53-page long handout has all the formulas for the AMC 10 and AMC 12. They went through thousands of past problems to make the list, and it's a great way to review and learn new topics for the AMC 10 and AMC 12. https://docs.google.com/document/d/1aRCEadlhKestAvrWe3Dz01sZlBh8ZCiMbz87ELhBbrE/edit?usp=sharing | ||

+ | |||

+ | * Sohil Rathi's [https://drive.google.com/u/1/uc?id=1tq-fMRM1j-K3dtG9k4cKd0NqBavt30No&export=download] is a great way to learn and review the topics on the AMC 8 | ||

+ | |||

+ | * Coach Monk's [http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf MathCounts Playbook] is a good place to start for MathCounts-level material. | ||

+ | |||

+ | * Coach Monk's [http://mathweb.scranton.edu/monks/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] | ||

+ | |||

+ | == 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 [http://www.artofproblemsolving.com/Classes/AoPS_C_About.php here]. | ||

+ | |||

+ | [http://www.artofproblemsolving.com/Classes/AoPS_C_WOOT.php 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: | ||

+ | |||

+ | * [http://www.promys.org/ PROMYS] | ||

+ | * [http://www.mathcamp.org/ MathCamp] | ||

+ | * [http://www.mathpath.org/ MathPath] | ||

+ | * [http://www.awesomemath.org/index.shtml AwesomeMath] | ||

+ | * [http://ideamath.org IdeaMath] | ||

+ | |||

+ | == Past AoPS Topics == | ||

+ | ''' * [https://artofproblemsolving.com/community/c5h520900 Stop Looking for the "Right" Training]''' | ||

+ | |||

+ | ''' * [https://artofproblemsolving.com/community/c5h420845p2379622 MellowMellon]''' | ||

+ | === Mathcounts === | ||

+ | * '''[http://artofproblemsolving.com/community/q1h1217349p6066794 How to Prepare for Mathcounts]''' | ||

+ | |||

+ | === AMC === | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=174272 Best Ways to Prepare] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=139495 AMC prep.] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=210890 too late to prepare?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=209162 Preparing book questions] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=208696 amc 10 prep course?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=194187 Torn between books] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=193636 studying for AMC 10 for next year] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=191604 advice for AMC success] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=189902 Advice for a newbie?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=187851 Which AOPS book to buy?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=186820 I need help with AMC10!] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=181662 How would you prepare for the AMC10 these next weeks?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=169323 Best way to prepare for the AMC 10] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=170137 amc 10??] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=148196 Algebra 2 and Preparation for AIME/AMC] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=136730 Getting High Scores on AMC] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=194022 Advice for AMC12] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=132677 getting better at math...fast] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=132374 Improving your AMC score!!!] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=133900 I finally got hold of the AoPS V2] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=181707 How to prepare for the AMC 12?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=184350 AMC 12 Preparation] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=174299 help w/ AMC12 and AIME?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=173664 Practicing By Taking Older Contests] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=152170 how should I study for the AMC 12/AIME?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=152414 Preparing for AMC12/AIME] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=1361994 attacking the later problems in the AMC 10/AMC 10 advice] | ||

+ | |||

+ | === AIME === | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=193085 AIME Prep: for newbies] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=173323 About preparing] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=169466 AoPS books are useless?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=170179 Book for AIME] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=136183 Concepts to know for the AIME...] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=136810 Trig functions to know for AIME?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=136536 Tips, Formulas, Must-knows, hard previous questions 4 AIME?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=132916 Easiest Subject to Improve In] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=132563 AIME prep...] | ||

+ | |||

+ | === USAMO === | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=199616 What to study] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=196885 Please Help!!!] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=172265 Recommended USAMO Prep?] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=149393 Memorization] | ||

+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=142476 USAMO Help!] | ||

+ | |||

+ | == 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] |

## Latest revision as of 13:52, 6 July 2022

## Contents

## Introduction

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

## Books

The **Art of Problem Solving books** are an excellent resource to help prepare for math contests. They cover a broad range of topics, from algebra to geometry to number theory to combinatorics and much much more.

- Art of Problem Solving Volume 1 - Mathcounts, AMC 8, AMC 10
- Art of Problem Solving Volume 2 - AMC 12, AIME, USAMO, MOP

The **AoPS textbooks** 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 here. Excerpts are provided, as well as pretests and posttests to see if the books are on the right level for you. **Alcumus** is a good resource even if you do not own any of the AoPS books.

