August 1, 2015Sng to BT 1Creation Cu 1: Cho, , a b cl cc s thc dng tha mn( )24 3 ac b a b c + = + + . Tm gi tr nh nht ca ( )( ) ( )22338( )2 2a cbPa cb a c a b c++= +++ + + +

Hng i: 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 . V ta d ona c = .Gii:( ) ( ) ( )2 2 222 ( ) 4 3 3 2( ) 3 a b c b b a c a c ac b a c b b a c + + = + + + + = + s + + + + s( 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( ) 3 a b c b + + s +n biu thc P: ( )( ) ( )( )2 22338( ) 42 2a cb a b c a cPa c a c a b cb a c a b c++ + + + | |= + > + |+ + + +\ . + + + + t a cta b c+=++ (ci ny kh m tm iu kin cht ca t, nhng ta ch cn iu kin t>0 l ) 21( )4P f t tt> = +21 1'( ) 2 04 2f t t tt= = = Bng bin thin: Theo bng bin thin,

34MinP = . Du bng xy ra khi1 11,2 2t b a c = = = =Cu 2: Cho cc s thc( | ; ; z 0;1 x y ev tha mn1 x y z + > + . Tim gi tr nh nht ca biu thc: ( ) ( )2 21 2 23 9 1xy x yPx y z xy z z+ + += + ++ + + +

August 1, 2015Sng to BT 2Creation Hng i: Nhn vo biu thc P ta thy rt chong vng v kh c th suy ngh ra c iu g. Nhng rt may bi li cho( | ; ; z 0;1 x y ev1 x y z + > + nn ta d on1 x y z = = =(D on trn ch da vo cm gic m khi cc bn tip xc vi BT thng xuyn) V cng da vo gi thit, ta s tm cc mi lin h vi P. i vi dng c iu kin( | ; ; z 0;1 x y eta thng c BT sau:( ) ( ) 1 1 0 x y >hoc ( )( )( ) 1 1 1 0 x y z > . Nhng bi ton ny khng c xut hin biu thc lin quan, xyz xy yz zx + +hocx y z + +nn ta dng nh khng th s dng( )( )( ) 1 1 1 0 x y z >Nn ta th xoay quanh( ) ( ) 1 1 0 x y > . V ngu nhin t ta c BT1 xy x y + > + V c bit ta s cn s dng 1 kt qu m t ai ch , l 2z z s(v( ) ( | 1 0 0;1 z z z s e) Gii:( )( ) 1 1 0 1 x y xy x y > + > +v 2z z sV ta c2 ; 2 x x y z y x y z + > + + + > + +(v( | ; ; z 0;1 x y e ) Ta tip tc c 21 1 1 z z x y xy + s + s + s + T , ( ) ( )1 1 4.3 1 9 1 9 1xy x y z x y z xy x y zPx y z xy xy x y z xy+ + + + + + + +> + + = ++ + + + + + +

t 1 4 4 3, 0 2 .9 9 4xyt t P t tx y z t t+= > > + > =+ +

Vy 43MinP =khi 2 1 213 3xyt x y zx y z+= = = = =+ +

Cu 3: Cho| | , , 0; 2 a b cev3 a b c ++ = . Tm gi tr nh nht ca biu thc: ( )3 3 3 2 2 22 2 2595a b c a b cP a b cab bc ca+ + + + + = + + + + +

