February 22, 2015BT TRONG THI I HC 1L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. Bt ng thc trong i hc Cu 1: (Din n Ton ph thng) Cho, a bv c l cc s thc dng. Tm gi tr nh nht ca: 2( )( ) 4( )a b c ab bc ca bcPabc b c+ + + += ++

Hng i: (Nhim v ca hng i l gip bn hng t duy v d on du bng) Vic u tin ta s d on du bng ca n. i vi bi ny, thng thng du = xy ra khib c = . Khi 22( 2 )( ) ( 2 )( 2 )1 1 2 5 1a b ab b ab a b b a a bPab ab b a+ + + + + | |= + = + = + + + |\ .

n y ta c th d ona b = (theo Cauchy). T y ta s bit ca b c = = Gii:( )2 2( )( ) 4 1 1 1 4( ) ( )a b c ab bc ca bc bcP a b cabc b c a b c b c++ + + | |= + = + + + + + |+ +\ .

Ta dng Bt ng thc ph:( )( ) a bx y ax by + + > + ( )( ) 2 2 a bx y ax by abxy ay bx abxy + + > + + + >(lun ng theo Cauchy).Du = xy ra khi a xb y=. Tip theo ta s lng biu thc( ) 1 1 1a b ca b c| |+ + + + |\ . vo BT ph trn, v iu u tin l ta s kh i na . ( ) 1 1 1 1 1 1. ( ) 1b ca b c a b ca b c a b c bc+ | | | |++ + + > + + + = + ||\ . \ . n y ta kim tra, du = xy ra khi 211 1( )a a bcaa bcb c b c a b cb c= = =+ + ++. iu ny i ng vi hng vch ra la b c = = . February 22, 2015BT TRONG THI I HC 2L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. ( bc ny ta c th lm nh sau ( )( ) ( )2 21 1 1 ( )1 1 2 1b c b ca b c b c b c b ca b ca b c bc bc a bc bc bc+ ++ + + + | |++ + + = + + + > + + = + |\ .) ( )224 1 11 1 4 ( ), 0;2b c bc bcP t f t tt b c bc b c+ | ( > + + = + + = = e

(+\ + 332 2218. 11 8 12'( ) 8 0 ( )12tf t t f tt t= = s = nghch bin trn 10;2| (

(\ 1( ) 42f t f | | > = |\ . . Du = khi 12t a b c = = = Vy4 MinP =khia b c = = . Cu 2: (nguoithay.vn)Cho, a bv c l cc s thc dng tha mn 2 2 2( ) 4 4 a b c a b c + + + + = . Tim gi tr ln nht ca ( ) ( )2 21a b c b a cPa c b c c+ += + + + . Hng i: Vi bi nh th ny, ta s mp m on raa b = . Khi , gi thit s l2 22 2 a ac c + + =( ) ( )2 22 21 1 1 1 12 ( ) 4 4 4 4 12 2a a c a a cP a a c c ca c a c c c c c c+ +| |= + = + = = + + s |+ +\ . (cauchy 3 s) Du = khi 1, 12c a b = = = Gii: Cha kha bi ny chnh l bt ng thc ph ( )22 2x yx ya b a b+s ++

y l bt ng thc u tin m bn nn hc nu mun chinh phc 10 im thi THPT Quc gia. N c rt nhiu tn gi, mt trong s l Schwarz. Cch chng minh BT ny c rt nhiu trn mng, cc bn c th ln google search. V l t nc trn con ng hi nhp nn cch bn hy t trang b cho mnh cc k nng, v mt trong l k nng tm kim thng tin. Ti ch xin ni ra du bng ca n. February 22, 2015BT TRONG THI I HC 3L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. Du = xy ra khi x ya b= T y ta s c 2 22 2 2 21 1 1 1 1( ) 4 4 4 4 12 2b aP a c b c a b c a b c ca b c c c c c| | | || |s + + + = + + + = = + + s |||\ .\ . \ .

