NHỮNG VIÊN KIM CƯƠNG TRONG BẤT ĐẲNG THỨC ( TRẦN PHƯƠNG )

MỤC LỤC: CHƯƠNG 1: NHỮNG VIÊN KIM CƯƠNG TRONG BẤT ĐẲNG THỨC CỔ ĐIỂN S1: BẤT ĐẲNG THỨC AMGM VÀ CÁC KĨ THUẬT CHỌN ĐIỂM RƠI S11: BẤT ĐẲNG THỨC AMGM S12: NHỮNG SẮC MÀU ĐIỂM RƠI TRONG BẤT ĐẲNG THỨC AMGM .......................................

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TRAN PHU’O’NG

C{)n_g téc vién: Trn Tuin Anh l\guyén Anh Cuimg; Bi1iVie_“:tAnh

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(Tdi bdn lcin thzi nhd’t)

NHA XUi\T BAN TRI THU’C

MUC ugc

CHU’O’NG I: NHU’NGVlEN KIM CU’O’NGTRONG BAT DANGTHLPC CO DIEN 11

§1 Bit ding thircAM - GM vi cic R9 thuitchqn aiém ro’i 12

§1.2 NhCrng sic miu diim rditrong bit ding thircAM —~ GM 20

§2 Bit ding thirc Cauchy - Bunhiakowski —Schwarz vi ky 123

§2.1 Bit dingthfrcCauchy —Bunhiakowski-—Schwarz 123

Kythuitchqndiim rditrong bit ding thfrcCauchy

-' -' BunhiaC6pski —Schwarz

126

§4 Bit ding mac Minkowski vi kv thuit sirdung 193

§5 Bit ding thirc Chebyshev vi ky thuit sirdung 201

CHUUNG "3 Nl-‘|l~)’NG VIEN KIM CU’O’NG TRONG BAT DANG THUC CANDAI 223

§6 Bit ding thirc hoin vi vi K9 thuit su’dung 225

§7 Bit ding thirc Schurvi ky thuit sirdung 249

§s Djnh ly Muirhead vi bit ding thirc d6i xu’ng 279

§9 Dinh ly Fermat vi (mg dung trong bit dingthvirc 309

§10.1 Dlnh ly Lagrange cho him mét bié'n vicac u’ng dung 339

§10:2 Cu'ctrl cuahim nhi'éu biin vi phu’dng phip nhin tu’Lagrange 347

§11 Bit ding thirc Bernoulli vi cic frng dung 373

§11.2 K?thuitchgn diimrdi trong bit dingthirc Bernoulli 374

§12 Bit ding thircJensen vi k7 thuit sirdung 393

§14 Bit ding thirc vé’i céc him s6 léi bén phii vi l6m bén tréi 425

§14.1 Cécdjnh Iv v‘é him s6l6i bén phii vi l6m béntrii 425

§14.2 KYthuitsir dgng bit dingthirc RCF, LCF, LCRCF 43O

§15 Bit ding thirc Popoviciu 457

§23.1 vé m<f>rdiy bit ding thircbic ba vi (mg dgng 805

§23.2 D6i bié'ns6di sing tio vi chfrng minh bit dingthirc 809

§23.4 Phnrdng phép tié'p tuyinchirng minh bit dingthirc 829

§23.6 Cécphép d6ibié'n thuin nghich theo cic dédii trongtam giic 879

§23.7 Phu‘dng phépdinh giicéc hé s6'cda dathc bing djnh IvViéte 891

§24.3 Ciu chuyénv‘é bit ding thc Nesbitt -Shapiro 987

§24.4 Bit dingthirc JackGarfunkel vimét s6'm6ring 1015

§25.1 Témtit nhfrng vién kimcnrdng vi cic bit dingthfrc co’bin 1055

PHU LUC: Th6ng ké biiviét, bii toin sir dung vi tii liéuthamkhio 1114

Tri I/1111' 1/i 1110 /1'/101' 61'.

