H W; HK’ > 4/EZL ‘“ 5% 1” » may ‘ MW H w ‘ 5% mt Wu"QaW $3W; K‘ PY’ AQWhm? H \ (V V W ; L “V; C7)J QX Mam3‘wti’K/ ; / UHW;V MK‘ V I/H WW V TRAN PHU’O’NG C{)n g téc vién Trn Tuin Anh l\guyén Anh Cuimg;[.]
_ H W; K’ H >_ _ 5% ‘“ _ 1_»_ may ”_ M ‘_ W _ 5% H_w ‘_ mt "Q $3 W; a Wu _ ‘W P Y _ ’ hm? QW A K V _H_\ _V W ;_ _( J _ ‘w ti3 /_;_/ ’K _ U W; H V ‘_ M _KV /H WW _I V_ Q _M am X L _ _ C7) “V; _ /EZL _ TRAN PHU’O’NG C{)n_g téc vién: Trn Tuin Anh l\guyén Anh Cuimg; Bi1iVie_“:tAnh i NHU’NG VIEN KIM CU’O’NG TRONG BAT DANG THU’C TO/§\N HQC (Tdi bdn lcin thzi nhd’t) NHA XUi\T BAN TRI THU’C MUC ugc CHU’O’NG I: NHU’NG VlEN KIM CU’O’NG TRONG BAT DANG THLPC CO DIEN §1 Bit ding thirc AM - GM vi cic R9 thuit chqn aiém ro’i §1.1 Bit ding thirc AM — GM §1.2 NhCrng sic miu diim rdi bit ding thirc AM GM §2 Bit ding thirc Cauchy - Bunhiakowski — Schwarz vi ky thuit chgn dié'm rdi §2.1 Bit ding thfrc Cauchy — Bunhiakowski Schwarz Ky thuit chqn diim rdi bit ding thfrc Cauchy ' ' BunhiaC6pski —Schwarz §3 Bit ding thlfrc Holder vi ky thuit chgn diim ro’i §3.1 Bit ding mac Holder §3.2 Ky thuit su’ dung bit ding thllrc Holder —~ -— §4 §4.1 Bit ding mac Minkowski vi kv thuit sir dung Bit ding thirc Minkowski §4.2 Ky thuit su’ dung bit ding thirc Minkowski §5 Bit ding thirc Chebyshev vi ky thuit sir dung §s.1 Bit ding thirc Chebyshev §5.2 Ky thuit sir dung bit ding thirc Chebyshev CHUUNG "3 Nl-‘|l~)’NG VIEN KIM CU’O’NG TRONG BAT DANG THUC CAN DAI §6 Bit ding thirc hoin vi vi K9 thuit su’ dung §6.1 Gidi thiéu v'é bit ding thirc hoin vi §6.2 Ky thuit sir dung bit ding thirc hoin vi §7 §7.1 §7.2 §7.3 Bit ding thirc Schur vi ky thuit sir dung Gidi thiéu vé bit ding thu’c Schur Ky thuit sir dung bit ding thllrc Schur Ung dung bit ding thin: Schur chirng minh bit ding thirc - don xu’ng ba blen §s Djnh ly Muirhead vi bit ding thirc d6i xu’ng §s.1 Gié'i thiéu dlnh I9 Muirhead §8.2 Ky CHU’O’NG Ill: thuit s\'r dgng dinh ly Muirhead" §9 thvirc §9.1 Gidi thléu dinh 19 Fermat §9.2 U'ng dung dlnh ly Fermat §10 Djnh ly Lagrange vi cic (rng dung bit ding thirc §10.1 Dlnh ly Lagrange cho him mét bié'n vi cac u’ng dung ' §10:2 Cu'c trl cua him nhi'éu biin vi phu’dng phip nhin tu’ Lagrange §10.2.1 Cgrc trj khéng c6 dléu kién ring buéc §10.2.2 Cu’c tri c6 dféu kién ring bu0 \"111111 s0cL111g111'.