Trigonometrical identities and inequalities Finbarr Holland January 14, 2010 1 A review of the trigonometrical functions These are sin,cos,&tan. These are discussed in the Maynooth Olympiad Manual, which we refer to as MOM! We assume that you know the following addition formulae: for all x,y sin(x±y) = sinxcosy ±cosxsiny, cos(x±y) = cosxcosy To get such a pragmatic mastery of inequalities, you surely need a comprehensive knowledge of basic inequalities at first. The goal of the first part of the book (chapters 1-8) is to lay down the foundations you will need in the second part (chapter g), where solving problems will give you some practice. It is important to try and solve 1 Basic (Elementary) Inequalities and Their Application 3 As a consequence of the previous inequality we get following problem. Exercise 1.5 Let a,b,c?R. Prove the inequalities 3 Olympiad Inequalities xxxxxxgggSSS999KKK——— III Evan Chen s? April 30, 2014 The goal of this document is to provide a easier introduction to olympiad inequalities than the standard exposition Olympiad Inequalities, by Thomas Mildorf. I was motivated to write it by feeling guilty for getting free 7's on The inequalities are extremely beautiful and sharp, and the book covers various topics from 3 and 4 variables inequalities, symmetric and non-symmetric inequalities to geometric inequalities. Many of the exercises are presented with detailed solutions covering a variety of must-know old and new techniques in tackling Olympiad problems. This implies the AM-GM inequality. Some generalizations of this inequality include the Power Mean inequality and the Jensen's inequality (see below). Here are several problems from the Putnam exam, which can be solved using the AM-GM inequality. (Note that some of the problems can be solved by di?erent methods too). Problem 1. 2 Olympiad Training Materials, imomath.com Theorem 2. If a,b ? R then: a2 +b2 ?2ab. (1) The equality holds if and only if a =b. Proof. After subtracting 2ab from both sides the inequality becomes equivalent to (a?b)2 ?0, which is true according to theorem 1. 2 Collection of Inequalities - a collection of inequalities published in various math journals (a majority in Crux Mathematicorum) and books, compiled by Eckard Specht. At this moment it contains 1454±? problems. (Updated: October 22, 2006) Inequalities and 'Maximum-Minimum' Problems Henry Liu, 26 February 2007 There are many olympiad level problems in mathematics which belong to areas that are not covered well at all at schools. Three major examples are geometry, number theory, and functional equations. Such areas must be learned outside class Edit. I posted a new "proof" (now deleted), before I realized I was adddressing the wrong question. I think the original proof is ok, I messed up something in trying to simplify the approach. Hint. Since this is an olympiad problem, it is likely that there is a proof without using calculus. This feature is not available right now. Please try again later. Functional Inequalities Kin Y. Li In the volume 8, number 1 issue of Math Excalibur, we provided a number of examples of functional equation problems. In the volume 10, number 5 issue of Math Excalibur, problem 243 in the problem corner section was the first functional inequality problem we posed. That one was from the 1998 Bulgarian Math Olympiad. Functional Inequalities Kin Y. Li In the volume 8, number 1 issue of Math