Du bng xy ra khi 1, 12c a b = = =Cu 3: (diendantoanhoc.net) Cho, x yvzthc dng tha mn 2 2 26 4 ( ) x y z z x y + + = + . Tim gi tr nh nht ca ( ) ( )2 2 3 32 2x y x yPzy x z x y z+= + ++ +

Hng i: Qua hai cu trn, chc chn cc bn d on cx y = T gi thit suy ra( )( 3 ) 0 x z x z = . n y ta s th 2 trng hpx y z = =v3 x y z = =ln lt vo P. Thy khix y z = =th P c gi tr nh hn. Vy du bng khix y z = =Gii: thy y l BT l thun nht (c ngha l ng bc)nn ta s s dng php t nh sau: t,x ya bz z= =(t y hng i ca chng ta u quy v1 a b = =) ( )22 2 2 22 2 22 214( ) 6 6 4 46 4 ( ) 6 024( ) 2 6a b a b a b x y x yx y z z x yz z z za b ab+ = + + > + ++ + = + + + = + > + 2 a b + > v2( ) 3 a b ab + > + ( ) ( ) ( ) ( )2 2 3 3 3 32 22 2 2 21 1x y x y a bP a bzy x z x y z b a a b+= + + = + + ++ + + + n y, ta thy s tng quan gia ( )321ab a + v ( )321ba b +. Nhim v by gi ca chng ta l kh mu hoc lm cho chng cng chung mu s. By gi ta 2 chia ra hai hng. Hng 1: Kh mu. Ta thy t l bc ba, iu lin tng cho ta cauchy 3 s. V ta c th kh theo hai cch nh sau: February 22, 2015BT TRONG THI I HC 4L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. Cch 1.1: Ta c( )321 38 8 41a a ab b ab a+ ++ + >+(lu khi1 a b = =th ( )321 18 8 41a a ab bb a+ += = =+nn ta chn nh th). Tng t ( )321 38 8 41b b ab a ba b+ ++ + >+ ( ) ( )| |3 32 22( ) 2( ) 3 12( ) 1 14 4 21 1a b a ba b a b abb a a b+ + + + > > =+ +

Suy ra 2 21 1 122 2 2 2a bP a b+> + + > + > + Cch 1.2: Ta c ( )( )3 223 222 1 3;4 16 412 1 34 16 41a b a a ab ab a b b ba b+ ++ + >++ ++ + >+( ) ( )( ) ( )3 32 22 23 1 1 3 1 1 1 1 1( ) ( ) 4 4 6 ( )8 16 8 8 16 8 8 4 21 1a ba b a b a b a b a bb a a b + > + + = + + > + + >+ + Suy ra 2 21 1 122 2 2 2a bP a b+> + + > + > + Hng 2: Lm cho chng cng chung mu s ( ) ( )3 3 3 3 3 3 3 32 2 3 32 2 54( ) 27( )2 ( 1)( 1) 2 (b 1)(b 1) (2 2 2) 4( 1)1 1a b a b a b a bb a a a a b a bb a a b+ ++ = + > =+ + + + + + + ++ + Mt khc: 33 32( )27a ba b++ >(BT ny rt d dng chng minh bng tng ng, nhng bn hy th chng minh theo hng khc nh, mi bi ton ln u cn 1 bi ton nh th ny) ( ) ( )3 33 33 32 1(3 3 ) (2 2 2)4 4.27 27.4 27a b a ba b a ba b+ + ++ + + + > = > =February 22, 2015BT TRONG THI I HC 5L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. T ta suy ra ( ) ( )( )( )33 32 2 32 127.12721 1 4 1a ba bb a a b a b+ ++ > =+ + + + Suy ra 2 21 1 122 2 2 2a bP a b+> + + > + > +Cu 4: (Boxmath) Cho, a bvcl cc s thc dng tha3 ( ) a a b c bc + + = . Tm GTNN ca b cPa+= Hng i: Ta d onb c = . Khi , ( )23 ( 2 ) 3 2 3 a a b b b a + = = + Vy du bng khi ( )3 2 3 b c a = = +Gii: Ta thy y l BT thun nht. t;b cx ya a= =(t y ta s xoay quanh3 2 3 x y = = +) Theo gi thit: ( )23 3( ) 6 4 34x yx y xy x y++ + = s + > + 6 4 3 P x y = + > + Du = khi ( )3 2 3 3 2 3 x y b c a = = + = = +Cu 5: (THTT) Cho cc s thc dng,, x y ztha 2 2 21 x y z + + = .Tm GTNN ca 2 21 1 2 31Pzx xy y xy= + +++ +