S17 c/In

Tfrnlzzivzgpliép soszinh trong cugicséng

The‘ gioi 1'51 ch51 v11 111111 1112111 trong 111 1111 111611 11111 do11g \'1‘1 hién (161 1\'11ong ngimg 1&0 nén

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111 $11‘ kh11c 11161 C1121 P1151 gizio 1161 g111o 1'11 T111611 c111111 g111o: s11‘ 1111110 11111111 g11r11 0111‘ 111111 \'1'111 11o11

Don"C ~Tv4 V11 C111‘ 111' 111611" C 11'1é1 hoc - c1111 11110 ~ 11611 111$ C ~'io'1.

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chod1'1‘11g lrong 111$gioi 1u611 11111)‘ 1161 n1‘1y 111 n11o' g111 111 Q1111 no nhung 11g1ro'i 111 11111'o"ng khong 11111111 1'11ring moi \'z_11chi co 111?: 11h:_1n gi11 11"1 1ro11g qu1111 11$ so s111111. Chfnh quun 11$ do 1111 1110 r11 czic b§1 d2‘111g 11111'c c1'111cuoc song.

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hon 1000 11111 so 1111 1 111n 111151 111111‘11g 111111 1161 vé 111131 111 g1111111 1111 g11p 11o'11 10 111n 1\/1o1 tong 111611;

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Con néu >161 vé lhu nhép 11i111g 1151111. 300.000 USD/1151111 01111 tong 111ong N1)” Bush c111 béng 11161

phn h111 trém 1§n so \'o'1 my du11 xc cong 111111‘ I 1161 11é111g 1hé gioi Michael Sch1111111chcr vo'1 111u nhép khoémg 60.000000 USD/nm Vz_1y 111 khi so $111111 11111611 hay khong 11g11‘o'i 111 01111 11110 d1n11 1111f1"11g

nhn 16 11111111gia v£1o so s11n11. luongh011c11c p11é111 Q11111 \'1‘1quy \'@r mot don 1'1.

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11111 p1111n 113 v1‘1 co em q11y11c‘>11g 1115111511110 s11n11 11111 p1111n so 151161111kénh

Quydéng: 13.§:135><197 265951189:189><143 27027 ma Z6595 < 27027

ic 11511 10a1n so 5111111 C151 111 vé $1111 11gZ1ycZ111g k110 11011 1'01 $11‘ 1110'réng c1111 czic phép 10;1n T211 ca'

czic 11h€1 101in'h0c déu 00 c111111g 11101 qu;111 diém 121 "C(ic kétqlui co’ bzin czia toxin /we thzrirng

g nh11'11‘011g cuéc séng 11gu‘0‘1 121 111611 gp s11‘ khzic 11hz111 g1f1'z1caic $11‘ 11:51 hién 1u'0'11g vil

nguy trong ban 1112111 n151 511' \'_é1 hién 111'Q'ng cfmg 111511 1101 111c0111'ng 1 Qiy ph1'11 T1131 111é néu szlu

dé dzinh gizi 11001 déng C1121 d0z1n11 nghiép h0:_11 d{111g \11111 nhjp 1\'11f1u 1111 11'11'0'11g ch1'1"11g 1;110z111 1211

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tzic gizi 111u0n b;111 6100 (:01 V1111 115 bi» sc1111"111h 1121) 11hu‘ 11101 diéu thét gin gi \*i1 1hi111 thién

B612

z?u'g'c

B6 5111'/1 /51 /11451 /113 r/Ming p/24111 /041/' c"<i(' p/zzrmzg /1/1:1/1 1'/111’/lg mi/1/1 />11’! (717/1g 1/1111' J11)" J11 W1 I1

/111 /1/1111." G0111 5 c1111‘0‘ng \‘0'1 Z5 chuyén dé B011 c1111'011g (11111 tién gidi 1h1{:u czic vién k1111 c11'0'ng

Trong 11151 chuyén <15? déu c0 phn din 11211 15111 1112111 11111111 110:1 bZ11 151p 111'giai 1'01 téng Iuqng czic

b E11 101111 1611 dén 2000 béi Céng phu 1111111 c0 1116 ké11é11 p11z1n 1r1'c11 g1c'>'1 thiéu 15 k_\" thuzjts11‘ d1_111g

bét ding 111110 AM — GM (1ré11 léng $0 30 k_€' thuzfat), nhfrng szing 130 \'Z1c11c kinh ng111é111 C1111 ctic

téc gizi c11u'21 xu:11111én 131 11511 c1'1'11101cu6n szich ni10 k11a1c.

