Category Archives: Math Competitions

35 Sets of Previous Official AMC 10 Tests with Answer Keys (PDF files)

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The best way to prepare for the AMC 10 is to do lots of practice problems either on your own or with a small group and then check your solutions with an answer key. For this reason, we provided all 35 sets of previous official AMC 10 contests (2000-2017) with answer keys and also developed 20 sets of AMC 10 mock test with detailed solutions to help you prepare for this premier contest.

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35 Sets of Past Real AMC 12 Tests with Answer Keys (PDF files)

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All problems from past AMC 12 (2000–2017) form our “big data” system. Based on  artificial intelligence (AI), machine learning, and deep learning, we devised a data mining and predictive analytics tool for math problem similarity searching. Using this powerful tool, we examined the types, styles, frequencies, characteristics, and traits of questions in all past AMC 12 exams, and then completely “decoded” the AMC 12.

For all questions on the recent AMC 12 contests, we can find their “ancestors” and “roots” from the old AMC 12 problems. Therefore, the best way to prepare for the AMC 12 is to practice by solving old AMC 12 problems!

We provided 35 sets of past real AMC 12 tests (2000-2017) with answer keys and also developed 20 sets of AMC 12 mock test with detailed solutions to help students prepare for the AMC 12 contest.

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14 Sets of Previous Real AJHSME (AMC 8) Tests with Answer Keys

The American Junior High School Mathematics Examination (AJHSME) for students in grades 8 and below, begun in 1985, was the predecessor of the AMC 8, administered by the American Mathematics Competitions. Since 1999, the name has been changed into the American Mathematics Contest 8 (AMC 8).

https___cdn_evbuc_com_images_23502174_94898100017_1_original.gifAll problems from past AJHSME exams (1985-1998) and AMC 8 (1999–2016) form our “big data” system. We used data/text mining and predictive analytics to examine the types, styles, frequencies, and characteristics of questions in all these materials, and then completely “decoded” the AMC 8. For all questions on the recent AMC 8 contests, we can find their “ancestors” and “roots” from the old AMC problems. Therefore, the best way to prepare for the AMC 8 is to practice by solving old AMC problems!

We provided 14 sets of previous real AJHSME tests (1985-1999) with answer keys and also developed 20 sets of AMC 8 mock test with detailed solutions to help students prepare for the AMC 8 contest.

All past official AMC 8 problems (1999-2016) can be found at:

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18 Sets of Past Official AMC 8 Tests with Answer Keys

The best way to prepare for the AMC 8 is to do lots of practice problems either on your own or with a small group and then check your solutions with an answer key. For this reason, we provided 18 sets of past official AMC 8 contests (1999-2016) with answer keys and also developed 20 sets of AMC 8 mock test with detailed solutions to help you prepare for this contest.

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Detailed Solutions of the 2016 AMC 8 Problems

copyright-small Henry Wan, Ph.D.

 This Solutions Manual contains detailed solutions carefully developed by Dr. Henry Wan, for the all 25 problems on the 2016 AMC 8 exam, and shows that all the problems can be solved using material normally associated with the mathematics curriculum for students in grades 8 and below. At least one solution was provided for each problem and all problems were solved without the use of a calculator. When more than one solution was provided, this was done to demonstrate an important contrast in approaches, e.g.,

  • algebraicvs geometric,
  • elementaryvs advanced,
  • computational vs conceptual,
  • explicit vs implicit,
  • analytic vs discrete,
  • “forward-solving” vs “back-solving.”

You can find some problems on the AMC 8 and their solutions on the internet. However, it is imperative to know that these solutions are not reliable, as they are not written by professionals. Most of the answers found on the internet are written by students, and lack expert (peer) review. They contain many mistakes and may be incomplete. Thus, they are not a good source for learning the strategies and shortcuts for solving AMC problems.

AMC 8 is a 25-question, 40-minute, multiple choice examination in middle school mathematics designed to promote the development and enhancement of problem solving skills. The problems generally increase in difficulty as the exam progresses. Usually the last 5 problems are the hardest ones.