\0: 11115111 k» b,l8),bH3> A~_,_,13_1 50 sanh phan bu cua phan so T1110 H3+]43_1_l97 +197 ic 11511 czic 11h€1 10a1n so , vé 5111111 C151 111 101in'h0c déu 00 — $1111 giy 11101 11151 k11611g 1h§ 111 dé dzinh gizi n151 1112111 gié déng 11001 chfnh I/zz2‘c phai C0 1u'd11y tzic gizi 111u0n b;111 6100 dé 111é 101 (:01 vé [hay séng ph€11 111511 11211 ding s11‘ khzic '1 11011 c1111 czic phép 10;1n quy 11g11)'é11 1)‘ gizi 1h1'1'c 101111 1100 1\'11f1u czic 101 1111 T211 ca' 111' néu szlu cfmg df111g 1h1'1'c 11'11'0'11g ch1'1"11g 1;110z111 gin gi 1211 1r011g 01100 5011;: qucn du 151111 diéu thét 11:11 vil 1u'0'11g ch1]11g Ia 115111 $11‘ 1111111 1101 101 hién $11‘ 11:51 ph1'11 T1131 111é sau 31) 11z_1p N01 nhfrng 11hu‘ 11101 1121) Qiy 1u6n s1‘1'd11ng nhjp 1111)‘ \'£1 11111111 g1f1'z1caic 111c0111'ng 111511 1101 h0:_11 d{111g \11111 111‘ 11hz111 h1}1 c111'111g1z1 bi» sc1111"111h V1111 115 gp 1hc0 (101 I111 nghiép réng $11‘ 1110' 11.5184) HZ3 111.1 "C(ic két qlui co’ bzin czia toxin /we thzrirng chi? khéng phzii b(1ng n/zl7'ng (Hing tlz11'c” Dié-11 (10, 111611 121 11'011g (101 d0z1n11 C1121 V1 th1 ce1n 1\'h611g khzic M511 111 hién 511' \'_é1 déu c111111g1u diém c111111g 11101 qu;111 g nh11'11‘011g cuéc séng 11gu‘0‘1 nguy ban Q > 11gZ1ycZ111g k110 11011 1'01 biéu tlzj br1ng nhzivzg bf1t (hing dzrgrc 11611 nhu' \=:_'1) thién \*i1 1hi111 B612 I 1'1’ 11/11!‘ 1’lg3t b6 sciclz trgm vgn vd sziu s(1c vé bzic p/1ll’0'IIg' pluip chzhzg nzinh br1I (Ting thzic Id n/u7’ng vién kinz cu'0'ng [M11 dg tlz1?’c torin lzgc B6 /111 /1/1111." 11100 1111111 111' 11101 1\1111c11‘0'n-1 0111'1'ng 1116 1a 1511 0111 1161 1'6 11611 1'11 111160 11111h 1111)’ 11100 011011 116p 06111 ling” /1110 141p 11111611 11111111 >11 1111611 011111 01111 111111 quy 11100 1111111 11111111 pluip 11111111 011111mg >711 111 0110 11171111é11g1111'1'0 16111111 1'11 11111" 11111 1'6 11161 11611 116 g11‘111 1111f1'11g $111111 1'11 151111 1115111 gia 11161 111111 11161 01111_\'611 116 11511 11)“ 1160 0110 ng116‘i 1160 1111111 116 \'21111'11g 1\'§ 1111161 nhin p112111 p11a1 01111 g16'i 01111111 1116 M131 cring lri/1/1 tlzgrc nglziém v15plz11"0'ng d1r0'0 06 g11i \i10 111611 111'1)'11g 1111 1611 111111'11g s11 1111110 011161 1611 111160 1121111 111111 $11011g16ng1111u'11161011u}"611 \‘16n 021 116 1161 11g11'0'1 gi6i gia B6 101111 Q1160 s110h n1‘1y 11‘1dZ1n11 gm 1-21 Q cho h00 0110 s1n11 gizio vi6n day toén céc cép cho nhfmg nhii nghién nhu' thé sé mang dén cho d6c gizi tiép xL'1c v6'i Tozin h6c toén h6C vii niém 11161 1i11 V510 $11‘ 111‘ c1'1'u tin czich nghiém 11’1c Yé tozin l16c p116 (T60 szich sé u'6'c 1110116‘ kha néng hién thtrc hozi M_6t bgi szich khzing djnh u'O'c 06 111211111 1110' 1h6ng Tzic gia hi \>6ng 1116 khoi gqi I160 ot11hf1'ng cm sinh gioi tofm Qu6c 11161 sinh 1116i thiinh 111121 I160 16 b6 szich 51) tri tug? sdng tgm Néu nhu' viéc gidi ton lit (ii tim czii tinh czii d6ng thi st!