Hng i: Ta d on du 2 22 1 x y x z = = .Thay vo P ta c 22 2 311Pzz= ++

Gii:Theo gi thit( ) 0;1 z e ( ) ( )( )2 2 2 2 22 2 441 1 2 3 2 2 3 21 124Pz zx xy y xy x xy y xyx xy y= + + > + >+ ++ + + ++ + February 22, 2015BT TRONG THI I HC 6L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. ( )2 2 22 2 2 3 2 2 2 3 2 2 3( )1 1 11 2fzx y z z zz x y= + > + = + =+ + + + + ( )( ) ( )22 22 2 3 2 3'( ) . 0(1 ) 1 1 1 1 1 1 1z zf zz z z z z z z z| |= = =| |+ + + + \ .

( ) ( )3 2 2 3 211 1 3 3 3 1 3 3 (2 1)(2 3 3) 02z z z z z z z z z z z z z += + = + + = = Bng bin thin: Z 0121 Theo bng bin thin,

8 33MinP = . Du bng xy ra khi 3 1,2 2x y z = = = f(z) 0 f(z) 2 2 3 + +8 33 Lu rng s nhiu bn gp kh khn khi tm 2 12 2 3lim11xzz| |+ | |+\ .. lm tt thi THPT Quc Gia, ti bit cc bn s phi chn lc nhng th cn hc, v d nhin phn gii hn lp 11 s b b qua, nn gp nhng trng hp th ny, cc bn c lm theo cm tnh, c tng tng, z cng tin ti 1 th 221 z cng ln.Nn ta c t tin m cho 2 12 2 3lim11xzz| |+ = + | |+\ . Cu 6: Cho cc s thc khng m,, a b ctha1 a b c ++ =v khng c hai s no ng thi bng 0. Tm GTNN: ( )( ) ( )1 11 3( )( ) (c )P c a ba b b c a a b= + + + + ++ + + +

Hng i: Ta d on du = khia b =khi 2 1 a c + = Khi ( ) ( )( )( )( ) ( )( ) ( )221 1 4 41 2 3 1 4 3 42 2 1 1 1P c a c c c ca a c a a c c c c= + + + + = + + = + ++ + + Gii: Theo gi thit| | 0;1 ce February 22, 2015BT TRONG THI I HC 7L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. ( ) ( )1 1 11 4( ) (c )P c ca b b c a (= + + + (+ + + (thm 1 ln na, ta thy c sc mnh ca Schwarz) 2 21 4 1 4. 3 4 . 3 41 2 1 1c c c cc a b c c c> + + = + + + + +2243 4 ( )1c c f cc= + + =

( )( )( ) | |228' 2 1 1 0 0;11cf c c cc= + + > e

Suy ra hm s nghch bin trn| | ( ) ( ) 0;1 0 8 f c f > = Vy8 MinP = khi 1, 02a b c = = = Cu 7: (nguoithay.vn) Cho cc s thc dng,, x y ztha 2 2 25( ) 9( 2 ) x y z xy yz zx + + = + + . Tm GTLN ca( )3 2 21 xPy zx y z= ++ +

Hng i: D ony z = . Khi ( ) ( )2 2 25 2 18 4 x y xy y x y + = + = Do 32 1216.Py y= . Kho st hm s ny ta thy Pmax khi 112y =Vy du = khi 1 1,3 12x y z = = = Gii: T gi thit, ta s hng n cc i lng i xng (yzhocy z +hoc 2 2y z +)( ) ( ) ( )22 2 2 2 2 25( ) 9( 2 ) 9 5 5 18 2 x y z xy yz zx x y z x y z yz y z + + = + + + = + > +( )222 9 ( ) 5 0 y z x y z x + + + >(y l bt phng trnh ng cp) ( ) 2 y z x + > l nhng g ta c c t gi thit, v ta khng th quy v n x kho st hm s. Do ta s quy v 1 n khc. V cha kha chnh l ny z + February 22, 2015BT TRONG THI I HC 8L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. ( )( )( ) ( ) ( )33 2 3 3 2 22 1 1 4 1 14 ( ), 02727 272y z x tP t f t ty z y z y zx y z y z y z y z+= s = = = = >+ + ++ + + + + 2'( ) 4 0 69tf t t = = = Bng bin thin T0 6+ Theo bng bin thin, 16 MaxP = . Du bng xy ra khi1 1,12 3y z x = = = f(t) 0 f(t) 0 + 16 Cu 8: (toanhoc24h) Cho,, a b cl cc s thc dng tha mn( )24ac b a b c + = + + . Tm GTNN ca ( )( )22 228 8 11a c bbPa ca b c+ + += +++ +

