There are four pillars of mathematical contests and olympiads: algebra, counting, number theory, and geometry. These areas are addressed in the four books of the Rigorous Elementary Mathematics, Second Edition series, authored by Samer Seraj. While there are many books that are geared towards helping students to learn how to solve problems on contests, those books tend to assume prerequisite knowledge without proofs, and they instead focus on applying theorems to problems. This four-volume series fills the gap in the literature by proving many of the relevant theorems in a logically sequenced framework from the ground up. Throughout this series, three ideas are reinforced: writing the same object in multiple ways, breaking up equality using antisymmetry, and using equivalence relations. The books contain hundreds of substantive problems for the reader’s practice, with complete solutions provided. This second edition includes additional theorems and problems, refreshed diagrams, and corrections. It is suggested that the books be read by those who have prior experience with mathematical proofs and problem-solving. The intended audience consists of outstanding students who want a second look at the material, teachers of mathematics who want to solidify their own fundamentals, and other mathematical enthusiasts.