# Detailed Solutions of the 2016 AMC 8 Problems

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!