         As the most influential mathematics competition in the United States, the AMC competition is a comprehensive exploration of the upgrade path of mathematics competitions in the United States. We hope you understand the climbing route of American mathematics competitions, especially the path initiated by AMC.

        Firstly, let's understand why we should participate in the AMC competition.

AMC serves as the gateway to high-level mathematics competitions.

The MAA (Mathematical Association of America) is the official organization for the American Mathematics Competitions. Through annual competitions such as AMC8, AMC10/12, and AIME, there is a scientific pathway. The most outstanding students may even be selected for the United States national team. Those with excellent AMC10/12 scores can enter AIME, and by succeeding in both AMC10/12 and AIME, they can proceed to USAMO. Advancing to IMO requires overcoming additional challenges.

        As the most influential youth mathematics challenge, AMC attracts many students every year. They are demonstrating significant international authority. Students achieving outstanding scores in AMC10/12 are invited to participate in the AIME (American Invitation Mathematics Examination), competing alongside exceptional students worldwide. Regardless of the destination, the victory lies in overcoming oneself.

        Passionate about mathematics, I participated in the 2023 AMC10A and AMC10B tests, achieving scores of 115.5 and 144, respectively. Good scores allowed me to enter the AIME invitation exam smoothly.

This book is based on my experience preparing for AMC10 day by day. I will outline in this book the 16 essential knowledge points that must be mastered for AMC10, focusing on key breakthroughs; the whole eBook is divided into two volumes,this ebook is volume 1, the volume 2 will be published next year.

Number theory basics: prime numbers, prime factorization, divisor number theorem, greatest common divisor, least common multiple, Euclidean algorithm.

Congruence and divisibility: Congruence, divisibility, indeterminate equations.

Advanced theorems and number systems: Euler's theorem, Fermat's little theorem, Wilson's theorem, Chinese remainder theorem, digit and number systems, infinite repeating decimals.

Geometry basics: triangles, area, perimeter.

Advanced geometry: similar triangles, point-line relationships in triangles.

Circles: basic knowledge of circles, advanced theorems of circles.

Solid geometry: lines, planes, angles, solid geometry in coordinate systems, polyhedral.

Analytical geometry: lines, circles.

Geometric transformations: translation, reflection, symmetry, rotation.

Addition and multiplication principles: multiplication principle, addition principle.

Permutations and combinations: permutations, circular permutations, combinations, groupings, inclusion-exclusion principle, etc.

Probability: classical Probability, geometric Probability, Markov chains, recursion.

Sequences: arithmetic sequences, geometric sequences, and other sequences.

Polynomials: fundamental theorem of algebra, general form of Vieta's theorem, rational root test, synthetic division, long division, Descartes' rule of signs, remainder theorem, factor theorem.

Functions and their graphs: standard functions and graphs, Gaussian and their graphs, ceiling functions and graphs.

Inequalities: linear inequalities, high-degree polynomial inequalities, quadratic inequalities.

Jeffrey(Fengyi)Jiang

October 22, 2023