Thot nhn biu thc ca P tht phc tp, nhng hy nhn k, cc biu thc s phn ra 2 phn, 1 phn i xng gm (a,c). V 1 phn khng i xng gm b. Nn cc bn ng qu lo lng. Ct li bi ton s nm nhng th . Hng i: Ta tip tc d on nh nhng bi trna c = . V cc bn hy lm th cc bc tip theo. Ti tin ti y cc bn c th vch ra hngi trong u m khng cn ghi ra hng i trn nhp. Gii:( ) ( ) ( )2 2 222 ( ) 4 2( ) 1 a b c b b a c a c ac b a c b b a c + + = + + + + = + s + + + + s(bc ny tng t cc cu trn, ta s quy cc biu thc vachoca c +hoc 2 2a c + . V bn ng ngi, c th ht 3 ci , th no cng c ci gn nht) V 1 t ta s cn th ny2( ) 1 a b c b + + s +n biu thc P: ( )( )( ) ( )( )( )( )2 2 2 22 22 2 28 8 1 4 1 8 11 2 2 2 24.1 11 1a c b a c b bb b bPa c b ba b c b b+ + + + + + += + > + > ++ + + + + t 11btb=+ (ci ny kh m tm iu kin cht ca t, nhng ta ch cn iu kin t>0 l ) February 22, 2015BT TRONG THI I HC 9L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. 22( ) 8 P f t tt> = +22 1'( ) 16 02f t t tt= = = Bng bin thin: T 012 + Theo bng bin thin, 6 MinP = . Du bng xy ra khi1 1 1,2 3 6t b a c = = = = f(t) 0 f(t) + + 6 Cu 9: (nguoithay.vn) Cho,, a b cl cc s thc dng tha mn( )( ) 2 4 a b b c bc + + =v3a c > Tm GTNN ca 2 22 a bPac+= Hng i: Nh nhng nh hng cc bi trc, ta s on cb c = . V bi ny cn d hn cc bi trc l c thm d kin3a c > nn ta c th gi nim tin du bng s xy ra khi3a b c = = Nhn vo gi thit v c biu thc P ta u ngh ti php t n ph,a bx yc c= = . Gii: T gi thit ta c 13ac > ( )( ) 2 4 2 1 4 2 2a b c a a ba b b c bcc c b c b c| || |+ + = + + = + + = | |\ .\ . t,a bx yc c= =13x >V 22 2 1 1 12 21 3 3 2x y yx y x yy y+ + = = > s s+

February 22, 2015BT TRONG THI I HC 10L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. ( ) ( ) ( )2 2 22 2 22 23 2 32 1 12 2 2( )1 1 1 1y y yy y y yy y y y yP x fyx y y y y + + + += + = + = = =+ n y o hm hi vt v.( )( ) ( )( )( ) ( ) ( )( )22 2 22 2 2 22 22 21 3 1 2 2 2 3 2 31 9 4 3 2 3 2 31 1'( ) 0 ;3 21 1y y y y y yy y y y y yf y yy y ( + + + + + + + ( = = > e ( Vy( ) fyng bin trn1 1 1; ( ) 13 2 3y fy f ( | |e > = | ( \ . Du bng khi 1 13 3 3cy x a b = = = =T y, ta thy hng ca ta vch ra l khng ng, y l mt bi ton kh hay, cho ta thy c s bt bin ca BT, by gi cng bi trn, cc bn hy th tm GTLN xem! Cu 10: (H Vinh) Cho cc s thc dng,, x y ztha( )2 2 22 3 x y z xy x y z + + + = + + . Tm GTNN ca 20 202P x y zx z y= + ++ ++ +

