Chlrrrlzg IV 1\‘l117''''ng rién /cim clrrmg Irong I211! (hing I/1121'''' /zifn 11111 5 19 §1s PHU''''()''''NG PIIAP DON BIBN (MIXING VARIABLE) T6111 tét n51 dung 1 11511 1 511 114% II B511d5111g1h11''''c 11 1 111611[.]
IV: Chlrrrlzg rién /cim clrrmg Irong 1\‘l117'ng I211! (hing I/1121' §1s PHU'()'NG PIIAP DON BIBN tét T6111 II 5-19 11111 (MIXING VARIABLE) dung: n51 11511 1-511 /zifn 114% B511d5111g1h11'c 11:1 111611 \0'i @111" 111 11111 1111 gi1111"1 1115111151 \11'11g 1.B1§1d1'111g1l11'1'c 1\'11511g 111151110511 Z D511 bién D511 111511 111'5'11g III D511 IV B211 111'111g 111110 v v11.1)611 bién 111111111f.\~1v> 11i.\1'1\'1 11151113111) 111511 11'0ng111n1 gizic 11-1e11 11611 1111-i~11 111j1¢ 111111 15ng D11111 1)‘ 11511 111511 XIII 111é11 115111 1111111 101111 1111511 \'0'i \11f~1 XII 116111 111611 1111 111511\"§g11111'111"u11g1111111 EMV \'0‘i X1 d11'5'c c1_1'c 111 11111 h§11g11Z1111 151 bién bng quy D511 E.\11V Z g111c 111111 B5111.111g1111'1~1~ 111é11 11-11111111 1111 VIII D5n D511 11'011g bién \'0'i 1111 s.\1\1> VI D5n hién X gizic 111511 b§111g 1\'§111uf1111Z1111 911111 1y IX kién \/0'1 1151 112'11g 1h1'1'c 00111511 \Ih111 \*1‘1 11511 1111 (G.\IV1 1111111 15p DAT VAN DE ' C1ic 1117111‘ c1ic'b111 1h1"1n diing mén kinh d11 1111511 111111‘ c1'111c111'111g 111 x1'mg h11y hon 1151 1111'1'c nhuu M51 vi 1r011g 111511 111 11111 d€111g1h1'1‘c s1111xé1y1"11c1é1ng1h11'c Bi1i 1.1 C110 1- ° S5 1u'5'11g c11c 11111'c $111 xziy 15111 1'11 11Z1)' c0 dy B511 11111111 111 11151 1111 211 Khi 111é 111511;; c110 vid1_11i€u T1‘011g b€11d1-“Ing C0 C15/111g \'i1c111'k11i \*= \1 \*i{.>c 111101‘ 1101 1111'1'c 111111 ‘\'= : b51 dzing xz1y chng 1h1'1'c 11510 11151 Ia c1ic bién $5 bng hgn \1'0'i /1 : 111 1-_ C0 11511 (1: 15111 Q0 101111 1111111 111% Q6 \a_\' c111111g 121111 111 11151 111 diéu 1u511 1i11 ring 111511 11111611 _\".;e 21.1-’+_1»1 + 1111'1'c 1111)-' 1111 c1?111 -P X61 x1"1)' 111111; 11151 11:11 11§111g 111111‘ 1151 x1'1'11g hién $5 1\'11511g 1'11 151 11311; 111111 11151 1111111 C15 1'51 c1111_\511 =9.1.('2() Chung minh ing: _k_ Q - -'+~!l~+~?Đ%\1Â1+/v+ 31> 1' vi) (‘zic han c(v1]1C' hiét mil liun chm déng lhirc xay ru =()(\'Z1c;'1_c dang hozin c6 gi O) dzfac nhu" diéu dfxc biét G6 121 thuc khéng fun nén bién hing 1Z1 hing c6 mél bién () ciwn du‘‘c Vi gia goi bién IE1 vi nfra 151 czic ban sé cbn gp nhfrng bait dfing lhirc mil du "='" xny nhiéu h0'n mét Chg (khéng tfnh czic hozin vi) Chdng déu 151 czic bnl ding lhirc dqp vii kh('>_ chng hgm nhu'm1 bin lozin rl n6i liéng sau dy: ° ~M©t diéu thL'1 T Bili 1.4 (Iran TST 1996) Cho i Phu'0'ng phzip dén bién duqc dél ru dé gizii quyél czic Bing xay tln1"6>'11g 111 lhco V + Tru'C>'ng hqp tét cé czic bién $5 bng la sc + Truimg "=" du'Q'c du'\‘.:) 2/ (/.I.I) f(.