Among the final 5 problems on the 2016 AMC 8 contest, there are 2 discrete math (which contains number theory, counting, and probability) problems: Problems 21 and 24. There are also 3 geometry problems: Problems 22, 23, and 25. When solving these difficult geometry problems, in addition to using the regular geometry approaches (plane geometry and coordinate geometry), we can also use a ruler, protractor, and compass, as well as graph papers to easily and quickly find answers to these problems. All of my students have systematically learned all unique strategies, skills, and techniques, which are also described in my article titled “Using the Ruler, Protractor, and Compass to Solve Geometry Problems on the AMC 8 and MathCounts.” My students have used these techniques to get the 3 hardest geometry problems correct and benefit greatly by receiving top scores on the AMC 8.

This manual is designed to be the silent mentor for students working on their own. From the detailed solutions, you can gain better insight into mathematics and problem solving techniques. Most importantly, you will get the most out of this guide by seeking full understanding of the whole process of choosing a good path toward a beautiful solution.

Focus your energies there and just beyond. In some cases, this may require additional research outside of this guide. In those cases, I particularly hope the solutions display enough to help you get started.

 Enjoy!

Mathematics is fun and beautiful. It is an art.

Learning is really fun!

20 Sets of AMC 8 Mock Tests with Detailed Solutions

MAA AMC 8

A Data-Driven Approach to AMC Success

We have gathered every problem from past AMC 8 exams (1999–present) and its predecessor AJHSME (1985–1998) to build a powerful “big data” system that helps us design the most accurate and effective AMC preparation materials available. Using data mining and predictive analytics, we analyzed the types, styles, and frequencies of questions across all these exams—allowing us to completely decode the AMC 8 and understand what truly leads to success.


Smart Design, Real Results

Based on the insights uncovered from this “decoded” AMC 8 data, we developed 20 original AMC 8 Mock Tests, each containing 25 brand-new, thoughtfully crafted problems. Every test mirrors the difficulty, balance, and style of the official AMC 8 exam, giving students a realistic and confidence-building experience.

These mock tests are one of the most effective tools to measure readiness, strengthen problem-solving skills, and gain a competitive edge before taking the real contest. Each test set also includes detailed step-by-step solutions that help students understand the reasoning behind every answer and master advanced strategies for future problems.


Rigorous, Expert-Reviewed Mock Exams for Top Students

Each year, we rebuild and update our AMC 8/10/12 and AIME mock exams from the ground up to reflect the latest contest questions, styles, and trends. Every new version is peer-reviewed by two independent experts, ensuring the highest levels of accuracy, clarity, and alignment with official MAA/AMC standards.

Our mission is simple — to help ambitious students develop the confidence, discipline, and mastery required to excel in national math competitions.

Our academic team goes even further — carefully studying the work of 16 official AMC 8 problem writers, including their past contest questions, published problems, and research papers. By analyzing how these experts think and design problems, we create authentic, contest-style mock exams that train students with precision and help them develop the mindset of top AMC performers.


Proven Accuracy You Can Trust

Over the years, our mock tests have proven to be remarkably accurate in predicting real AMC 8 questions. In fact, several problems from our earlier mock exams have later appeared on the official AMC 8 contests:

These examples highlight the precision and reliability of our data-driven approach—and show why so many students trust our materials to help them excel on the AMC 8.


Learn, Practice, and Excel

Taking our AMC 8 Mock Tests is one of the best ways to maximize learning and boost performance. Each test offers a rich opportunity to explore key ideas, sharpen techniques, and build the confidence needed to perform at your best on competition day.

Click HERE to see a typical AMC 8 Mock Test Sample.

Click HERE to view the detailed solution of the AMC 8 Mock Test Sample.

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We offer free USPS Priority Mail shipping for all domestic orders.
(Standard international shipping rates apply.) Orders are shipped only to the verified address that appears on the PayPal transaction and typically arrive within 2–7 business days after we receive and process your payment.