Hng i: Bi ny thc s l 1 bi kh kh, y khng phi BT na i xng, v ta kh m on c im ri ca n. Nhng kh ch khng phi l khng th. k, ta s thy s xut hin ca 2 biu thc 1x z +v 12 y +. iu ny gip ta lin tng n vic kh mu v rt thy, h s ca chng u l 20, nn rt c th chng bng nhau. Khi chng bng nhau th( )2 2 2 2 22 2 3 2 1 0 x z y x y z xy x y z x y x y + = + + + + = + + + =V 402( ) 2 P x zx z= + + +. Kho st hm s ny th hm s s c GTNN l 26 v4 x z + = 2 242 1; 2; 32 1 0x zx z y x y zx y x y + = + = + = = =+ =

T ta bit c im ri ca BT. February 22, 2015BT TRONG THI I HC 11L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. Gii:( ) ( )22 2 2 23 2 x y z x y z xy x y z + + = + + + = + +(d thyx y z + =nn ta lm tip) ( )22x y z + +> 6 x y z + + s ( )( )( )420 20 40 40 22 22 2 2P x y z x y z x y zx z y x y z x z y= + ++ + > + ++ > + ++ + + + + ++ + + n y th n gin, c nhiu cch s l, v ti s chn cch gn gi vi cc bn nht. t2 2 2 t x y z = + ++ s ( )240 22 P f t tt > = + ( )32 240 2 2 40 2'( ) 2 0 22tf t t t f tt t= = s s nghch bin trn (2; 2 2( Vy( )( )2 2 26 f t f > =.Vy26 MinP =khi1; 2; z 3 x y = = = Cc bn hy lm th bi ny: (moon.vn)Cho cc s thc dng,, x y ztha( )2 2 22 3 x y z xy x y z + + + = + + . Tm GTNN ca 54 547 5P x y zy z x= + ++ ++ + +

Cu 11: (Tilado.edu.vn) Cho,, x y zl cc s thc khng m tha mn0 xy yz zx + + >v 23 x xy yz zx + + = . Tm GTNN ca 16 25 x y zPy z z x x y= + ++ + +

Hng i: Thot nhn, ta thy y l bt ng thc thun nht, v th khng cn suy ngh nhiu, ta s dng php t n ph.Gii:0 x = : 23 0 0 x xy yz zx yz xy yz zx + + = = + + = (v l) February 22, 2015BT TRONG THI I HC 12L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. Vy0 x >t;y za bx x= =(cc bn c th t khc ti),0 a > 223 1 13 1 3 1 31 3y yz z bx xy yz zx a ab b a bx x x b + + = + + = + + = = >+ ( )( )( )2 216 3 1 25 116 25 1 16 25 11 1 4 1 41b bx y z a b bPy z z x x y a b b a b bb ++= + + = + + = + ++ + + + + + + + n y ta s xt hm( ) fb , nhng thc phc tp o hm, ta hy d on du bng. trn kia ta thy 13b >nn ta hy th vi 13b =th khi 343P = ( )( )( ) ( )( )2 2 2 212516 3 1 25 1 16 3 11 1 34 334 1 4 4 1 4 31 1bb b bb bPb b b bb b| | | + + + | |\ .= + + = + + + + |+ + \ . + + ( )( )( )( )( )2 2 2 216 3 1 25 3 11 34 16 4 25 343 3 14 1 12 3 4 1 12 31 1b bb bbb b b bb b ( + + | |= + + + = + +( |+ + \ . + + ( ( )( ) ( )4 3 22225 138 320 710 265 34 343 13 312 1 4 1b b b bbb b b+ + + = + >+ + (v 4 3 2125 138 320 710 265 03b b b b b + + + > > ) Vy 343MaxP =khi 1, 0 0, 33b a y x z = = = = Bi ton ny i hi s bin i cn thn, cc bn hy th theo con ng kho st hm s xem. Bit u s nhanh hn. Cu 12: (THTT) Cho cc s dng,, a b ctha 2 2 214 a b c + + = . Tm GTLN ca ( )( )2 2 2 244 5 33 28 7 ( )a caPa c a bc a b ca b+= + + + + + ++