Ta d on du bng khi 1 s bng 0, 1 s bng 1 v s cn li bng 2. Gii:Cch 1: Bi ton c tnh hon v gia cc bin nn gii quyt bi ton ny ta phi gi s { } | | max ; ; 1; 2 a a b c a = e. Nn ta c BT( )( ) 1 2 0 a a sAugust 1, 2015Sng to BT 3Creation Mt khc( ) ( ) ( ) ( ) ( )3 3 33 3 3 3 3 3 23 3 9 3 3 a b c a b c bc b c a b c a a a a + + = + + + s + + = + = +n y ta c th kho st hm s tm max ca ( )29 3 3 a a + . Tuy nhin, nu v du bng th ta s bin i ( ) ( )( )29 3 3 9 1 2 9 9 a a a a + = + s3 3 39 a b c + + s Tng t ( ) ( ) ( ) ( ) ( )2 2 22 2 2 2 2 2 22 3 2 6 9 2 1 2 5 5 a b c a b c bc a b c a a a a a a + + = + + s + + = + = + = + sV cui cng l n ( ) ( ) ( )22 2 2 2 2 2922 2a b c a b c a b cab bc ca+ + + + + ++ + = = > Vy 9 5 5 199 52 5 7P+> =+. Du bng khi ( )( ) ( ) ( )01 2 0 ; ; 1; 0; 23bca a a b ca b c= = = + + = cng cc hon v. Nhng trc quan hn ta c th bin i v 1 bin kho st hm s nh sau: Cch 2:( )( ) 13 225ab bc caP ab bc caab bc ca + +> + + + + +

t | | ( )2 13, 2; 3 25tt ab bc ca t P t f tt += + + e > =+ ( )( )( )223' 2 05f t f tt= + > + ng bin trn| | 2;3( )1927P f > =Cu 4: Cho cc s thc khng m tha1 x y z + + = . Tm gi tr ln nht ca biu thc: 4 522 1 1zP z xy zx y= + + + + +

Hng i: Nhn vo bi ton ta d on1 2 x y z x = = +August 1, 2015Sng to BT 4Creation ( )( ) ( )( )( )2 2 1 2 2 15 12 1 2 1 32 2 1 1x xP x x x f xx x+ += + + + = =+ +

( )( )2 3' 31 1xf xx x+= + +. Thy0 x =l nghim ca( ) ' 0 f x = Vy ta d on0; 1 x y z = = = Gii:V d on0 x y = =v biu thc P c cha1 1 x y + + + nn rt nhiu kh nng ta s s dng BT ph1 1 1 1 x y x y + + + > + + +(BT ny c s dng nhiu trong cc k thi HSG) Mnh s chng minhbng cch tng ng: ( )( ) 1 1 1 1 2 2 1 1 2 2 1 x y x y x y x y x y x y + + + > + + + + + + + + > + + + + +( )( ) 1 1 1 0 x y x y xy + + > + + >(lun ng) Suy ra1 1 1 1 1 x y x y z + + + > + + + = +Mt khc( )( )( ) ( )2 22 11 1 1 14 4x y zz x y z xy x y xy x y+ + += + + + = + + + = + + s = ( ) ( )4 5 4 31 12 2 1 1z zP z z z f zz zs + + = + =+ + ( )( )( )( )( ) ( )( )2 2 24 8 3 2 1 1 3 52 4 3' 021 2 1 2 1z z z z zzf zz z z+ + + ++= = = =+ + +

1 z = V bng bin thin, ta c( )312MaxP f = =khi0; 1 x y z = = = kh v hay hn mnh xin a ra 1 cu vi t duy tng t: Cu 5: Cho cc s thc khng m tha1 x y z + + = . Tm gi tr ln nht ca biu thc: 2 22 21 2zP z xy zx y= + + + + + August 1, 2015Sng to BT 5Creation Gii:1 2 1 2 1 1 x y x y z + + + > + + + = + +( )( )( ) ( )2 22 11 1 1 14 4x y zz x y z xy x y xy x y+ + += + + + = + + + = + + s =( )( )( )( )2 2 1 12 21 2 1 2 2 1 1 1 21 1 1 1zzP z z z z zz z+ ( s + + = + = + + ++ + + + ( )21 2 4 2 2 4 2 2 z = + + s Vy4 2 2 MaxP = khi0; 1 x y z = = =Cu 6: Cho cc s dng; ; a b c . Tm gi tr ln nht ca biu thc: ( )( ) ( ) ( )3 3 331 21 1 12 1Pa b ca b c= + + ++ + +