- 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

Here are a few more **books good for preparation for higher level contests** such as AMC 12, AIME, and USAMO (though some can be found online):

- Art and Craft of Problem Solving by Paul Zeitz

- Problem-Solving Strategies by Arthur Engel

- Geometry Revisited by H.S.M. Coxeter & Samuel L. Greitzer

- 102 Combinatorial Problems by Titu Andreescu & Zuming Feng

- 103 Trigonometry Problems by Titu Andreescu & Zuming Feng

- 104 Number Theory Problems by Titu Andreescu & Zuming Feng

- Euclidean Geometry in Math Olympiads by Evan Chen

### AMC 8:

Algebra: Introduction to Algebra

Geometry: Introduction to Geometry

Combinatorics: Introduction to Counting & Probability

General: Math competitions-Middle School

-Note:(Not much Number Theory is asked on the AMC 8).

### 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:

Algebra: Intermediate Algebra

Geometry: Introduction to Geometry

Combinatorics: Intermediate Counting & Probability, Introduction to Counting and Probability

Number Theory: Introduction to Number Theory

General: Volume 2

Extra: Precalculus

## Practice Problems

Old practice problems (with solutions) sorted by contest and year are available on the Wiki and the Resources section. Many practice problems are also available on the forums.

Clevermath provides weekly problems. (Currently, Clevermath is not available.)

Here are some old contest archives that may be useful for practicing with:

American Mathematics Competitions:

- 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.

American Invitational Mathematical Examination:

- AIME Problems and Solutions - AIME is a contest administered to those who qualify with a high score on the AMC 10/12.

United States of America Mathematical Olympiad:

- USAMO Problems and Solutions - USAMO is a proof-based contest which must be qualified for through a combination of AMC & AIME scores.

Harvard-MIT Mathematics Tournament:

- HMMT is a nice contest on a hard AIME level.

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

The AIME cutoff on the AMC 12 have ranged throughout (84-96) in the recent years. The top 5% of scorers qualify. In order to get a score in the range, a simple way is to answer 13 questions right and leave the rest blank which earns a score of 96. In the past, cutoffs have been around 100.5 but it’s very rare as contests are getting slightly harder year by year. This means, since the first 10 questions should be solved by nearly anyone, they are 60 “free” points. Throughout questions 10-20, answering 3-5 shall be enough.

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 (96-120) 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 they use AMC 10 scores.

Top 10 most Difficult math Competitions(National) in The USA:

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

9. AMC 10 - Intermediate Algebra, Geometry, Advanced 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

2. IMO - Very Advanced Algebra, Very Advanced Geometry, Very Advanced Number Theory, Advanced Combinatorics, Advanced Pre-Calculus

1. PUTNAM - Advanced Algebra, Geometry, Number Theory, Advanced Combinatorics, *Extremely 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 Middle School forum is for MathCounts and AMC 8/10-level problems.

- The High School forum is a good place to find AMC10/12-level and AIME-level problems.

- The Olympiad forum is a forum for problems at the olympiad level.

- The LaTeX and Asymptote forum is a place to get help with , which is what you use to type things like on the forums. It's also for Asymptote, which is what we use to make diagrams, like:

## Cheat Sheets

Many great reference guides are available for free on the internet.

- ExploreMath's 53-page long handout has all the formulas for the AMC 10 and AMC 12. They went through thousands of past problems to make the list, and it's a great way to review and learn new topics for the AMC 10 and AMC 12. https://docs.google.com/document/d/1aRCEadlhKestAvrWe3Dz01sZlBh8ZCiMbz87ELhBbrE/edit?usp=sharing

- Sohil Rathi's [1] is a great way to learn and review the topics on the AMC 8

- Coach Monk's MathCounts Playbook is a good place to start for MathCounts-level material.

- Coach Monk's 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 All of Math in 3 Pages.
- The Noah Sheets

## 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 here.

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:

## Past AoPS Topics

** * Stop Looking for the "Right" Training**

### Mathcounts

### AMC

- Best Ways to Prepare
- AMC prep.
- too late to prepare?
- Preparing book questions
- amc 10 prep course?
- Torn between books
- studying for AMC 10 for next year
- advice for AMC success
- Advice for a newbie?
- Which AOPS book to buy?
- I need help with AMC10!
- How would you prepare for the AMC10 these next weeks?
- Best way to prepare for the AMC 10
- amc 10??
- Algebra 2 and Preparation for AIME/AMC
- Getting High Scores on AMC
- Advice for AMC12
- getting better at math...fast
- Improving your AMC score!!!
- I finally got hold of the AoPS V2
- How to prepare for the AMC 12?
- AMC 12 Preparation
- help w/ AMC12 and AIME?
- Practicing By Taking Older Contests
- how should I study for the AMC 12/AIME?
- Preparing for AMC12/AIME
- attacking the later problems in the AMC 10/AMC 10 advice

### AIME

- AIME Prep: for newbies
- About preparing
- AoPS books are useless?
- Book for AIME
- Concepts to know for the AIME...
- Trig functions to know for AIME?
- Tips, Formulas, Must-knows, hard previous questions 4 AIME?
- Easiest Subject to Improve In
- AIME prep...