‘ szing 1310 \'2_‘111 dung cftc phtrcmg phzip gizii chfnh 151 phzit huy czii d6ng ctii linh Hui 111511 d6i 1z_'1p dft hoii \'Z10 IE1111 11161 b6 szich ny A mii Cr 161 V6 B51 ding thirc m6t iinh vtrc khé nhung czii khé kl16ng n§1111 génh néng vé lu"6ng kién th1'1'c szit_ linh cam tinh 16 \"i1 s1'1‘c szing 1&0 d6i diio cua ngtréri giai M61 b6 szich phéi din rz1 11111c tiéu tr(11l1£1nh 11161 n1i6n dzit nu6i du'c hc — Btich khoa Bdch khou H€1N(>i Kho;1h(_>c 11_1'nhi6n T1"11*(>'11g D511 Y hQC — 201 I T1"11'O'11g Hz} N61 DHQG TP HCM DLl'Q'C C511 Tho" H21 H21 Dai hQc Blois Céng Tfnh h(‘)u Phzip u — Tin khéa 2008 — Z012 T1*u'Ong D211 Hung Sinh vién khoa Toain khézl 2005 ~ 2009 12 B:_1ch Nggc Thénh Céng Ldp Dai 2006 1ru'(‘mg THPT Nguyén Trung Thién Thach nzing kh6z12()07 Z002 T1"11‘O‘11g D211 h11g 511' G13 11:11 th11'c a’_iapl1u'0'ng) 11‘011g so" kién xziy du béng" )" Q1111 06 111111 \"i§1' D11 1111111 hu'1'1‘11g 1111511‘ 1113 1h1': g1)i gqi 1161 \1'1'11g CLILI 11111; (I11) 111111 1111" 11111111 1111111 1111" 111151 §' 11131 diéu 11111 111'11‘11g — \‘£1 P 1111111“ 1h11"1 111111151 Buniakowski 1\'iQ11 111311; 111111 \Z1 \1‘1 11£1)' 111111 [111 111'1_111g 111111 "Ki 151 Schwar; $11011 "A B" xa) = 1'11 111111‘: 1'11 d5111g 1h1'1‘1‘ 11:11 thut kiém tra diéu "K_§ 151: — 111111111 g1;1 1111' 11111111 xay 1\'1§*11 11111111 c{1p 111111; 11$ 1111'1'_1'1‘ >5 1115111 AM—G1W: Cauchy 1115111 phu‘11'11g ph11p 11*u11g g11111 1111510 b1} p11:_111 11611 Q13 1115 r01’ 211 211 bng 1\"1111‘>11 11111 1'111'Q'c b§111g P g11111 111151‘ 1'111'1151 115111; 111111" 111} 11112111 111112111 C1111 u'0‘11g: bi» phzin 115111;: 111111‘ 21;: X} 1; 1,1, 1:111 1'11 1111‘ 11211 11511111 111111‘ Néu 112111 51211211 thuzt ch1_)n diém b§1td§ngth1'1-c" Trong 1:hu'O'11g c1111ng 101 S5 phn gidvi lhiéu vé ch1_m b:“1t diém r0'i" lrong czic Trong déng ding 11211 1111'1'c khzic nhuu 111111 IV — nhirng vi 15p 111§u 1111nh ch11‘0'ng dng bz'11 th1'1'c A111 111' c11L'1ng x1'1'11g hoéc 111 gidi 11111311 1-51 1111'1'c GM" 1111 (11111 11-; vé "1’i thiéu vé bét ding thtvc AM - l Dans _ l6n2 & Czic tru'b’ng hqp déc biét Chtrng minh GM q uzit NhCvng séc méu diém rcvi - GM bét ding thu»c AM I Diém ro‘i lrong dzinh gi£1u‘1'A;\/I sung G.