Tuyển tập đề thi vô địch bất đẳng thức thế giới P1

Preface This work blends together classic inequality results with brand some of which devised only a few days ago What new problems, could be special about it when so many inequality problem books have already been written? We strongly believe that even if the topic we plunge into is so general and popular our book is very different Of course, it is quite easy to say this, so we will give some supporting arguments This book contains a large variety of problems involving inequalities, most of them difficult, questions that became famous in competitions because of their beauty and difficulty And, even more importantly, throughout the text we employ our own solutions and propose a large number in this book and of new memorable original problems solutions There as well This are memorable is why this work problems will clearly appeal to students who are used to use Cauchy-Schwarz as a verb and want to further improve their algebraic skills and techniques They will find here tough problems, new results, and even problems that could lead to research The student who is not as keen in this field will also be exposed problems, ideas, techniques, for mathematical contests to a wide and all the ingredients Some of the problems variety of moderate and easy leading to a good preparation we chose to present are known, but we have included them here with new solutions which show the diversity of ideas pertaining to inequalities Anyone will find here a challenge to prove his or her skills If we have not convinced you, then please take a look at the last problems and hopefully you will agree with us Finally, but not in the end, we would to the proposers of the problems featured like to extend our deepest in this book and appreciation to apologize for not giving all complete sources, even though we have given our best Also, we would like to thank Marian Tetiva, Dung Tran Nam, Constantin Tanadsescu, Calin Popa and Valentin Vornicu for the beautiful problems they have given us and for the precious comments, to Cristian Baba, George Lascu and Calin Popa, for typesetting and for the many pertinent observations they have provided The authors Contents Preface Chapter Problems Chapter Solutions 25 Glossary 121 Eurther reading 127 CHAPTER Problems Problems Prove that the inequality JES OF + VPS oP + EF ap > holds for arbitrary real numbers a, b,c Komal | Dinu Serbanescu | If a,b,c € (0,1) prove that Vøabe+ w(1— a)(1— b)(1— e) < Junior TST 2002, Romania | Mircea Lascu | Let a,b,c be positive real numbers such that abc = Prove that b+e cta Tat atb et Fe Vat vb+ ve +3 Gazeta Matematica If the equation x* + ax? + 227 + be + = O has at least one real root, then a? +b? > Tournament of the Towns, 1993 Find the maximum value of the expression z° + y? + 2° — 3xyz where x? + y? + z* =1and z,y,z are real numbers Let a,b,c, x,y,z be positive real numbers such that «+ y+ z= aœ + bụ + cz + 2V/( Prove that + 0z + z#)(ab + be + ca) Prove that Vat + a2? +bt4+ V/b* + b?c? + ct + V(ct + ea? +0! > > av 2a? + be + by 2b? + ca + eV 2c? + ab Gazeta If a,b,c are positive real numbers such that abc = 2, then a& +h +2 >aVvb+ce+bVetatevatb Matematica Old When and New Inequalities does equality hold? JBMO 2002 Shortlist 10 [ loan Tomescu | Let x,y, z > Prove that +z Sma: (1 + 32)(x + 8y)(y + 9z)(z + 6) When we have equality? Gazeta Matematica 11 [ Mihai Piticari, Dan