Hng i: Nhn vo biu thc ca P ta thy, , a b cc vai tr nh nhau nn ta d on1 a b c = = = Khi ( )( )( )3331 212 3 1P f aaa= =++

( )( ) ( )( )243 333 6' 013 1 2 3 1af aaa a= + =++ + . n y ta dng my tnh SOLVE th thy1 a =l nghim ca phng trnh trn. V vy ta d on1 a b c = = = Gii: Trc tin ta cn chng minh ( )( ) ( ) ( ) ( )( )33 3 3 3 3 2 3 2 2 20 04a ba b a b ab a b a a b b b a a b a b++ > + > + + > > ( ) ( )20 a b a b + >lun ng. Tng t ta chng minh: ( )33114cc++ >(v ta d on1 c =) ( ) ( ) ( )3 3 33 3 31 11 1 14 16a b c a b c a b c ( + + + > + + + > + ++

August 1, 2015Sng to BT 6Creation V ta dng AM-GM cho ba s:( )( )( )( )331 1 127a b ca b c++++ + + sDo ( )32 5413Pa b ca b cs + + ++ ++ t( )32 543, 32t a b c t P f tt t= + ++ > s = ( )( )( ) ( )( )24 2 22 42 162' 0 81 2 0 9 18 9 18 02f t t t t t t ttt = + = = + + =369 3 172ttt

=

=

=

M3 6 t t > = V bng bin thin th ta thy( )164P f s =khi1 a b c = = = Cu 7: Cho, , 0 a b c > tha1 c sv3 a b c ++ = . Tm gi tr nh nht ca biu thc: ( )( ) ( )2 23 11 1 1 8 8a bP ca b c+= ++ + + Nhn vo BT ta c th d ona b =v chc chn bi ton s a v dn bin c. Gii:Cch 1: ( )( ) ( )( )( ) ( )( )2 22 22 2 33 1 3 18 8 8 82 1 5 1a b cP c c f ca b c c c+ > + = + =+ + + +

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )22 24 22 3 5 1 3 2 5 1 51'( ) 0 2 085 1c c c c c c cf cc c| | ( + + + | = + < | + |\ . (khc ny o hm hi thm) Nn( ) fcnghch bin. ( ) 1 0 P f > = khi1 a b c = = =August 1, 2015Sng to BT 7Creation Nhng cch ny chng ta s mt thi gian cho cng vic o hm v tm nghim o hm. Chnh v vy mnh s a thm 1 cch na cc bn c th so snh vi cch 1. Cch 2: ta thy1 c sv d ona b =nn ta th ngm1 a b c = = =v ta c cch no sau ( ) ( ) ( )( )( )( )2 22 2327 33 1 3 1 3 11 1 1 8 8 8 8 16 8 82 3a b ca bP c c ca b ca b c+ += + > + = ++ + ++ ++ ( )( )21 34 3016 16c cc c += = >Vy0 MinP =khi1 a b c = = = C hai cch u c nhng u im v khuyt im, nhng quan trng vn l ng dng ca chng ta vo chng, hy th sng to theo hng ca chnh mnh ri cc bn s hiu ti sao li c kt qu nh th. Cu 8: Cho 3 s thc dng; ; x y ztha iu kin 2xy z > . Tm gi tr nh nht ca biu thc: 2 2 2 22 2 2 2 2z z x yPx z y z z+= + ++ +

