# Sample problems from Olympiad Inequalities Book ?· Sample problems from Olympiad Inequalities Book…

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<ul><li><p>Sample problems fromOlympiad Inequalities Book</p><p>This book is intended as a useful resource for high school and college stu-</p><p>dents who are training for national or international mathematical competi-</p><p>tions. But anybody who is interested in elementary mathematical inequali-</p><p>ties may find this book useful. This problem solving book is divided into</p><p>six chapters, containing more than fifty topics of interest to mathematical</p><p>olympiad contestants and coaches, demonstrating ideas and strategies in solv-</p><p>ing inequalities. The reader will find in the book clever applications of well</p><p>known results as well as powerful original methods, each is explained and</p><p>illustrated by carefully selected problems.</p><p>Chapter 1 Inequalities between means</p><p>This chapter starts with the fundamental fact x2 0, upon which many interestinginequalities are derived. It also serves to emphasize that powerful results can be obtained</p><p>by little means.</p><p>Chapter 2 Cauchy-Schwars</p><p>The classical result of Cauchy-Schwarz is revisited with examples illustrating sophis-</p><p>ticated, at time surprising, ways to apply the inequality. Other classical inequalities such</p><p>as those of Chebyshev and Holder are also discussed and shown how they might cooperate</p><p>with Cauchy-Schwarz inequality.</p><p>Chapter 3 Convexity</p><p>This chapter utilises calculus in solving inequalities. Based on simpe properties of</p><p>linear and convex functions, systematic methods are derived to tackle some advanced</p><p>problems. Also discussed is tangent line method, which gives a geometric interpretation</p><p>of bounds.</p><p>1</p></li><li><p>Chapter 4 Homogenous inequalities</p><p>Homogeneous inequalities constitutes a large class of inequality problems. This chap-</p><p>ter discusses various approaches to solving this class of inequalities, including the tech-</p><p>niques of homogenization, normalisation, the application of Rolles theorem to reduce</p><p>the number of variables, the use of limits and partitions, quadratic estimations, and estab-</p><p>lishing new bounds through isolated fudging. Especially in focus are powerful techniques</p><p>to solve inequalities by the change of variables p = a + b + c, q = ab + bc + ca andr = abc and by transforming them to one of the following forms</p><p>(1) x(a b)(a c) + y(b c)(b a) + z(c a)(c b) 0,</p><p>(2) x(a b)2 + y(b c)2 + z(c a)2 0,</p><p>(3) M(a b)2 + N(c a)(c b) 0.</p><p>All three, four variable symmetric polynomial inequalities can be solved using ideas in</p><p>this chapter.</p><p>Chapter 5 The method of Mixing Variables</p><p>The method of mixing variables has been used in various forms for decades - an ex-</p><p>ample is G. Polyas delightful proof of the AM-GM inequalities. This chapter examines</p><p>this idea in depth with extension in different directions. The first three sections explain</p><p>why mixing variables work, give hints to find approriate variables to mix by taking equal-</p><p>ity cases into consideration. The most important results in this chapter are two theorems</p><p>which facilitates solutions for a large class of multi-variable inequalities.</p><p>Chapter 6 Further Topics and problems with solutions</p><p>The chapter starts with miscellaneous indenpendent topics touching upon various</p><p>aspects of solving inequalities. The discussion includes the interplay between trogono-</p><p>metric and algebraic substitution, absolute values, inequalities with special equality cases</p><p>and inequalities with ordered sequences.