\'.>\".:)—‘/'(I.1.:):.\"+_\‘+:—3 Q/.\_w": —§ Bu'6'c 2: Chung mmh _/(1.1.:)= 21+ Thé1t\=c_1y _\~ _\< 1: N/Ty ; Tu > Q6: Z1 S \'+ _\- suy 1'11: 2/+:—3~{/.\1\":J:.\~+_\~—212() ‘ -3-"\/I"; 20 " lucéi _/'(1.z.:)2() 20 v \'é'i (Z1+;)' -2711-_2(1(1—_-A):(8r+;)2() (dung) Kétluzfm: _/(1-.v\~.:_)2/"(/.1.;)z0 Crich 3: Bu'6'c 1: K)’ Néu lhuz_31l chuén héu lhco tfmg d6i \'+_\‘+:=/\'>O thi(l)(_\'+>\'+:)‘2Z7.\1\': k} lhuql chmh dung dc don hm blen C(ich1.‘(1) _/'(.\"._\"_:):.\"+_\’+:—3-{.\j\‘: *V_\"._\*.:>() Bu'0'c 1: Chung mlnh: ‘/ (.\‘.>\'.:)2 [rung dgai lm_)'ng Ch 12'/lg I1 , I lix llmu hién hui l\Lrb'c ll‘L1‘O'c xél m (), _\" I xi Ihi diéu kién déng {hire 20‘ ' nén ta Chl phai xcm xél su lhuy déi cu;1_\"+_\'+ t Theo = \‘ \" , 1) A ' (*).khid(1(1) Q =1 \ » \' ~ Q \ hm) man urc du'(_v'c Bu'6"c3:Thav:=1—21u1c6:f(r_r.:):1—27z3(1—21)=(l—6z)(l~3r)' V0‘1Cl1eu k1¢n("-‘)lh1CIangLhu‘c 20 Z7.\‘_\": Thco \-_\': Q» _f(.\- II _/‘ (_\" _\" {hay déi cuu SL_l' , :1‘ , \\ dé :> (dpcm) 1' Q 2f:-1‘ Q \"= 6 hi 1hu)§1 phuc h0"i minh \"1'dg1 tI111'z~ don gian nén pl1u'=2(z1+1)l Dc dimg chung mmh _g>(11)2()bang kl1:10s1i1l1Z1m Lhco bién D€inglhL'1‘c xay ru vil chi 11:/1:010 11 : Gidi NĐul2+1*;Â() chun lwglc :>_r'11.1.1->21ằ \'Z1ch1 khi11=/2 111$! 1hL'1 hit dng qu;i1_ —9("_1 —1811~+(21‘ —13+8).T;1c(>: Bili 2.3 [Viét Nam TST 1996] Khéng giéunti11l11611g * ,_ . \ +-l(\/11+\/w) Â/201111)" _/'(11./>_1")2 _/'(\11/a J11/2.1) \/Đ1)'bi1iI0z'1n chL'1'ngminhxong D;'1ngIhL'1'c xa); ' nén chi cn 1" 1/=_/'(.(')—‘/’(\/_1",)—(\'11~\/w) ;_2(\11+\/>1 A'=(91)’ -(11 +2)' (21‘ -11 +14):-(11 -1): dévn /1 11 kh(mg m = (11: + 2)(/7: + 2)(1': + 2) —9(11/1+/¢1'+1'(13 \' @511 chL'1'ng 1:/%5 minh: +6/n'+u : —>u(/>+)2() , , Git! sir u = min{u./2.0} :> Z/>3 + 20: +8/7('+u: —5(1('l>+¢') = I/73 + Z0: +3/70-411: +5(/7~u)(—(')1 +\///2+(('—u): I (1 +/1 +0 = Chfrng minh +\f('+((l—/7): \/J3 ring: ‘ { C/'11?'ng D31 VT = f(_u./2.0) Gi2'1 sir :1 = {(1.110} minh D0 41+/>+ 111 Q6 \'Z1i 11111111 suy >111z1.x{ O x61 dé 111 1h0z1n1Z1111111111c_>i /1" g1:111 111111 /1 >0 > /1 111311 11511 101111 cf111g1hQz1n1Z111 vi: 1111+/1 1h1 caic 1h0z1111f111 111é1111 115 101111 11211 suu 151 111'1'\'11g d11"o*11g: (i) (1)d1'111g (ii) (1)d1'111g 161 111111 11./1.~: [3l@111gll11'1'c chtrng minh dL1\_)'C It I \Z1 Â/ xilo /M/././1\ J \'Ief1Ikhzic.lI1u\' +/n'+m)E ,(u+/1)‘ Tu)" nhién nilu czic ban thu' dén bién thco czic rt phirc tap Bili 2.9 I K 2% “* {rung binh thi czic ban [1100 sé thy viéc nily diéu kicfrn I + 3a/2r‘ 1" C11/rug minlz Dill + l K/>+'11g 1f\i cht'rng minh kl1(mg khzic hil gi xu \"O'i hl dflng lln'1'c dgli \(‘\ nén 0' d[1§ chx xin néu ru Ina}! \‘L1i \*1'd1_1 dé minh hQL1L‘l1U \"(rc mqnh C1111 phuvng pluip {rung dung ht dfmg um nix} Clzing Bi1i2.l0.ChoAABC l‘imgi;iu‘i ‘I +1-0.