y l dng bt ng thc rt quen thuc v ta d dng d onx y =khi y ta li c thm 2 2x z > . Nh vy, rt nhiu kh nng bi ton s a y ta vx y z = = Gii:t 2 2; 1 2 2x ya b ab a b abz z= = > + > > 2 22 2 2 22 2 2 2 2 21 1 1 1 41 1 21 1x yP a b a bz z a b a bx yz z| | | |= + + + = + + + > + + ||+ + + +\ . \ .| | | |+ + ||\ . \ . t 2 242, 4 2 t a b t P tt= + + > > + n y ta c th kho st hm s hoc dng Cauchy (AM-GM) nh sau: 4 4 3 32 2 2 .4 2 3 44 4 4tP t t tt t| |> + = + + > + = > |\ . Vy3 MinP =khix y z = = Cu 9: Cho cc s thc dng; x ytha mnx y xy + = . Tm gi tr nh nht ca biu thc: August 1, 2015Sng to BT 8Creation 2 2 2 2 3 32 2 2 21 1 2 2x y x y xy x yPx y x y y x+ += + + ++ + + +

R rng BT ny s a ta n2 x y = = . T ta s i g ri t t. Gii:0 2 4 1 x y xy xy xy xy xy = + > > > ( )2 22 22 2 2 21 1.1 1 2 2y xP x y xyx y x y y y| | | |= + + + + ||+ + + +\ . \ . n y ta thy1 xy >nn dng BT 2 21 1 21 1 1 x y xy+ >+ + +

V vy ( )( )( )22 22 22. .1 2x yP x y xyxy x y x y+> + ++ + + + ( )22 24 41 2 1xy x y xy xyxyxy x y xy xy+> + = ++ + + + t 4, 41 tt xy t P tt= > > ++ ( )24'( ) 1 0 ( )1f t f tt= + > + ng bin trn| ) 4; +Vy 36(4)5P f > =khi2 x y = = Cu 10: Cho cc s dng; x ytha( )2 2312x y x y + = + . Tm gi tr nh nht ca biu thc ( ) ( )4 2 2 4 28 2 8 4 x x x y y y x yPy x y x x y x y+ + += + ++ + +

Gii: Ta s thm cho P mt cht th ny: ( ) ( ) ( )4 2 2 4 2 28 2 8 4 x x x y y y x yPy x y x x y y x y+ + += + ++ + + V bi ton ny nu k th ta s thy bc ca biu thc trn t ca P chia thnh 2 loi l bc 1 v bc 4. T ta c cch nhm sau August 1, 2015Sng to BT 9Creation ( ) ( ) ( ) ( )4 2 2 4 2 22 48x x y y x y x yPx x y y x y y x y x x y| | | |+ += + + + || ||+ + + +\ . \ . ( )( ) ( ) ( ) ( )22 22 248x yx y x yx x y y x y y x y x x y| |+ | | |= + + + | | | + + + +\ .\ . P c chia thnh hai loi l i xng ( ) ( )x yy x y x x y++ + v bt i xng ( )( ) ( )22 22 24x yx yx x y y x y+++ + V vi bt c hc sinh no th n con ng ny ta s liu c chox y = (ta da v gi thit bi ( )( )222 231 1 22 2x yx y x y x y++ = + > s + sv biu thc ( )( ) ( )22 22 24x yx yx x y y x y+++ + p dng BT Scharwz.) ( )( ) ( )( )( )( )24 2 2 2228 16 16.4x y xyx y x y x yPx x y y x y x y y x x y x yx y+ ++ + | |> + + > + = + |+ + + + + ++\ . t | |216, 1; 2 ( ) t x y t P t f tt= + e > + = n y ta c th kho st hm s hoc s dng Cauchy (AM-GM) nh sau: 2 238 8 8 83. . . 12 P t tt t tt> + + > = Vy12 MinP =khi1 x y = = Cu 11: Cho, , a b cl cc s thc dng. Tm gi tr nh nht ca biu thc:( )2249 1 1 4.50 2 3 4 2 4 3 48 2 5 6 32Pc a b a c bca b c= + + + + + ++ + +

Gii: Nhn vo biu thc P ta ngh ngay n vic thot cn ca4 bcv( )222 5 6 32 a b c + + . Tht vy: 49 1 1 4.50 4 4 4 2 3 4 2 8 5 6 4Pa b c c a b a b c> + + + + + + + + +