Hng i: Bi ton ny ti nh gi rt kh, to ra bi ton ny, chc hn tc gi phi to ra im ri trc ri mi nu ln tng. Cn i vi chng ta, im ri vn cn l du chm hi. Nhn xt thy cc biu thc trong P c chia thnh hai nhm l February 22, 2015BT TRONG THI I HC 13L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. +Nhm 1: 24 37 ( )aa bc a b c+ + +c cha c 3 bin,, a b c +Nhm 2:( )( )2 2 2453 28a ca ca b++ ++ch cha 2 bin s Ta s x l nhm 1 trc. By gi, ta lng ghp gi thit vo P xem th th no. ta s thy ( )23 12( ) a b ca b cs++ +. Du = xy ra khia b c = + ( )2 2 2 2 224 8 87 3 23a a aa bc a bc b ca b c= =+ + + + ++ + Nh cc bi trn, ta cn a chng v cng dng mu s vi biu thc ( )212a b c + +v phi ch a b c = + nn ta lm nh sau: ( ) ( )( )2 2 2 2 2 28 8 8 42 23 2a a aa a b c a b ca b c a a b c= s =+ + + +(+ + + + + V ta chia thnh 2 nhm nn ng ngha vi vic ta s bin i P thnh hai hm s c lp (kiu nh( ) () P f x g y = +) nn ta cn tm ln lt GTLN ca cc hm s thnh phn. Ta s tm GTLN ca ( )22 2 4 3 4 12 1 1 1 1127 ( ) 6 3 3aa bc a b c a b c a b ca b c| | > = + > |+ + + ++ + +\ . ++ Ti y, ta p bi ton vo du = khi2 2 236 2114a b c aa b c bca b c = + = + + = = =+ + =. Gi ta s i gii quyt bi ton. Gii: Ta c ( )23 12( ) a b ca b cs++ + v ( ) ( )( )2 2 2 2 2 28 8 8 42 23 2a a aa a b c a b ca b c a a b c= s =+ + + +(+ + + + + Suy ra ( )22 2 4 3 4 12 1 1 1 1127 ( ) 6 3 3aa bc a b c a b c a b ca b c| | > = + > |+ + + ++ + +\ . ++ Tip tc, ta li c: February 22, 2015BT TRONG THI I HC 14L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. ( ) ( )( )( )( )2 222 24 4 43 3 1 3 2828 1 3 284 4 3a c a c a ca ca c a c+ + += s+ + | |+ + + + + |\ .(bunhiacopsky) ( )1 625 25a c s + +. (y l BT tip tuyn). ( )( )25 2 325 5a ba b s + +. (Tip tc ta s dng BT tip tuyn. bit thm v BT tip tuyn, cc bn xemhttp://diendantoanhoc.net/forum/index.php?app=core&module=attach&section=attach&attach_id=6785 ) T ta c ( )( )( )( )2 2 2 2 2 22 2 23 2 14 5 3 2 9 9 13 28 25 25 25 25 5a b ca c a b ca ca b+ + + ++ + + s s =+ ++ Vy 815MaxP =khi3, 2, 1 a b c = = = Cu 13: (H Vinh) Cho,, x y zl cc s thc khng m tha mn ( ) ( )2 2 25 6 x y z xy yz zx + + = + + . Tm GTLN ca( ) ( )2 22 P x y z y z = + + + Hng i: Ta d ony z = . Khi ( ) ( )2 2 225 6 52x yx y z xy yz zxx y

=

+ + = + + = Khi 25x y =th 2252 32P x x = (kho st ta thy ra xu nn ta c cho l n sai i :3 ) Khi2 x y = th 222xP x = Kho st hm s ny th c 3min2P = khi 11;2x y z = = = Gii: ( ) ( )( )( )( )( ) ( )2 222 2 2 2 255 6 6 5 5 6 6 5 6 04 2y z y zx yz x y z y z x x y z x x y z y z+ += + + + s + + + + + s ( ) ( ) ( ) ( )( )225 5 0 5 0 x x y z y z x y z x y z x y z ( ( + + + + s s