Popescu | Prove that 5(a? + b2 + e2) < 6(aŸ + bỶ + c3) + 1, for all a,b,c > O witha+6+c=1 12 [| Mircea Lascu | Let 71, %2, ,%, đa + +0„ = a and #7 +z2 + +2 € R,n > anda > such that x + < T Prove that x; € [ *) , for all ¿€{1,2, ,n} 18 [ Adrian Zahariuc | Prove that for any a,b,c € (1,2) the following inequality holds b/a + cvb + a/c 4bf/e—cJa 4eVWa-aVvdb 4aVb-bVc >1 — - 14 For positive real numbers a,b,c such that abc < 1, prove that a b be +-+—=>a+b+c ec oa 15 [ Vasile Cirtoaje, Mircea Lascu | Let a,b,c,2,y,z be positive real numbers such thata+a>b+y>c+zanda+b+c=2+y+2z Prove that ay+bzr > ac+2z 16 | Vasile Cirtoaje, Mircea Lascu | Let a,b,c be positive real numbers abc = Prove that 1+ œ+b+c_— > so that ‹ ab+ac+bc Junior TST 2003, Romania 17 Let a,b,c be positive real numbers Prove that a3 bồ c aa b c2 JBMO 2002 Shortlist 10 Problems 18 Prove that ifn > and x%1,%2, ,%, > have product 1, then l+zi+4+ 122 + 1+ 222+ %2%3 f+ + ——— > 1+2%,+%n21 Russia, 2004 19 [| Marian Tetiva | Let 2, , z be positive real numbers satisfying the condition a? ty +27 4+ 2Qaeyz = Prove that a)) ryz yz —1 JBMO, 2003 23 Let a,b,c > with a+6+c=1 Show that a +b b+e Hee c+a tal yg a+b — 24 Let a,b,c > such that at + 64 + ct < 2(a?b? + bc? + c?a”) Prove that aŠ + bŠ + e? < 2(ab + be + ca) Kvant, 1988 16 Problems 64 | Laurentiu Panaitopol | Let a1,a2, ,a@, be pairwise distinct positive integers Prove that 2n+1 aj +a5+ -+a;,>23 (a, tag + -+4y) TST Romania 65 [ Calin Popa ] Let a,b,c be positive real numbers such that a+b+c¢= Prove that b/c cự avb 3v3 a(Vầ + Vab) b(Vần+ Và) c(VAb+ ve +7 66 [ Titu Andreescu, Gabriel Dospineseu | Let a, 6, c,d be real numbers such that (1 + a?)(1 + 67)(1 +c?)(1 + d?) = 16 Prove that —3 < ab+bce+cd+da+ac+ bd — abcd < 67 Prove that (a + 2)(b? + 2)(c? + 2) > 9(ab + be + ca) for any positive real numbers a, b,c APMO, 68 | Vasile Cirtoaje | Prove that ifÍ < #z < a) (L— #p)(1— z)(1 — #z) > 0) 3,2 b) ay abc Prove that at least two of the inequalities 6 ~454->6,54+-4+->6,-+-4+->56, a oboe b ¢ a c ab are true TST 2001, USA 70 | Gabriel Dospinescu, Marian Tetiva ] Let 2, y,z > such that EtYyYtZe= xLyzZ Prove that ( — 1)(wT— 1)(z— 1) < 6V3 - 10 Old and New Inequalities 17 71 | Marian Tetiva | Prove that for any positive real numbers a, b,c, a®—b a+b bì—c3 b+e -a? c+a + (a — b)? + (b-c)* + (c—a)? < |— Moldova TST, 2004 72 | Titu Andreescu | Let a,b,c be positive real numbers Prove that (a° — a? +3)(0? —b7 4+3)(2 —c +3) > (at+b+e)® USAMO, 2004 73 [ Gabriel Dospinescu ] Let n > and 21, 2%2, ,%p, > such that S2) (>: ¬ k=1 k=1 %k =n?+1 , Prove that 2) rat - S2) k=1 Ly >n®P+4+— n(n — 1) 74 | Gabriel Dospinescu, Mircea Lascu, Marian Tetiva ] Prove that for any positive real numbers a, b,c, a2 + b2 + e° + 2abe+ > (1+ ø)(1+b)(1+ e) 75 [ Titu Andreescu, Zuming Feng | Let a,b,c be positive real numbers Prove that (Qa+b+c)? 2a2 + (b+c)? (2b+a+c)? 26?+(at+c)? (2c+a+b)? 2c? + (a+b)? ~~ USAMO, 2003 76 Prove that for any positive real numbers x,y and any positive integers m,n, ty pyr" ag), + (mtn-1 (ary +ary™) > mnlartr (n—1)(m—1)(a™*"ty™*") Austrian-Polish Competition, 77 Let a,b,c, d,e be positive real numbers such that abcde = Prove that a+abc ltab+abed b + bcd 1+6bc+bcde c+ cde 1+cd+cdea d+ dea 1+de+deab Crux 1995 > 10 e+eab l+ca+ cabe — Mathematicorum 18 Problems 78 [ Titu Andreescu | Prove that for any a,b,c, € (0, 3) the following inequality holds sỉn ø - sin(œ — Ö) - sin(œ —e) sin( + c) sinb-sin(6 — e) - sin(b — ø)_ sin(e + ø}) sinc- sin(e — a) - sin(e — b) >0 sin(a + b) — TST 2003, USA 79 Prove that if a,b,c are positive real numbers then, Va4 + b1 + c+ + Veh? + Pet 2a? > Va3b+ Bet Gat KMO 80 | Gabriel Dospinescu, Summer Vab? + be? + ca Program Test, 2001 Mircea Lascu | For a given n > find the smallest constant k, with the property: if a1, ,@, > have product 1, then a1 Q2 + (a? +a2)(az +a) a203 (a3 +.a3)(a? + a2) AanG1 fives $ — (a2 + a1)(a? +an) F< ky — 81 [ Vasile Cirtoaje | For any real numbers a, b,c, x,y, z prove that the inequality Ww] az + bụ + cz + v/(a3 + b2 + c2)(2 + 2+ z?) > holds s(œ+b+e)(z+ +2) Kvant, 1989 82 [ Vasile Cirtoaje | Prove that the sides a, b, c of a triangle satisfy the inequality 3(0 4242-1) b ¢ oa 50(2 4622) a b oe 83 | Walther Janous | Let n > and let 71, 2%2, ,%, > add up to Prove that Crux 84 | Vasile Cirtoaje, Gheorghe Eckstein | Consider Mathematicorum positive real numbers #1,Z2, ,„ such that #+zs #„ = l Prove that m-lt+a, + n-1+22 + +®———— nm—-14+ 2p È Vietnamese IMO Training Camp, 1995 89 | Dung Tran Nam ] Let #,,z > such that (x + y+ 2)? = 32zyz Find the ei ty* +24 minimum and maximum of ——————_ (ct+y+z) Vietnam, 90 | George Tsintifas ] Prove that for any a,b,c,d > 2004 0, (a+ 6)?(b+ e)3(e+ đ)®(d+ a)? > 16a?b?c2d?(œ + b+ e+ đì! Crux Mathematicorum 91 [ Titu Andreescu, Gabriel Dospinescu | Find the maxinum pression value oÊ the ex- (ab)” | bc)” | (ca)” l—ab l1-be l—ca where a,b,c are nonnegative real numbers which add up to and n is some positive integer 92 Let a,b,c be positive real numbers Prove that a(l+6) + b1 +c) + >> c(I+a) — VYabc(1 + Wabe) 93 [ Dung Tran Nam ] Prove that for any real numbers a, b,c such that a? +6? + c2 =9, 2(z+b+c)— abc< 10 Vietnam, 2002 20 Problems (of) (OE i)e(oe£a) (ced ra(oed 1) (ba)ea 94 | Vasile Cirtoaje | Let a,b,c be positive real numbers Prove that 95 [ Gabriel Dospinescu |] Let n be an integer greater than Find the greatest real number m, and the least real number Ä⁄„ such that for any positive real numbers £1,02, -,%n (with ty, = %,%n41 Mn = #1), < "Sd : < My 1)#¿ +.” +92(nT— 96 | Vasile Cirtoaje | If x,y, z are positive real numbers, then 1 #2 + #U +9 3T 2 tụz+z zt z2 + zz + z 52 (z++z) Gazeta 2° Matematica 97 | Vasile Cirtoaje | For any a, b,c,d > prove that 2(aŠ + 1)(b3 + 1)(eŸ + 1)(đ + 1) > (1+ abeđ)(1 + a”)(1 + 02)(1+ e2)(1+ d?) Gazeta Matematica 98 Prove that for any real numbers a, b,c, (a1 + b* + €) sa) (z+b)*+(b+e)*+(e+a)°®> Vietnam TST, 1996 99 Prove that if a,b,c are positive real numbers such that abc = 1, then 1 l+a+6 lI+ö+c + < ltet+a724+a + + 2406 2+c Bulgaria, 1997 1.2 100 | Dung Tran Nam | Find the minimum value of the expression ¬ + + : where ø, ô,e are positive real numbers such that 2lab + 2be + 8ca < 12 Vietnam, 2001 101 [ Titu Andreescu, Gabriel Dospinescu ] Prove that for any x,y, z,a,b,c > such that xy + z + zz = 3, a (y+z)+ btew b c+ -+z)+— Cc +b > (œứ+)>3 ... our own solutions and propose a large number in this book and of new memorable original problems solutions There as well This are memorable is why this work problems will clearly appeal to students... and popular our book is very different Of course, it is quite easy to say this, so we will give some supporting arguments This book contains a large variety of problems involving inequalities, most...Preface This work blends together classic inequality results with brand some of which devised only a few