T1100 1161 110 szich 111 06 1115 nhgin 111§)' s1_1' sip xép 11111" 11_1'g1f1';1 caic c1111‘011g \'Z1 czic chuy

an

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Q1111. Riéngc1111‘0'11g III1510 giri k116ng 01111611 11100 11101 gizm 11151 dial tén 1hc0 dzflc diémz "N11f1'ng vién

1ra.::g 111511 11101 gian: C0dién cén dai v51 hién dai C1111‘0ng V g01n 3 chuyén déz Chuyén C15 1110‘ du

C 11:1 n%:i§11p11u‘0ngp1151p c1101 cimg 151 czic bi11 105111 111e'1c11 1h1'1‘c.

1

P1111n 1110“ dau 11161 01111_\'611 116 111 >711 1116 1a 1511 0111 1161 1'6 "vién kim cm/11g" 06 110ng 01111y6n

116 s11u 116 111 0110 11511 1611 1115111 11111111 11611 1'11 01161 01mg 111 0110 11111 [zip 11111111 0110 111111 1160 111giai K111

1111111 11Z1}' 0h110‘11g 1 1110 gia 11111611 11111111 >11 1111611 011111 01111 111111 1100 110ng 11161 11113111 \'1 1611g 11‘-'11 11611

0f10h 1161 1116n 1111111 11100 quy 11161 01111 11111311 111110 “'I‘1‘r (I011 girin 17611 plz1?'1' tgrp" \i1 11111611 11511 101111

ling”. C111h6 110ng 11111 011uy611 116 1110 diiu \"6 11111 1161 Ling 111110 1211 1111011 11161 111 AM ~ GM 1'11 11111

dzing 111110 Cauchy — Bun111k0ws111 — S011\x'111z 1110 gizi 1151 11111111611 11161 phong 011011 1161 11161 1'6 0110

111110 111 111611 1111111 0110 1‘ 1u'11'11g 11161 011011 06 11111 111111 11161 116 5111' d6u 111 n1111'11g 11)“ 111'1ng qu1111 11-61151

day toén céc cép. cho nhfmg nhii nghién c1'1'u Yé tozin l16c p116 1h6ng Tzic gia hi \>6ng 11161 b6 szich

nhu' thé sémangdén cho d6c gizi $11‘ 111‘ tin khi (T60 szich sé 06 1116 khoi gqi ot11hf1'ng cmI160 sinh 1116i

tiép xL'1c v6'i Tozin h6c 11161 czich nghiém 11’1c u'6'c 1110116‘ 111211111 I160 sinh gioi tofm Qu6c 16. thiinh 111121

trong b6 szich ny

mii Cr yéu ceiu v6 6c quan szit_ linh cam tinh 16 \"i1 s1'1‘c szing 1&0 d6i diio cua ngtréri giai M61 b6 szich

161 V6 béitding th1'1'c phéi dinrz1 11111c tiéu tr(11l1£1nh 11161 n1i6n dzit nu6i du'<"mg v21 11111‘ thzich tr1’ 1116sing

1110 CL18 con ngudi Be‘ mm atéu <16 lfLl'C'C héi ban thin 1><f> Sach phzii thé hién 11161 SL'I'C tim tbi szing

tao.

czic “sting tao tozin i16c"’ c1'1z1 tzic gia vb czic c611g 1110 \'i611 Phn sting 1210 116i 1r6i <’111"6‘c thé hién

trong cl1u'0ng IV: “Nhfrng vién kim cu'o"ng trong L151 dzing 1l11'1"c 111611 dgii". D6 121 C£iC "\"i6n kimcu‘0'ng": S.O.S; MV; ABC; GLA; DAC Diy 121 nhf1"ng “vién kim cu'0'ng"’ du'6'c szing tao bO'i

ngudi Viét Nam v€1 IE1 nhfrng c6ng cu manh nheit dé ch1'r11g 111inh bait déng 1h1'rc

gizi C15 dng célin loai b0 nhfrng kién 1h1'1‘c kh6ng phii hqp \'<'1i t1'nh chtit ch611 16c cua cu6n sétch