5 x y z x s + s February 22, 2015BT TRONG THI I HC 15L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. ( ) ( )( ) ( ) ( )2 22 2122 1 12 2 2y z x y z y zx y zP x y z y z + + +( ++ + + = + + + s = + sVy1 MaxP =khi 11;2x y z = = = . Bi ny vn cn rt nhiu cch khc, cch bn hy th tm 1 li gii khc cho bi ny nh! Cu 14: (moon.vn)Cho,, a b cl cc s thc dng tha mn 2 2 21 2 2c a b= + . Tm gi tr nh nht ca 2 2 2a b cPb c a ca b c= + ++ ++ +

Hng i: Bi ton rt n gin ta c th on c2 a b c = = V hng i chc chn phi l t n ph. Gii: 22 2 21 2 2 1 1 1 1 1c a b a b c a b| |= + > + > + |\ . t ( )21 1; 1 44x ya bx y x y xy x yc c x y+= = + s + s s + >Ta li c ( )( )222 2 2 22 42 1 1 14xy xy xyx yx y xy x y xy xy x y xy> >++ s + s + +s s V2 2 2 2 2 2 21 11 1 21 1 1 11 1a b c x y x yPb c a c y x y xa b c x y x y| || |= + + = + + = + + + + ||+ + + + + +\ . + + + + + + \ . ( )( )( )( )2 24 11 1 4 41 2 21 1 24 4x yx yx y x yx y x y+ + | |> + + + + > + |+ + + ++ + \ .

t( )24 4 44 22 4tt x y P f tt t+= + > > = + + February 22, 2015BT TRONG THI I HC 16L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. ( )( )( )( )( )( )22 2 2 22 2 24 3 14 3 4 1 2'( ) 4. 0 424 4 4t tt t tf t ttt t t ( + + (= = = > > (+ ( ) ( )543f t f > =Du = khi2 2 x y a b c = = = = Sau y l mts bi ton m ti thy hay v ng lm: (toanhoc24h.blogspot.com) trch trong thi th ca thy Khi. y l ngi thy m ti rt knh trng. Trong thy ti thy c s am m Cu 1: Cho,, x y z l cc sthc dng tha mn4 4 x y z + = . Tm GTNN: 2 22 4 2z x y zP z y zx y= + + + ++

Cu 2: : Cho,, x y z l cc sthc dng tha mn3 2 3 x y z + + = . Tm GTLN:2 2292 31x yP z zxy+= + + +

Cu 3: Cho,, a b cl cc s thc dng tha mn 2 2 2a bc b c + = + . Tm GTLN: ( )36 2 2 2 23 b c aPa c a bb c= + ++

Cu 4: Cho, x yl cc s thc dng tha mn3 7 x y + = . Tm GTNN: 2221 2 31x xyP xy y xy+= + + +

Cu 5: Cho,, a b cl cc s thc dng. Tm GTNN: ( ) ( )( ) ( )2 2 42 4 216 a ab aPb ac c a a b c a= + ++ + + +

Cu 6: Cho,, a b cl cc s thc dng tha mn( ) 4 abc a b c ++ = . Tm GTNN:( ) ( ) ( )2 21 88bcPbc b ca b a c= + ++ +

Cu 7: Cho,, x y z l cc sthc dng tha mn1 x y z + + = . Tm GTNN22 x y zPx yz y zx z xy+= + ++ + +

February 22, 2015BT TRONG THI I HC 17L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. Cu 8: Cho,, x y z l cc sthc dng tha mn 2 2x y z = + . Tm GTLN( )( )( )( ) 22 2 22 21 2311 1xy zzPx y zz z+= ++ ++ + Cu 9: : Cho,, x y z l cc sthc dng tha mnx z > . Tm GTNN ( )222 32x zxz yPy yz xz yz x z+= + ++ + +

Cu 10: Cho,, x y z l cc sthc dng tha mn 2 2 23 x y z xy + + = . Tm GTNN 2 2 22 2 2x y x yPy yz z x x z+= + ++ + +

Cu 11: Cho, x y l cc s thc dng tha mn 2 21 x y xy + = + . Tm GTLN 3 33 3241 1 2xy x yPy x x y= + ++ + + +

Cu 12: Cho,, x y z l cc sthc dng tha mn 2 2 23 x z y xy yz zx + = + + + . Tm GTLN ( )( )2122xPxy y zy z= ++ Cu 13: Cho,, x y z l cc sthc dng tha mn. Tm GTNN ( )3 3 22 2 2.2 24x y x y zPxy y xz x yzx y| | += + + |+ ++\ .