\/I II Diém roi dzinh gizi tir GM sang AM III Nguyén IV D510 V Phéi hqp hai b§1déngthL'1"c déng béc nguqc chiéu v1 Phu'o'ng philp Qhuém héa bét aging lhtrc ba bién $5 VII Bit ding VIII B5 dng lhfrc déng béc cht1"ucén1ht1"c IX Be‘1td:'ing lhirc khéng déng bf1c_ X D510 XI Diém XII Phu'0'ng phzip cim bémg XIII Ky 1y aésng béc l1'Ong bél déng lhtrc biét héa dual vé bit dflng lhfrc d5ng can bfac thfrc déng bgic dang céng mgm biél héa b§td§1ngthL'rc khéng déng béc 1"0'i khéng déi xrng thuét tzich phn thirc I aim vil dzinh gizi mĐu $6 XIV ve dÂp diém ro'i bi: XV M61 XVI Céc bili tgip dimh cho ban dQc tu giai bi tozin ắmg ma-¢ chn lc_>c _L'1'ng lu''ng dung diém ro'i gizic All — GM Chlrovzg cAc I: Nhng thlir c6 ziié §1.1.B/QT BANG THUC AM - GM vién kinz cumzg bt ring DANG BIEU DIEN BAT DANG THU’C AM Dgmg tozng qzuit: Giél sir D2_1ng (11 ,a2, a” IE1 — '—a 3"‘ - GM n s6 thuc khéng ém, dé ta cé: D:_mg I +42 + -+51” >,,/a H a, cz] k +a3 + +u” 211.4’/(z,a3 cz” ¢ /I H_é qud: ' Néu a, +512 a, N H 2ala3 aH II ‘ =a3 = =a” 20 l S + +a” =5 const thi Néu ala2 a” = P const thi Min(ai + Cdc trzr&ng h_0‘p dgic \7| S Mc1.\'(ala3 a”)= —’ xa'.y /1 Mi D2_1ng fl! +a, + +a” ~ ' Déng thfrc xziy l7 , = =c1” =— n + a” ) =n.Q/F xay a2 z1| 1:23 al =a3 = =a” = ’\'/F biét n n=2 Biéu kién Va,b2O Va, Lg-b 2\/E Ma +2 +£ 23/ubc Mwrbic +d 2%‘/abcd a+b+c'23.§/% a+b+c+d24.\*/abcd n=3 b, 12:4 (‘Z0 Va, 12, W120 Dang Dang a+b22.\/ab D2_1ng3 K-Lgbj Zab %Z+1;+Cj> Za/90 %1+bZC+dj Zabcd Déubéng a=b a=b=(" a=b=c=d Binh lu:§n: Khi chtrng minh ' ding thirc, néi chung ta rét it gép céc bit ding thL'rc c6 dang cén déi, déy dfl nhu caic dang duqc phzit biéu 1y thuyét mi thuivng gép czic bét ding thrc cé mét vé phtrc tap, mét vé rL'1t gqn Cflng giéng nhu‘ chirng minh déng thL'rc ta pheii dnh gié ttr \/ n+1 —c](p 11- ‘+p n~Z q+ +q n» I1 H —q ‘H : [(p!1_qpn—l)+(pn_qZpn—Z)+-“+(pn_([n—|p)+(pn_qn):’ H : [:p/1~I +pn—Z(p+q)+m+p(pn~Z +17/1—3_(]+m+qI1—Z)+(pn~I +p!1—Zq+-uq/2-1 M 12+ - ’ Nhu'v2_?1y(2)C1‘u' MinS = a -\ INguyén nluin sai lm: MinS =2 21 =i=l a méu thugm vé"i gizi thiét a IPhzin tich vd tim téi ldi girii: Xét bién thién cua a.l du doain Min dé v21 S a 113'456{7‘s}9’101112' L 1 a S 33 L l 1 1 1 44 55 66 1 77 giém dc} 11 i @1111 nén a cémg =3 1111 s 111113111 gié Min Do h n ding 11111 A au, nen 11ép @1111 1! t2_11 S6 11 S = L30 S6 V21 (ll) OL dzglt aé vi‘ 11% LL'1c ‘a aé 11,11 "Biém 1111": t2_1i 11 111 a cng S nhO 30 1.3@_‘ Sé 30 nhung dé téng cimg "Die§m r0‘i: a = v51 diéu 1