Refund and Return Policy

Please review the following choices carefully, refunds and returns can only be processed when eligible items are returned using the appropriate method. If your return is shipped to us after the timeframes indicated below, we regret that we cannot issue a refund to you.

We ask that you review your purchase carefully before submitting your order.

  • Refunds are available for items returned within 30 days of purchase, provided all materials are unused and in their original condition.
  • Refunds will be issued to your original form of payment within 5–10 business days after processing.
  • Used mock test papers, even partially completed, are not eligible for refund.

Due to copyright restrictions, our mock tests are normally provided only as hard copies shipped by mail via USPS.
In exceptional cases, we may authorize delivery of a PDF version of five or more mock tests to the buyer’s verified PayPal email address. Please note that all payments for electronic versions are NON-REFUNDABLE. We can provide either the digital version or the paper version of the product, but not both. Unfortunately, we are unable to provide both formats at the same time.


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Phone: (240) 406-3402
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Click HERE find out more about Math Competitions!

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Some Hard Problems on the 2017 AMC 10A/12A are Totally the Same as Previous Problems on the AMC 10/12

copyright-small Henry Wan, Ph.D.

We have developed a comprehensive, integrated, non-redundant, well-annotated database “CMP” consisting of various competitive math problems, including all previous problems on the AMC 8/10/12, AIME, MATHCOUNTSMath Kangaroo ContestMath Olympiads for Elementary and Middle Schools (MOEMS), ARML, HMMT, Math League, PUMaC, Stanford Math Tournament (SMT). The CPM is an invaluable “big data” system we use for our research and development, and is a golden resource for our students, who are the ultimate beneficiaries.

We have also devised a data mining and predictive analytics tool for math problem similarity searching. Using this powerful tool, we can align query math problems against those present in the target database “CPM,” and then detect those similar problems in the CMP database.

For those hard problems on the 2017 AMC 10A/12A, according to the database searching, we have found:

  • The 2017 AMC 10A Problem 18 (also known as Problem 15 on the 2017 AMC 12A) is the same as the following 3 problems:
    • 2015 AMC 12B Problem 9
    • 2016 AMC 12B Problem 19
    • 1981 AHSME Problem 26
  • The first part of the 2017 AMC 10A Problem 21 (also known as Problem 19 on the 2017 AMC 12A) is exactly the same as  the 2007 AMC 10B Problem 21

Click HERE to see my detailed solutions of the hard problems on the 2017 AMC 10A/12A.

In my AMC 10/12 Prep Class, I ever used Problem #19 on the 2016 AMC 12B, as a typical example, to efficiently solve discrete probability problems on the AMC10/12, including the complementary probabilities. Thus, when my students attended the 2017 AMC 10A/12A, they already knew how to solve the AMC 10A Problem 18/AMC 12A Problem 15 and its answer. So they took one second to bubble the correct answer and then got 6 points very easily!

Conclusions
All problems from past AMC 10/12 exams (2000-2017) and AHSME (1950–1999) form our “big data” system. The AHSME (American High School Mathematics Examination) was the former name of the AMC, before 2000. We have used data mining and predictive analytics to extensively examine the types and the frequencies of questions in all these materials, and then completely “decoded” the AMC 10/12. We always show all the “secret code” cracked from the above big data to students, and teach them to totally grasp and “control” the AMC. For all questions on the recent AMC contests, we can find their “ancestors” and “roots” from the old AMC problems. Therefore, the best way to prepare for the contest is to practice by solving old AMC problems.

Click HERE find out more about Math Competitions!

Click HERE to find out more about SAT Prep!

Copyright copyright-small Dr. Henry Wan. All rights reserved

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Using the Ruler, Protractor, and Compass to Solve the Hardest Geometry Problems on the 2016 AMC 8

copyright-small Henry Wan, Ph.D.

The AMC 8 is a 25-question, 40-minute, multiple choice examination in middle school mathematics designed to promote the development and enhancement of problem solving skills. The problems generally increase in difficulty as the exam progresses. Usually the last 5 problems are the hardest ones.