Thianh Viét cng nhu' m6t S6 ky 1111151 ma céc Sinh vién vs Qu6c B51 can Hoilng Trqng Hién Lé

Cu6n sétch 121m61c6ng trinhtémi1uyé1-c1'1z1tzic gizitrong nhiéu 115111. Hi v6ng 1611 cuzi tzic gia

Trén Phwowg

DANH sAcH céms TAC v1é1\1

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LFSA.

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Quéc lé1511 lh1'1'-13 1z_1i Vu'o'ng quéc Anh 1151111 Z002.

3.Nguyén Anh Cufrng

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1h6ng kh6;12()()7 ~Z01] Truimg Dqi h<_>c Bdch

h<_1c Kho;1h(_>c 11_1'nhi6n. DHQG TP. HCM.

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11.Nguyén QuécHung

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do nhil xuéll ban Gil xulban ném 2006 sang tiéng Viét dé cuén szich cT11Q'c hon

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.\l() -Olympic Tmin Q1160 gin.

I.\lO—OlympicTozin Q1150 lé.

APMO —Olympicchflu 131 Thzii Binh I)u‘o'ng.

VMEO—Ki {hi giai 101111 lrénnlgzng CULIlrung \\"\\"\\".dic111i;11110;111l10c.ncl.

APMO—Olympic cl1f1u A Thzii Binh Durmg

('h1r0‘ngI: .\'lz17'ng 1'1'en kim 01n/ug[rung11111 1711/1g 1/1111" cu 1111511

CHU’O’NG1:NHU’NG VIEN KIM CU’O’NG

TRONG BAT DANG THU’C CO DIEN

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biéu 11161 011011 1161 x11n; D11 116 1111611; 11011 11;1_10 11111611 khi 11111111 11111 0110 \'1' 1111 >6 111 0110

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11111 161 nhiéu 13211 115111; 1h110 1101111 )1 1111 1111110 ;11"'1i 1111)'61 ;611 ;1111

11151110110 111110 2.111 v11 31111611161116 h1>"11.

C110 111111 1111) 11110 01111

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+ \’I<§t vzjn déng \'ién thé thao cn xzic djnh ngily thi du dé diéu chinh khéi 1u‘Q'ng \":_§n

thzii sung sire vii hung phim nhél.

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dzinh dim cho hiéu qua \/If)! mu mim cng cn plmi chon dling diém ro‘i dé bit nhfrng

Football World Cup 1998 ln thL'1' 16 16 chirc tz_1i Phzipi ” Tra/lg lrgill cfqii ruyén Bra;/Y gap

Tuy nhién mi»! vi du dzfic sic v21 rt n lu'<_J'ng mil chflng {Oi muén dimh cho ban doc suy

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111c1> 11161 11611 QI1115111 115111 1113 dé 1111111 hu'1'1‘11g 1111111 11131 11111 \1‘1 \Z1 11111111 g1;1 1111' 11:11 d5111g 1h1'1‘1‘

11*u11g g11111 1111510 b1} p11:_111 11611 Q13 1115 g1)i 1111" §' 111'11‘11g 11£1)' 151: "K_§ thut kiém tra diéu

lrongczic 11211 15p 111§u 1111nh 11021.