\:¢\)(l + 1xhonl1f/atclux: (l)df1ngl1u)' _/'(‘~\.B.C)2_/'(ii;[i.iJ§'5.(‘)= §’(C') : i’(3—r'0\C): (1+z@.|-,1 \(3'i 4 Béng lhirc xay 111 AABC déu V5)’ _/‘ :1(;_f Bf1i2.l1 Cho lam gifxc ABC kh (‘~\)Z)' (/>+0) [hi lfnh d g(/22) ~: x :> f(m) blen :> f(m) dong l— °z f(0):(T“) Bu'6'c 3: Ta céin chtrng minh f(O) =l1(I)=(-$1212’) A g(()) = + O :> f'(/21) () K - K + I I) 2‘-F L 2 Vie /(*1 h'(r)= 2/+1"+11+/1+1‘ /1+1‘ 11+/7, 1‘+11 /7+1" ("+11 11'+/J :(11+/2+1,-)(%+i+l—)2(11+/>+1~)?—1i:2 l2+1* 1‘+11 11+/1, (/>+1')+(1"+11)+(11+/7) b) hqp T1‘L1'(7'11g I I1" — Z la thu du'Q'c btdz'111g Df1)'iE1mc}1b£1it0zin khzi thd Z,‘ :2 11 1»~.~1»\5b+C >2 Z11 /< \ ~ A I11 khéng phai I51 , - ba1 to-an 111111 t ( hing $6 151 - ~ rm hay da nhit nhu'ng mang 116 U l1+1' ) l:_ai +( A -A /~ dang tren du'(_>’c \/11+/9 1r+11 AM — GM: 11h1eu ) +( , , A b 1"+11 ~ C 11+]; ) k 2‘ 1 tap ch1 toan hqc Mac du k cho chilng ta IO'i gizii O Y(i)j Zi k > O.11,1'),1'2 :Đ rit dÂp: 1z+12+1=11+/liÊ+b+C23-I11Êb+C)_ :>( )32 I> \J 2 b+c 11+b+1‘ -_:*1b+1: Bi 3.2 Cho Z “+b+" 111”+b+C Day cung \‘1>+1' :2(11+l2+1'):7 211 , ; , 2~§_ ta co bat dang 1hu‘c sau: c) Truong hqp 1* "T ;,—“- + \/?7- + suu: \"1Đ'i111â1l1Jigiz'1i do111git'1n s1'1'd1111g \-'1 Z\/11(b+1") 111 1h1'1c vii 11+ b + 1' ring: = 3.ChL'1ng minh / N‘ _ _ (11b)‘ +(lJ17)k +(1"11)‘ Smax - T (1) J1 Ch 1271 g minh Khéng sé du'Q'c D231 B511 quail c6 thé giéi 111a‘t 1611;; 11111 chn 1:211 v21 $11‘ /2 (viéc léiy 1‘ Ta kho 1h1'1‘c 111% $211 (1) f(111) suy f(111)=11 vdi me [0, _/"(111)2O g(111)=11 Chi can > () V/1z€(_O; I] hoéc f’(_m) = f(m)di1én di xuévng hozflc f(/21“) —O+ czi Z Lrut'i r A re ‘U/l’ 1i :> g(I) dbng bi€n ,— |\j—l I1'@n (1 :> tL'1'c 15 a Phn cén lai Thay 121 f(a.b.c')Zf(5_s.'(') vdi s':\/ab minh f(.s', ChL'1'ng s, ¢-) (ft =£ s“c‘ ° = , ’° xv; ' \\/‘)(l+("‘) Gié str (1219 '(f)=6 Zc Xét a——_ ham m+2+c') > I S6 v g(l)- _/'(ta.€.c) — —l92(c1—— [- ll vdi IE B/ilg Ta C61 t:r13+c1/J+b—;}v(1+c‘:) I“ \ /— Vi t€[\/-Il,1]nén g'(¢) 20 néu (fa +€+c) _ v :+a[2+;)—;J(l+=141111-321-I -s1» +s x HO'!1I1f1'£1l1I3+(.\'+I): >-1.\~=3+ +,_ +3+('~i w w ,/ PI1 rrmzg D7(1 Thay 1': 3- Zs \/"Z10 1u'u 151 S I = 3s — f'(I) — s' /7‘ +/< S 1‘ 1 :£:i:().757924546 c111'Q'c cho bZ1i 105111: = 3.T1n1g111t1'1 161111h§lcuz1bi§11th[1‘c: 1' S(c1.!2.1c'):\[i+w#+ 41' +/1' Max ('1 \/_§1y c6 Lhé zip dung vi1(1+/J + O -4-, 3+(~'-=\ 3+(" Zm-(11+\-1(3+(':1