Cu 14: : Cho,, x y z l cc sthc dng tha mn 2 2 22 x x z x + + = . Tm GTLN( )2242 1 1x z z xPx y yx y+= + + + ++

Cu 15: : Cho,, x y z l cc sthc dng tha mn 2 2 24 x y z + = + . Tm GTNN( )2 2 2 222 3 3 222y x xy x yPy zxy z+ = +++

Cu 16: : Cho,, x y z l cc sthc dng tha mn 2 2 22 x y z xy z + + = + . Tm GTNN( )( )232 2 2 2 2 2 2 28 x y zPy z x z x z y z| |= + + |+ + + +\ .

February 22, 2015BT TRONG THI I HC 18L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. Cu 17: Cho,, a b cl cc s thc dng tha mn( ) ( ) 2 5 a b b c bc + + =v2a c > . Tm GTLN v GTNN 2 2a bPac+= Cu 18: Cho,, x y z l cc sthc dng tha mn( ) ( ) z x z z y z xy + =. Tm GTNN 2 2 2304 4 12x y x y zPy z x zx y z xy+ += + ++ ++ + +

Cu 19: Cho,, x y z l cc sthc dng tha mn 2 23 x y z xy + + =. Tm GTNN3 316x y x yPy z x z z+= + ++ +

Cu 20: Cho,, a b cl cc s thc dng tha mn 2 2a ab b c + + = . Tm GTLN ( )( )22 2 23 2 2 361 12 24 2 3ab cPa bab c+ += + + ++

Cu 21: Cho,, x y zlcc s thc tha mn,, 1 x y z > v3 x y z + + = . Tm GTLN ( )2 22 2 214 1 4 5x yPx y xy z z= ++ + + +

Cu 22: Cho,, x y z l cc sthc dng tha mn 2 2 25x y z yz + = +. Tm GTNN ( ) ( )2 22 z y z x yPx y x z x z x y+ += + ++ + + +

Cu 23: Cho,, a b c l cc s thc dng tha mn( )24ac b a b c + = + + . Tm GTNN ( )( )22 228 8 11a c bbPa ca b c+ + += +++ +

Cu 24: Cho,, x y z l cc sthc dng tha mn 2 2 22 x y z + + =. Tm GTNN( )22 2 22 2 2 542 2 2xy yz zxPz x yx y z+ + += + + ++ + ++ +

Cu 25: Cho,, x y z l cc sthc tha mn,, 1 x y z > v( )2 2 26 2 x y z xy x y z + + = + + + . Tm GTNN February 22, 2015BT TRONG THI I HC 19L Anh Tun_Trn Ph-BRVT_0966194630 Email: smart.mst@gmail.com Em no thang ph H Ch Minh c nhu cu mun anh l gia s th lin lc vi anh nha. 21 11 1x y x yPy z x z z+ + + | |= + +|+ + \ .

Cu 26: Cho,, x y z l cc sthc dng tha mn4 4 x y xy xyz + + =. Tm GTLN ( ) ( )( )32x z y z zP zx y z y x zx y+ += + + + ++

Cu 27: Cho,, x y z l cc sthc dng tha mn 2 2 22 x y z xy yz zx + + = + + +. Tm GTNN ( )( )( )2 2 2 2 22 2 23 82 2x x y z y zPy z xy xzx y z+ + += ++ + ++ +

Cu 28: Cho,, x y z l cc sthc dng. Tm GTNN ( )( )2222 2z xyx yPx z y zx y z= + ++ ++ +

Cu 29: Cho,, x y z l cc sthc dng tha mn 2 3xy z z + =. Tm GTNN( ) ( )( )32 4216x yz zPzx y x y+= + + +

Cu 30: Cho,, x y z l cc sthc dng . Tm GTNN( ) ( ) ( )32 2 28z x y x yPzx y y z x z| |+= + +| |+ + +\ .

Cu 31: Cho,, a b cl cc s thc dng tha mn iu kin2 a b c ++ = . Tim GTNN 22 21 1 2 2 6424b c a bPa b bc ca a b a| |= + + + + + |\ .