Trong ch11‘0'ng IV c11L'1ng 111 sé d11'1_1'c giéri 1111éu vé "Ky thu-{It ch1_1n 111121111 r0'i" 11111110111‘ 11111

déng 1111'1'c 111111 x1'1'11g hoéc h01111 \‘1 n1111'11g Q13 1111311 1\'11;§11 112111 b§111g 111; 121 1211 11111 11"Q11g 1h111

khzic nhuu

BIII(l'(IngtluicAll— GM

m TOM TATNQI DUNG

l. Dans l6n2 _ & q uzit

3. Chtrng minh

II. Diém roi trong dzinh gizi tirGM sang AM

III. Nguyén 1y aésng béc l1'Ong béldéng lhtrccan

IV. D510 biét héa dual vé bit dflng lhfrc d5ng bfac

V Phéi hqp hai b§1déngthL'1"c déng béc nguqc chiéu

nhau

VII Bit ding thfrc déng bgic dang céng mgm

VIII. B5 dng lhfrc déng béc cht1"ucén1ht1"c

IX. Be‘1td:'ing lhirc khéng déng bf1c_

XIII. Ky thuét tzich phn thirc aim vildzinh gizi m§u $6

XIV. ve d¢p diémro'i trongbi: aémg ma-¢ lu'<_>'ng gizic

XV. M61 bi tozin ch<_>n lc_>c _L'1'ngdung diém ro'i

XVI. Céc bili tgip dimh cho ban dQc tugiai

ChlrovzgI:Nhng vién kinz cumzgtrong bt ring thlirc6ziié l7

§1.1.B/QT BANG THUC AM - GM cAc DANG BIEU DIEN BAT DANGTHU’C AM-GM

1. Dgmgtozng qzuit: Giélsir a,,a2, a” IE1 n s6thuc khéng ém, khi dé ta cé:

2. Cdc trzr&ng h_0‘p dgicbiét

Biéu kién Va,b2O Va, b, (‘Z0 Va, 12, W120

Dang 2 a+b22.\/ab a+b+c'23.§/% a+b+c+d24.\*/abcd

Binh lu:§n: Khi chtrng minh bait ding thirc, néi chung ta rétit gép céc bit ding thL'rc c6

bét ding thrc cé mét vé phtrc tap, mét vé rL'1t gqn Cflng giéng nhu‘ khi chirng minh

dang nhanh v51 phéln trng linh hoat hon khi su‘ dung AM — GM Befmc biét 151 dang 3

khéng chtra cén thfrc nhéc chng tac6 thé sirdungAM - GM ngay cékhi khéng c6 dzéu

Phdn tich: Ta thumg khai trién (1’)

<:>16ab(a2 —2ab+b2)s (14 +4a3b+6a3b3 +4ab3 +29‘

Britzizfng th12'cAM-GM

Khi dé viéc phn tich hiéu gifra hai vé Lhiinh téng caic binh phu'o‘ng sé ggip nhiéu khé

hinh thtrc “chisd"dqmgAM— GMklzi mét véc6 cluia cdnth1'rc".Tuy nhiénnhbr C6

dang 3 mél gqi y cho chimg ta “c6tlzé; s1?dgmgAM—GMngaycc?khiedhaivé déu

Tm/alaz -5(,/al —\/(12) 20 :> T21/a,a3.{-Dangthuc xay 1a <=>a] =a2.

- Gia sL'1'b§tdz“ing thtrc <1) dng vdi /1 22.

Ta phz'1ichL'rng minh bit déing thtrc (1) dng vdi (n +1) 56: cz,.cz2, a”,a“H 20

SL1‘dung gié thiétquy nap chon S6: a],a2 a” 20 la 06:

a +a + +a +a 12.”Cl (1 a +a aa a = J ” ”*'+ ”*'

au ant xay ra<:> <=>a,-a2- -an-a”+,

Theo nguyén 1)? quy nap suy ra béit dingthtrc dimg véi m<_>i 112 2,ne N

C h lrmzgI:Nh17“ng vién kimcmmg trongbfitring th121' cox dién 19

Chli dn: Tén gcpi AM - GM 121 viét tit cua thuét ngfr tiéng Anh Arithmetic mean —

ll

Cési Ceich gqi nay Xuéuphzit mviéc nhilton h<_>¢ PhzipCési (Cauchy) 1:1 nguivi aéutién

d chtrng minh béit ding thirc nay vil éng d chtrng minh nébng mcfn phu'o'ng phép quy

Bzrc I: Kiém Iraménh dé dng véi 11 = 2.

