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Mi kin ng gp xin gi v

Nguyn Minh Tun

K62CLC Ton- Tin HSPHN

Gmail : popeyenguyen94@gmail.com

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Ti liu c pht hnh trn din n : www.k2pi.net.vn. Mi hot

ng s dng ti liu v mc ch thng mi u khng c cho php.

Xin chn thnh cm n

Nguyn Minh Tun (Popeye)

Mc Lc

T.Andreescu, V.Cirtoaje, G.Dospinescu, M.Lascu

Old and New Inequalities (Bn dch ca Dng Vit Thng ) 1

Li ni u 3

Chng 1. Cc bi ton 4

Chng 2. Cc li gii 24

T in thut ng 138

Ti liu tham kho 142

MathLinks Members - Inequalities Marathon 144

MathLinks Members

Inequalities From Around the World 1995-2005 199

Years 2001-2005 205

Years 1996-2000 259

Years 1990-1995 304

Supplementary Problems 320

Classical Inequalities 353

Bibliography and Web Resources 358

Nguyen Manh Dung, Vo Thanh Van

Inequalities from 2008 Mathematical Competition 362

Problems 366

Solutions 372

The inequality from IMO 2008 400

Hojoo Lee, Tom Lovering, Cosmin Pohoata - INFINITY 408

1. Number Theory 412

Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

Fermats Infinite Descent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .414

Monotone Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

There are Infinitely Many Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Towards $1 Million Prize Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .424

2. Symmetries 425

Exploiting Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Breaking Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

Symmetrizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

3. Geometric Inequalities 432

Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Conway Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

Hadwiger-Finsler Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Trigonometry Rocks! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Erdos, Brocard, and Weitzenbock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

From Incenter to Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

4. Geometry Revisited 456

Areal Co-ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Concurrencies around Cevas Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461

Tossing onto Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

Generalize Ptolemys Theorem!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .465

5. Three Terrific Techniques (EAT) 472

Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .472

Algebraic Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

Establishing New Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

6. Homogenizations and Normalizations 480

Homogenizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

Schur and Muirhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .482

Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

Cauchy-Schwarz and Holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

7. Convexity and Its Applications 491

Jensens Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Power Mean Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Hardy - Littlewood - Polya Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

8. Epsilons 498

9. Appendix 607

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

IMO Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

T. Andreescu, V. Cartoaje, G. Dospinescu, M. Lascu

Bin dch: Dng Vit Thng

Bt ng Xa v Nay

1

Mc lc

Li ni u 3

Chng 1. Cc bi ton 4

Chng 2. Cc li gii 24

T in thut ng 138

Ti liu tham kho 142

2

Li ni u

Quyn sch kt hp nhng kt qu kinh in v bt ng thc vi nhng bi ton rtmi, mt s bi ton c nu ch vi ngy trc y. Lm sao c th vit c iu gc bit khi c qu nhiu sch v bt ng thc? Chng ti tin chc rng d tiny rt tng qut v thng dng, quyn sch ca chng ti vn rt khc bit. Tt nhinni th rt d, vy chng ti nu vi l l minh chng. Quyn sch cha mt s ln biton v bt ng thc, phn ln l kh, cc cu hi ni ting trong cc cuc thi ti v kh v v p ca chng. V quan trng hn, trong cun sch chng ti s dng nhngli gii ca chnh mnh v xut mt s ln bi ton c o mi. Trong quyn schc nhng bi ton ng nh v c nhng li gii ng nh. V th quyn sch thch hpvi nhng sinh vin s dng thnh tho bt ng thc Cauchy-Schwarz v mun ci tink thut v k nng i s ca mnh. H s tm thy y nhng bi ton kh hc ba,nhng kt qu mi v c nhng vn c th nghin cu tip. Cc sinh vin cha saym trong lnh vc ny c th tm c mt s ln bi ton, tng, k thut loi va vd chun b tt cho cc k thi ton. Mt s bi ton chng ti chn l bit nhngchng ti a ra nhng li gii mi chng t s a dng ca nhng tng lin quann bt ng thc. Bt k ai cng tm thy y vic th thch cho nhng k nng camnh. Nu chng ti cha thuyt phc ni bn, xin hy xem nhng bi ton cui cng vhy vng bn s ng vi chng ti.

Cui cng nhng khng kt thc, chng ti t lng bit n su sc nhng ngi tra cc bi ton c trong quyn sch ny v xin li v khng a ra y xut x dchng ti c gng ht sc. Chng ti cng xin cm n Marian Tetiva, Dung Tran Nam,Constantin Tnsescu, Clin Popa v Valentin Vornicu v nhng bi ton p m h nura cng nhng bnh lun qu gi, cm n Cristian Bab, George Lascu v Clin Popa vvic nh my bn tho v nhiu nhn xt xc ng ca h.

Cc tc gi

3

Chng 1. Cc bi ton

4

1. Chng minh rng bt ng thc

a2 + (1 b)2 +

b2 + (1 c)2 +

c2 + (1 a)2 3

2

2.

ng vi cc s thc a, b, c bt k.

Komal

2. [Dinu Serbanescu] Cho a, b, c (0, 1), chng minh rngabc +

(1 a)(1 b)(1 c) < 1.

Junior TST 2002, Romania

3. [Mircea Lascu] Cho a, b, c l cc s thc dng tha mn abc = 1. Chng minh rng

b+ ca

+c + a

b+

a+ bc

a +b+

c + 3.

Gazeta Matemati

4. Nu phng trnh x4+ax3+2x2+bx+1 = 0 c t nht mt nghim thc, th a2+b2 8.

Tournament of the Towns, 1993

5. Tm gi tr ln nht ca biu thc x3 + y3 + z3 3xyz vi x2 + y2 + z2 = 1 v x, y, zl cc s thc.

6. Cho a, b, c, x, y, z l cc s thc dng tha mn x+ y + z = 1. Chng minh rng

ax+ by + cz + 2(xy + yz + zx)(ab + bc + ca) a+ b+ c.

Ukraine, 2001

7. [Darij Grinberg] Nu a, b, c l cc s thc dng, th

a

(b+ c)2+

b

(c + a)2+