</p><p>Authors: Phm Vn Thun, L V</p><p>Hanoi University of Science, Vietnam</p><p>332 pages, LATEX typset, soft cover</p><p>Price: 12 (twelve) USD</p><p>It is available for sale in La Thanh Hotel where deputy leaders and contestants stay.</p></li><li><p>3</p></li><li><p>4</p><p>b</p><p>b</p><p>b</p><p>b</p><p>Phm Vn Thun, L V</p><p>Olympiad Inequalities</p><p>Introduction to the art of solving inequalities</p><p>bbb</p><p>f (x) = (x a)(x b)(x c)</p><p>y = f (x)</p><p>b</p><p>b</p><p>O</p><p>a b c x</p><p>y</p><p>f</p><p>(</p><p>1 t3</p><p>)</p><p>f</p><p>(</p><p>1 + t</p><p>3</p><p>)</p><p>VIETNAM NATIONAL UNIVERSITY PRESS</p></li><li><p>5</p><p>The 11 out of 600 problems</p><p>Problem 1. Prove that if x, y, z are real numbers, then</p><p>7(x4 + y4 + z4) + 10(x3y + y3z + z3x) 0.</p><p>Problem 2. Prove that a, b, c are positve real numbers, then</p><p>a</p><p>b2 + 14 bc + c2</p><p>+b</p><p>c2 + 14 ca + a2</p><p>+c</p><p>a2 + 14 ab + b2 2.</p><p>Problem 3. Let a, b, c, d be non-negative real numbers such that</p><p>a2 + b2 + c2 + d2 = 1.</p><p>Prove that</p><p>a + b + c + d a3 + b3 + c3 + d3 + ab + bc + cd + da + ac + bd.</p><p>Problem 4. Suppose that p, q, r, s are real numbers such that the following equation hasfour roots (not neccessarily distinct)</p><p>x4 px3 + qx2 rx + s = 0.</p><p>Prove that (p2 2q)5/2 + 8ps 4(p2 2q)r.</p><p>Problem 5. Let n be a positive integers, n 2. Non-negative real numbers a1, a2, ..., ansatisfy a1 + a2 + + an = s, s < 2, define</p><p>f (a1, a2, ..., an) = 1i< jn</p><p>1</p><p>1 (</p><p>ai+a j2</p><p>)2.</p><p>Prove that</p><p>1</p><p>2n(n 1)/</p><p>[</p><p>1 (</p><p>s</p><p>n</p><p>)2]</p><p> f (a1, a2, ..., an) n 1</p><p>1 s2/4+</p><p>1</p><p>2(n 1)(n 2).</p><p>Determine cases of equality.</p><p>Problem 6. Let a, b, c, d be non-negative real numbers such that a + b + c + d = 2.Prove that</p><p>ab(a2 + b2 + c2) + bc(b2 + c2 + d2) + cd(c2 + d2 + a2) + da(d2 + a2 + b2) 2.</p><p>Problem 7. Prove that if x, y, z are postive real numbers then</p><p>x</p><p>y+</p><p>y</p><p>z+</p><p>z</p><p>x</p><p>(</p><p>x2 + y2 + z2</p><p>xy + yz + zx</p><p>)2/3</p><p>.</p></li><li><p>6</p><p>Problem 8. Let r, a, b, c be positive real numbers, put p = 2r 3</p><p>r + 2. Prove that</p><p>a</p><p>pa + rb + c+</p><p>b</p><p>pb + rc + a+</p><p>c</p><p>pc + ra + b</p><p>1</p><p>1 </p><p>r + r.</p><p>Problem 9. Prove that if a, b, c, d are non-negative real numbers then</p><p>1</p><p>a2 + b2 + c2+</p><p>1</p><p>b2 + c2 + d2+</p><p>1</p><p>c2 + d2 + a2+</p><p>1</p><p>d2 + a2 + b2</p><p>12</p><p>(a + b + c + d)2.</p><p>Problem 10. Let x, y, z be non-negative real numbers such that x2 + y2 + z2 = 1. Provethat</p><p>cyclic</p><p>1 xy.</p><p>1 yz 2.</p><p>Problem 11. Prove that if x, y, z R then</p><p>x(x + y)3 + y(y + z)3 + z(z + x)3 8</p><p>27(x + y + z)4.</p><p>Do you think problem 10 can be solved using only Cauchy-Schwarz inequal-</p><p>ity? If not, have a look at this book. Do you believe that a solution for problem 9</p><p>is just a few line long with only simple reasoning? Problem 4 and 5 look intimi-</p><p>dating but we have strategies to deal with such types. Problem 3 is selected from</p><p>the section on symmetric polynomials which contains a large number of original</p><p>and nice inequalities. There is a very short and clever solution to problem 6. Can</p><p>you figure it out? Problem 11 is a refinement of a well-known inequality.</p><p>This book offers a lot more beautiful problems together with powerful meth-</p><p>ods and strategies.</p><p>-</p><p>Do not miss this book when you are in Hanoi for IMO.</p></li></ul>

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