Bl!'é'C 3: Gié sirménh dé dng vé'i n =p,chL'1'ng minh ménh dé dng véi n = p- 1 '

~

a +a (x/”1"\/Q") a +a

Thét véy xét 2k $5 thuc a],a3, a‘.,ak+1, a2k 20. SL'1'd1,1ng gié thiétquy naptacé

al+a2+ +a3k Zl a]+a3+ +ak +ak+]+ +a3k

.

Giési1'b§I d:§1ngth£1'cdflngvéri n = p, ta sé chimg minhbitdéingthirc dng vdi n:p-1.

Thét vaiy xét(p — 1) 56: a,,a3, ap_]2O.Su'dunggia1hiétquy nap vdi n =p ta Q6:

-|

a1+a3 + +ap_| +1’ /a,a3 a_l

2"a, ap_l./’ /a]a3 ap_[ =/’_/ala3 ap_,P

20 Brit ling thzicA1’V[-GM

§1.2.NH13NG sAc MAU DIEM R01 TRONG BAT DANG THU'C AM - GM

Chng tactmg buéc véo "thé giéidiérn r0*i"\/G1 bfai tozin don giém v51quen biél sau dy:

ISailz"§mth1rd’ngggip: S=a+L22 /a -i=Z:>MinS=2

MinS=L30 dzglt t2_1i "Die§m r0‘i:a = 3".

Do 11111 ding 111111 AM - GM xéy 111 11511 being 1111 diéu 1<1e11 11¢ 16 11111111 gia phéi bang

11ép @1111 2 S6 11 V21 lH vi‘ 3 11% LL'1c nay 111 Sé gié 1111111 S11 dungbé1 aé11g111u~¢AM - GM

@110 C£§1p S6 (ll) aé 11,11 "Biém 1111": 11 =3"1111 i =i 1111 111111 C6 $11 a6 "1-ném 1111'" ;

Ch!rangI: Nhz7'ng vién kimcumzg trong bfitdrillgthzicco:diéll 7

m rm a = 2 va M1nS=Z la Clap so dung nhung cach

giéi tréndé mic sai lm trong viéc dzinh gizimiu s6:

“N”QUCZ z2m‘—2—=—1‘a' W 4 3 anh gl'2'1sa"‘l

Dé cnéu chinhldi giéi sai thénh lwi giéi dng ta can phéi bién déi s sao cho khi su clungbét ding thtrcAM- GM S5 khtr hél bién S6 (7 ¢- miu S6.

-L61‘gm"z d’ung: Bién* déi* S via sL'rdung{>51 dimethircAM —GMta c6:

S=a+—17=(£+£+Lq)+6l23.3/2-2» 1, +-6'-2=2 Vc5“ia=2thi MinS=2

22 Biit ring thzicAM—GM

' Léi gidithuggm:

Do 1=4 ¢> a =1;=% nén bién aéi lI‘L1'C[iép s sirdungbét déng thfrcAM— GMIa <26:

Ch lt'0'IlgI: Nh17'ng viénkim cuvng trollgbit(Trim gtluicc5ziié Z3

I

Binlz Iun vd l()'igiiii

'Sai llim thzrdngggip: S=la+ 2 ,5“——a + /7 ~~L/7 :21*MinS :"

' L6'igii dzing:

S:a+b+\/E:£a+b+\/EJ+3(u+b)>2 ,iu+1; y +3(a+b)_;

Z4 Bdt"(Tingth12'cAM—GM

¢ Phdntich vdtim tbi léi gidi:

S0dddiem r0’i:

1a:b:C:—

' vién kim czrrmg trorzg

Tim gié tri nhé nh2‘1l ctia b1

Binh lun vdl&’igiiii

6 Br}: mingma-CAM-GMé1i8. Cho Ll /1 (' C1> 0. Tim gié trj nho nhzit Cuzi biéu thtrc

S = b+c+d +c+d+a+a+l2+b+a+b+c+ (1 + [2 + (‘ +

c = d+a+b

d = a+b+c'

