The book presents a selected set of mathematical and geometrical topics useful for preparing for high school math competitions, Olympiads, SAT tests, and some college courses. Repetitive exercises and hands-on practice help to students to elaborate an efficient know-how capability in problem solving. The book consists of nine chapters, each divided to multiple sections.
Chapter 1 “Introduction” reviews basic fundamentals, including patterns of additive and multiplicative sequences and summations, integer numbers and prime factors, main geometric figures and their features, Venn diagrams for sets, factorial and Pascal’s triangles, permutations and combinations, main characteristics of graphs, piecewise sequences, and about 50 exercises on them. The next chapters study these topics more rigorously.
Chapter 2 “Sequences and summations” describes linear and quadratic, geometric and factorial, alternating and piecewise series of numbers, and shows how to find the totals of progressions. The recursive sequences and the first-order difference equations with solution for the initial value problem are presented in multiple examples, and more than 30 exercises are suggested. Chapter 3 “Proofs” considers algebraic proofs and proofs by induction for deriving formulas of sums and products for different consecutive number series, with about 20 exercises for self-training on these topics.
Chapter 4 “Integers’ Characteristics” deals with consecutive even and odd integers, their prime factorization and divisors, perfect squares and ending digits, and proposes about 40 additional exercises. Chapter 5 “Pascal’s Triangle Identities” defines combinations for coefficients of the binomial expansions, discusses various traits of the triangles’ horizontally- and diagonally-oriented characteristics, such as Pascal’s, power, square, and other identities, with more than 40 exercises given on them.
Chapter 6 “Geometry” describes various types of triangles and their properties, area and perimeter of different figures, proportions and their applications, and contains more than a dozen other exercises. Chapter 7 “Graph Theory” examines the features of graphs, including degrees of vertices and cycles, regular and semi-regular graphs, and Hamiltonian cycles, with additional two dozen exercises.
Chapter 8 “Answers to Chapter Exercises” presents the answers to the odd numbered exercises by all chapters. Chapter 9 “Appendices” presents the figures and formulas on the above considered topics. Bibliography of the author’s articles and Index finalize the book. There are multiple illustrations on the problems under consideration.
The book is interesting, innovative, and can serve as a good addition to the regular high school algebra and geometry manuals, but it is not a collection of advanced problems for students self-preparing to actual complex national and international Olympiads. More sources can be found in Lipovetsky (Citation2018, Citation2020, Citation2022).
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References
- Lipovetsky, S. (2018), “Math Tools: 500+ Applications in Science and Arts, by G. Glaeser,” Technometrics, 60, 263.
- Lipovetsky, S. (2020), “2∧5 Problems for STEM Education, by V. Ochkov,” Technometrics, 62, 557–558.
- Lipovetsky, S. (2022), “Math and Art: An Introduction to Visual Mathematics, by S. Kalajdzievski,” Technometrics, 64, 425–426. DOI: 10.1080/00401706.2022.2091873.