Among the final 5 problems on the 2016 AMC 8 contest, there are 3 geometry problems: Problems 22, 23, and 25. When solving these difficult geometry problems, in addition to using the regular geometry approaches (plane geometry and coordinate geometry), we can also use a ruler, protractor, and compass, as well as graph papers to easily and quickly find answers to these problems. All of my students have systematically learned all unique strategies, skills, and techniques, which are also described in my article titled “Using the Ruler, Protractor, and Compass to Solve Geometry Problems on the AMC 8 and MathCounts.” My students have used these techniques to get the 3 hardest geometry problems correct and benefit greatly by receiving top scores on the AMC 8.

Click HERE to see my detailed solutions of the hardest geometry problems on the 2016 AMC 8.

Click HERE find out more about Math Competitions!

Click HERE to find out more about SAT Prep!

Copyright copyright-small Dr. Henry Wan. All rights reserved

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Some Hard Problems on the 2016 AMC 8 are Exactly the Same as Previous Problems on the AMC 10/12 and the Other Competitions

copyright-small Henry Wan, Ph.D.

We have developed a comprehensive, integrated, non-redundant, well-annotated database “CMP” consisting of competitive math problems, including all previous problems on the AMC 8/10/12, AIME, MATHCOUNTS, Math Kangaroo ContestMath Olympiads for Elementary and Middle Schools (MOEMS), ARML, HMMT, Math League, PUMaC, Stanford Math Tournament (SMT). The CPM is an invaluable “big data” system we use for our research and development, and is a golden resource for our students, who are the ultimate beneficiaries.

We have also devised a data mining and predictive analytics tool for math problem similarity searching. Using this powerful tool, we can align query math problems against those present in the target database “CPM,” and then find those similar problems in the CMP database.

For those hard problems on the 2016 AMC 8, based on the database searching, we have found:

  • 2016 AMC 8 Problem 21 is completely the same as 2001 AMC 10 Problem 23/2001 AMC 12 Problem 11 
  • 2016 AMC 8 Problem 24 is totally the same as the following problems:
    • 2004 International Schools Mathematics Teachers Foundation (ISMTF) Junior Mathematics Competition – Team Event Problem 17
      • Click HERE to find this problem.
    • 2005 BRITISH COLUMBIA COLLEGES Senior High School Mathematics Contest, Final Round, Part B (May 6, 2005) Problem 1.
      • Click HERE to find this problem.
    • 2008 Bermuda S2 US Grade 10 Mathematics Olympiad (May 4, 2008), Part B Problem 1
      • Click HERE to find this problem.

In my AMC 8/MathCounts Prep Class, I ever used Problem #23 on the 2001 AMC 10, as a typical example, to elegantly solve discrete probability problems on the AMC10/12, including the complementary probabilities. There were 23 middle schoolers to attend my AMC 10/12 Prep Class. When they attended the 2016 AMC 8, they already knew how to solve this problem and its answer. So they took one second to bubble the correct answer (B) and then got 1 point easily!

1136217959-c032701ebba946fc4a01ff9a1aa9f3cdThis year’s AMC 8 has a similar difficulty level as last year’s AMC 8; however, it was much more difficult than the AMC 8 of the years before 2015. Because the AMC 8 problems are getting harder, we must practice not only previous AMC 8 problems but also easy or medium difficulty level problems from previous AMC 10 to do well on the AMC 8.

Click HERE to see my detailed solutions of the hard problems on the 2016 AMC 8.

Click HERE find out more about Math Competitions!

Click HERE to find out more about SAT Prep!