QPhéntich viztim tbil0*igidi:

Bé tim Min s ta Can Che y s 151 mét biéu mm; déi xirng vc’>'i (I, 11, C C1do d6

MinS (hozfacMax S) néu 06thurng dgt tai "Diém roi t1_1' d0'_': a =b=c =d > 0.

xélyra 151tép conciladiéu kién dudoein: a =12 = c=d > O

ChavingI: Nhzivzgvién kinz cum:gtrong bf]!17511 gthzic 05riié 2

' Phn tich vd tim téild'igirii:

Dos 1:1 biéu thtrc @161 xrng mi <1, 1;, 1-, 11 nén C111‘L1O£11l 1\/1m s d;1tu_1i'dién1 10-1 111110;

l\) U.)

Ch1r0’ngI:Nhz?ng vién kimczrmzgtrong bt ring thzicc5 ciié 29

Nhgmxet: XetbatdangthucAM— GM: MZQ/¢,l¢, _A(1,' Vcz].(13, a” 20

Dé 3'; ring trong vé phzii (vé yéu) cua bit ding lhirc trén 1110 121 biéu thtrc G11/I C6 $6 czic

thira 56 trong cén thtrc dng béng Chl $5 cén thtrc (cimg béng n)v D0 <36. khi gap bét

ding thtrc mé vé yéu ca bait ding thirc c6 chfra cim lhL'1'<:"\/51 $6 céc thira $6 G trong cn

thirc nhé horn chi $5 cén thtrc thi la cn nhén thém cafc hing sé thfch hqp dé $5 céc thira

$6 trong cén thtrc being chi $6 cua cén Lhtrc. D5 xéc dinh du'Q‘c céc héng s5 thich hqpchng ta phzii dgr doén du‘Q'c du béng CUB bit ding thfrc nén ky‘ thuét néy cé tén g(_>i

diém roi trong dzinh gizi tr GMsangAM.

a+b+c=l

Phdn tich vdtim tr)i l()’1'gi(ii

' Sai mm tizzrdngggip:

'Dgrdodn vd tim diémr0'icda MaxS:

Vi s 121mét biéu thtrc déi Xumg vc'1i a, b, C nén Max s am tgi cnéu kién

Phén rich vdtim tdilifi gidi

Vi s 1:1 mét biéu mu-C déa Xifng vé‘i a b, C nén Max s dat tgi cnéu kién

C I1mingI: Nlzvzg vién kim czrmzgtrong bfit drillgthzic05riié

' Dy‘ dodn vdtim diérn roicziaMax S:

:> SIHQ/b2+C2 +17‘;/C:+a2 +05/a:+b2 S:1) 1O(a'+b“6+c")+24:l2

Véi a=b=c=2, MaxS=12

bc\/a— 7 +cu- 3/b—6+alri/0- 12

33$ 1 abc'+ubc"+ abc 5y~+ I

32 ‘ B(it (Trillgth1i'c AA/I — GM

Chu'0‘ngI:Nhz?ngvién kimclrmzgtrorzg b't ring thzic05 rfiézz 33

Czic bi: dang thtrc aé Cép trong ky thuét ny ducyc minh hqa bdi céc him da mu-¢ v<'>‘i ba

sé tiépczin ky thuél nfly [tr nhfrng vén dé don gian nhét

v6 $6 céc d0'nthL'1'c bzic k cho lru'c'>"c biéu dién qua 3 bién $6 a b_ c.

