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In Unit 1 we learnt several classical inequalities. While the inequalities themselves can be easily memorised, it is often not easy to apply them to prove other inequalities in practical situations. As a matter of fact, very few problems concerning inequalities can be proved by a direct application of one of the inequalities we have learned. Also, sometimes it is not too apparent which classical inequality to apply in dealing with a problem. Ingenious tricks are often needed to obtain a nice solution. Repeated trials and deep thoughts are also demanded. In this unit we shall look at some techniques that are frequently employed in proving inequalities. However, it should always be remembered that there is no standard way of proof and there is no general rule in choosing which technique to use. Practice and experience, however, often help. We often have to substitute expressions into the classical inequalities we learned in Unit 1 to prove new inequalities. Different substitutions often yield different results. It is therefore important that we apply known inequalities suitably in order that the desired result can be obtained. Of course we may not be so lucky to have applied the inequality in the correct way the first time.

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The problems can be presented and discussed at mathematics workshops and sessions of preparation for contests and olympiads as well as in the classroom. These problems offer a range of difficulty levels. This comprehensive presentation includes the detailed steps to be executed.

The reason for this deliberate abbreviation is to allow the solver to discover the exact identification of the objects and working tasks. As the solver moves progressively from a partial solution to a more complete one. The material. The problem books which follow will be presented in the same structural format.

The problem-solving assumes theoretical and analytical skills. These are arranged consecutively. The present work is the first of a series that will also operate in other domains and subdomains of mathematics. The problems were selected with care so that each problem meets all the intended methodological criteria. The concept of such a problem book successfully supports the development of these skills of the solver and meanwhile offers mathematics instructors models for teaching problem-solving as an integral part of the mathematics learning process.

The topics of the problems belong to algebra II and algebra of the first two years of college. The problem book actually contains three categories of problems. Most of the solutions are not immediately obvious and do not represent direct applications of an isolated theoretical result. This series is part of a far-reaching publishing project whose goal is the involvement of mathematics instructors and graduates in editing such interactive problem books.

Rusu n AL1. Andrica Missing part ……………………… AL1. Prove that there exists an irrational number a such that numbers axi. Marchidan AL1. Tomescu ……………………. Niculescu ……………………. Missing part ……………………… Solve the new system in a and b formed by the last equation and the one obtained after adding together the two given equations. Find a — b. Consider the four consecutive terms that must exist. Complete squares in the resulting equation. Rearrange the terms and factor out in the second factor of the product.

Subtract the second given equation from the first. Do a new subtraction and arrange the terms to put in evidence a — b. Show that the left-hand side is smaller than Show that the ratio of these two numbers is irrational. Reductio ad absurdum: Find that each power from the left-hand side is smaller than a multiple of and add them together.

Show that there exists a row containing only irrational numbers: Prove by reductio ad absurdum: Assume that each row holds at least one rational number.

Show that a. Show that there exists at least one column holding two rational numbers. Show that the last digit of N cannot be 7.

Consider all numbers m. Assume N is a perfect square. Show that the last digit of N is 7.

Show that k cannot be Apply the previously proven property to these squares. Express f x. Show that there exists g x.

Write the given relation by replacing the coefficients according to this assumption. Consider five integers and their squares.

Group the factors conveniently. Free hosting has reached the end of its useful life Express the difference of these two squares and show that the sum or the difference of their square roots is divisible by 7.

Show that at least two of these squares have the same remainder upon division by 7.