Copyright copyright-small Dr. Henry Wan. All rights reserved

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2017 Fall – Competitive Math Courses

Fall is the great time to prepare for the American Mathematics Competitions!
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Competitive Math Program — Fall 2017 Schedule

Please see: 

Class Day/Time Grade
AMC 10/12 Prep 11 Classes (Time: 6:00 – 8:00 pm), Total: 22 Hours
9/10, 9/17, 9/24, 10/1, 10/8, 10/15, 10/22, 10/29, 11/5,  11/12,  11/19
View Course Outline
7-12
Online Intensive AMC 10/12 Prep 11 Classes (Eastern Time: 6:00 – 8:00 pm), Total: 22 Hours
9/10, 9/17, 9/24, 10/1, 10/8, 10/15, 10/22, 10/29, 11/5,  11/12,  11/19
View Course Outline
7-12
AMC 8/Mathcounts Prep Advanced Level 11 Classes (Time: 6:00 – 8:00 pm), Total: 22 Hours
9/9, 9/16, 9/23, 9/30, 10/7, 10/14, 10/21, 10/28, 11/4, 11/11, 11/18
View Course Outline
5-8
AMC 8/Mathcounts Prep Regular Level 11 Classes (Time: 3:00 – 5:00 pm), Total: 22 Hours
9/9, 9/16, 9/23, 9/30, 10/7, 10/14, 10/21, 10/28, 11/4, 11/11, 11/18
View Course Outline
5-8
Online Intensive AMC 8/Mathcounts Prep 11 Classes (Eastern Time: 6:00 – 8:00 pm), Total: 22 Hours
9/9, 9/16, 9/23, 9/30, 10/7, 10/14, 10/21, 10/28, 11/4, 11/11, 11/18
View Course Outline
5-8
Math Kangaroo Prep 11 Classes (Eastern Time: 3:30 – 5:30 pm), Total: 22 Hours
9/24, 10/1, 10/8, 10/15, 10/22, 10/29, 11/5, 11/12, 11/19, 12/3,  12/10  
View Course Outline
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Featured Math Instructors

All of our sessions are taught by highly qualified instructors who are excellent experts on preparing students for the exam. We distinguish ourselves by the high quality of our instructors. Finding top-quality instructors is no easy task. We’ve hand-picked some of the best, including graduates of Ivy League institutions. Meet some of them here:

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Our Students

In 2017, we have 61 students who are qualified to take the AIME either through the AMC 10A/12A or AMC 10B/12B. One of our students was among the 28 Perfect Scorers worldwide on the AMC 10A: Austen M., and two of our students were among the 65 Perfect Scorers worldwide on the AMC 10B: Ashwin A. and Brad Z. Remarkably, eight middle schoolers and one elementary schooler qualified for the AIME, which is geared toward high school students. Very impressively, Bryan Z., a 6th grader, gained a score of 132 out of 150 on the AMC 10B.
Read more at: 2017 AIME Qualifiers Announced — 61 Students Qualified for the AIME

In 2016, we have 36 students who are qualified to take AIME either through AMC 10A/12A or AMC 10B/12B. One of our students was among the 23 Perfect Scorers worldwide on the AMC 10A: Joel (Junyao) T. Particularly, seven middle schoolers and one elementary schooler qualified for the AIME, which is geared toward high school students. Pravalika P., a 6th grader, got a 115.5 out of 150 on the AMC10B, which is very impressive. Read more at: 2016 AIME Qualifiers Announced — 36 Students Qualified for AIME

From 2011 to 2015, in total, 37 students scored above 120 on the American Mathematics Contest 10 (AMC 10) and qualified for the American Invitational Mathematics Examination (AIME); 26 students scored above 100 on the American Mathematics Contest 12 (AMC 12) and qualified for the American Invitational Mathematics Examination (AIME); 3 students qualified for the USA Mathematical Olympiad (USAMO), the highest level of math competition for high school students in the USA. Read more at: Notable Achievements of Our Students

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Our Uniqueness

We have a long history of close collaboration with the MAA‘s American Mathematics Competitions (AMC), which are dedicated to strengthening the mathematical capabilities of our nation’s youth, and are the first of a series of competitions in high school mathematics that determine the United States team for the International Mathematical Olympiad (IMO).

We are only one in the Washington DC metropolitan area to offer elementary, middle, and high-school level competition math courses. Our students have received top scores and awards at prestigious national and math competitions.

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Click HERE find out more about Math Competitions!sat-logo-3

Click HERE to find out more about SAT Prep!

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