2. T(lisaophdisosdnh cdc biéu thév: dng bgicGié sirta phéi so sénh 2 da thtrc khéng déng béc chéng han xét bZ1i tofm sau dy:

Bziitozin: Chzivzgmirzh rring Y/a,b.c > 0 tac6: awn” + bwoo +02000 2:1 + b + c ("“)

tac6 (*)<:> 3x300”2 3x <:> x2°°° 2x

D0 bail dingthirc X2000 2 x dng néu x 2 1 v xzw“ 2 x sai néu xe(O_ I) nén suy ra:

bit dine thtrc rét tém thu'C7n2:- amo+ b2°””+ cm“2 z1+b+c Va,b_c 2 I

W ' Néu co hqrpmién xéc d1nh:a,b,c e (0,1) thi 21 00°< :1, b'° < b, c ‘O0 < c vi tacfmg <36

2 2 00 " z<

bit ding thirc nittm thu'<‘7ng: 212000 +b2°°°+ cm°< a+b+c Va,b,ce(0,1)

° Két lugin: Kh6ng nén dét vén dé so szinh czic Cia thirc khéng déng béc trén mién xéc

dinh 151RT Vi cc’) v6 $6 czich biéu dién céc don thirccé cimg béc k cho tru'<3"c nén czic bili

tozin dudi déy am vein aé la so szinh céc ham da mu-C ciéng béc mi bzfxc ctla czic don mu-C

duqc biéu dién b0“i nhrng trang théi khzic nhau

bét ding thirc bé phén D0 bét déng thtrc da cho xéy ra ding thtrc tai czic bién s6 béngnhau v51 chgy khép 113* nénczic bit ding thfrc bi» phémcng xéy ra ding thirc tai czic bién

among léy dai dién mét don thfrc <1 vé lam r5i céng thém vé'i czic don thfrc dang béc vdidonthtrc dai dién néy Céc béi tip sau déy sé minh hqachoczic bét ding thrc ba bién $6

ChlumgI:Nh/ng viénkim czr0’ngtrongbt (Tring th12'c 05dién 35

S11‘dung bit ding thtrcAM —GM ta cé

Nhzfmxét: ca hai vé 1a czicbiéu mu-C baa 2 nén biéu thtrc céng thém cling 1a baa 2

sudung bitdang thtrcAM —GM tac6

36 B17’)?(flingtluicA54—GM

Nhginxét: ca hai vé IE1céc biéu mac béc 0 nén bié/u mac céng thém cfmg151 béc0

Nhzfmxét: Hai vé 121 caic biéu mac b2f1c (-3) nén biéu mac ccfmg thém cé béc (_3)

ChmmgI:Nhzhzg vién kinz czrcrngtrongbitring thzi"c 05 riiérz 37

Nhilnxét: Hai vé 151 czic biéu thfrc déng béc béc (-3) nhung chng ta sé dzinh gié m§u $6

mikhéng sir dung k§' thuzit céng thém czic biéu thtrc déng bzfac.

B5 é‘: x3 +)3 =(.\'+_\')(x2 +_\': —_\j‘)2(.\'+_\‘)(2.x3* —.\3‘)=(.\'+_\').\‘_\‘ ‘V’.\'._\*2O

p {mg 203+/)3+abc g((1+b)c112+zzb¢' ;ab(a+b+c) abc(a+b+c) £156

Bai 11 [APMO 1993] Chtrng minh: (1+£)(1+B)(1+£)z2+3("‘i‘), Va.b.c>0(l)

ChzrmzgI: Nhvzg vién kimczrzrngtrongbfétring thzic c5riién

Ta chi can Chirngminh 5(a1 +111 +02)—4(ab+bc+ca)2ab+I2c+ca

Ding thin: xéy ra <:> a=b=c.

w § < -_ =. — i -v ,_‘_:,

ea.-~_‘~- ‘— :- 1-? it “ "@v*=-~-_: “‘“5~""i3-_- ‘ ~ -7 -“:’w 1'1 "1;‘ "1; Ha: +a3) (a” +al)— Z

, _ _ ’ ' ‘-' ‘U1 as)“: +“4)-~-(0 +u,) 1' T"-d-\.’= 1. 5 ; ;~'— ¢ k&a>\1A, > 3 +5.-4;-"a +§}2n => Max{A,,A,}>; _

Béi 19 Chirngminh:a" +0" +6" z(“+32")” +("‘g2C)” +("+32“)” , Va,b,c>O;neN

a"+b,1+Cn:Z:a"+b3”+b"2Z(a+‘g+b) :(a—g2b) +(b-I520) +(c+32a)

('_\'(' ('_ ‘('