Elementary Mathematics Selected Topics And Problem Solving G Dorofeev M Potapov N Rozov.rar
Elementary Mathematics Selected Topics And Problem Solving G Dorofeev M Potapov N Rozov.rar - https://blltly.com/2ttFRI
Elementary Mathematics Selected Topics And Problem Solving: A Review of the Book by G Dorofeev, M Potapov and N Rozov
If you are looking for a comprehensive and challenging book on elementary mathematics, you might want to check out Elementary Mathematics Selected Topics And Problem Solving by G Dorofeev, M Potapov and N Rozov. This book covers a wide range of topics in arithmetic, algebra, geometry, trigonometry and calculus, with plenty of examples and exercises to test your skills and knowledge. The book also includes a section on problem solving techniques and strategies, as well as some interesting and non-trivial problems from various mathematical competitions and olympiads.
The book is intended for students who have a solid background in elementary mathematics and want to deepen their understanding and improve their problem solving abilities. The book is also suitable for teachers who want to enrich their curriculum and challenge their students with more advanced and creative problems. The book is written in a clear and concise style, with detailed explanations and proofs for the main results. The book also contains hints and solutions for some of the exercises and problems.
Elementary Mathematics Selected Topics And Problem Solving by G Dorofeev, M Potapov and N Rozov is a valuable resource for anyone who loves mathematics and wants to learn more about its beauty and elegance. The book is available in PDF format online, or you can order a hard copy from various online retailers.
In this article, we will give a brief overview of some of the topics and problems that are covered in the book Elementary Mathematics Selected Topics And Problem Solving by G Dorofeev, M Potapov and N Rozov. We will also provide some examples and exercises for you to try on your own.
Arithmetic
The book starts with a review of some basic concepts and properties of arithmetic, such as divisibility, prime numbers, greatest common divisor, least common multiple, modular arithmetic and congruences. The book then introduces some more advanced topics, such as Diophantine equations, linear congruences, Chinese remainder theorem, Fermat's little theorem, Euler's theorem and Wilson's theorem. The book also explores some applications of arithmetic in cryptography, number theory and combinatorics.
Example: Find all positive integers x and y such that x + y = xy.
Solution: We can rewrite the equation as xy - x - y = 0, or (x - 1)(y - 1) = 1. Since 1 has only two positive divisors, 1 and itself, we have two possible cases: x - 1 = 1 and y - 1 = 1, or x - 1 = -1 and y - 1 = -1. The first case gives x = 2 and y = 2, and the second case gives x = 0 and y = 0. However, we are looking for positive integers, so the only solution is x = 2 and y = 2.
Algebra
The book continues with a review of some basic concepts and properties of algebra, such as polynomials, rational expressions, equations and inequalities, systems of equations and inequalities, functions and graphs. The book then introduces some more advanced topics, such as quadratic equations and inequalities, complex numbers, roots of unity, Vieta's formulas, symmetric polynomials, binomial theorem, arithmetic and geometric progressions. The book also explores some applications of algebra in geometry, trigonometry and calculus.
Example: Find all real values of x such that (x + 1)^3 + (x - 1)^3 = 8x^3.
Solution: We can expand the left-hand side using the binomial theorem and simplify the equation as follows:
(x + 1)^3 + (x - 1)^3 = 8x^3
(x^3 + 3x^2 + 3x + 1) + (x^3 - 3x^2 + 3x - 1) = 8x^3
2x^3 + 6x = 8x^3
6x(1 - x^2) = 0
This equation has three possible solutions: x = 0, x = -1 or x = 1. However, we can check that only x = -1 satisfies the original equation. Therefore, the only real value of x that satisfies the equation is x = -1. 248dff8e21