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## Über dieses Buch

The techniques presented here are useful for solving mathematical contest problems in algebra and analysis. Most of the examples and exercises that appear in the book originate from mathematical Olympiad competitions around the world.

In the first four chapters the authors cover material for competitions at high school level. The level advances with the chapters. The topics explored include polynomials, functional equations, sequences and an elementary treatment of complex numbers. The final chapters provide a comprehensive list of problems posed at national and international contests in recent years, and solutions to all exercises and problems presented in the book.

It helps students in preparing for national and international mathematical contests form high school level to more advanced competitions and will also be useful for their first year of mathematical studies at the university. It will be of interest to teachers in college and university level, and trainers of the mathematical Olympiads.

## Inhaltsverzeichnis

### Chapter 1. Preliminaries

Abstract
We will assume that the reader is familiar with the notion of the set of numbers that we usually use to count. This set is called the set of natural numbers and it is usually denoted by $$\mathbb{N}$$, that is,
$$\mathbb{N}=\left\{1,\;2,\;3,...\right\}$$
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 2. Progressions and Finite Sums

Abstract
In antiquity, patterns of points played an important role in the use of numbers and cosmological conceptions. The Pythagoreans used to represent some integers as a set of points arranged in polygonal or polyhedral forms. These integers, presented as spatial arrays, are known as figurate numbers. In this section we will study some of these numbers.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 3. Induction Principle

Abstract
In Chapter 2 we deduced several formulas for finite sums of numbers. Thus we learned that the sum of the first n natural numbers is given by the identity
$$1+2+\cdots+n=\frac{n(n+1)}{2}$$
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 4. Quadratic and Cubic Polynomials

Abstract
Consider an expression of the form
$$P(x)=a_3{x}^3\;+\;a_2{x}^2\;+\;a_1{x}\;+\;a_0$$
, where $$a_0,\;a_1,\;a_2\;\mathrm{and}\;a_3$$are constant numbers.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 5. Complex Numbers

Abstract
The roots of a quadratic equation are not always real numbers. For instance, the roots of the equation $$x^2+2x+10=0$$, which can be calculated using directly equation (4.6) of Section 4.2, are
$$\frac{-2+\sqrt{{-36}}}{2}\quad \mathrm{and}\quad \frac{-2-\sqrt{{-36}}}{2}$$
.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 6. Functions and Functional Equations

Abstract
The concept of function is one of the most important in mathematics. A function is a relation between elements of two sets X and Y , which we denote by f : X → Y , that satisfies.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 7. Sequences and Series

Abstract
A sequence of numbers {an}can be thought of as a function f defined on the set of positive integers and whose images are a set of numbers A.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 8. Polynomials

Abstract
There are some special names for polynomials whose degree is small. A polynomial is linear if it has degree 1. We have already studied the quadratic and cubic polynomials, which have degrees 2 and 3, respectively. If the polynomial has degree 4, it is called quartic.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 9. Problems

Abstract
Problem 9.1. Find the irrational numbers a such that a2 + 2a and a3 − 6a are rational numbers.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Chapter 10. Solutions to Exercises and Problems

Abstract
The first eight sections of this chapter contain all solutions of the exercises in the first eight chapters. In Section 9, you can find the solutions to the problems of Chapter 9. The difficulty of the problems in Chapter 9 is usually greater than the difficulty of the exercises that you find in the first eight chapters. However, solving the problems of this last chapter would be an excellent training in preparation for international mathematical competitions.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

### Backmatter

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