August 1, 2015Sng to BT 10Creation ( ) ( ) ( )2 2 27 7 20 4400 400 400 2 3 4 2 8 5 6 4 a b c a b c c a b a b c= + + + + + + + + + + + +

( )234 4400 5 6 4 2 8 5 6 4 a b c a b c> + + + + + + t5 6 4 2, 0 t a b c t = + + + > ( )234 4400 6P ftt t> =+

( )( )( )( )226 204 74 204' 0 34400 6t tf t tt t += = =+

V bng bin thin ta suy ra( )334200MinP f= =khi 5 6 4 24 15 6 4 2 34 4a b c ac b ba b c c+ = = = = + + + = =

Cu 12: Cho, , a b cl cc s thc dng tha2 ab c = . Tm gi tr nh nht ca biu thc: 231 510aP ca b+= + +Ta s d on1 a b = =v 12c = Gii:Cch 1: Nhn vo biu thc P ta thy c xut hin 5b v 2 31 1a a+lm ta lin tng n vic bin i AM-GM sao cho 2 31 1a a+v dng 5a. V th ta lm nh sau: 2 2 23 2 31 5 1 1 5 5 510 1 1 1 10 3 10 3aP c c ca b a a b a b+ | |= + + = + + + + + + > + + |\ . 2 210 5 210 3 10 3 c cab c> + = + August 1, 2015Sng to BT 11Creation t( )2 452 40 32ct c t P t ftt= = > + =( )325 1' 160 02f t t tt= = =V bng bin thin v ta thy 1 192 2MinP f | |= = |\ .

Vy 192MinP =khi 11;2a b c = = = Nhng vi cch gii trn ta s thy tht g b v khng t nhin nn mnh xin dn cch 2 nh sau: Cch 2: 2 23 2 31 5 1 1 510 102a aP c ca b a a c+= + + = + + +Bi ton tr thnh hai bin v vic cn li ta s s dng AM-GM loi b ht h savc . Ch 22 31 1 122 2 4 2aca a c= = = = ( )2 2 1052 2192 2 3 3 2 31 1 1 1 1 1 1910. 5.2 19 . . . 22 2 2 2 4 2 2 4 2a aP c ca a a a c a a c| || || |= + + + + + > = | | |\ .\ .\ .

Vy 192MinP =khi 11;2a b c = = = Cu 13: Cho, , 2 a b c > . Tm gi tr nh nht ca biu thc: ( )2 2 2 2 2 22 2 2 2 2 28a ab b bc c caP abc a b cb c a + + += + + + + ++ Gii:( )2 2 2 2 2 22 2 2 2 2 28a ab b bc c caP abc a b cb c a + + += + + + + ++( ) ( )2 2 21 1 12 2 8a b cab bc ca abc a b cb c a a b c| |= + + + + + + + + + + + |\ .

Ta c: 2 2 22 2 2a b cb a c b a cb c a+ > + > + >August 1, 2015Sng to BT 12Creation Cng v theo v ta c 2 2 2a b ca b cb c a+ + > + +( ) 1 ( )( ) ( ) ( ) ( ) , , 2 2 2 2 0 2 4 8 0 a b c a b c abc ab bc ca a b c > > + + + + + >( ) ( ) 2 4 8 abc ab bc ca a b c + + > + + + ( ) 2 T( ) 1 v( ) ( ) ( )1 1 1 182 5 2 8 5 8 P a b c a b ca b c a b c| | > + + + + + + > + + + + |+ +\ . t( )18, 6 5 8 t a b c t P t f tt= + + > > + + = ( ) ( )2 22 25 18 5.6 18' 0tf t f tt t = > > ng bin trn| ) 6; + ( ) 6 41 P f > = Vy41 MinP =khi2 a b c = = = Cu 14: Cho, , 1 x y z > . Tm gi tr nh nht ca biu thc: ( )2 1 12 2P x y z xy yz xz xyzx y x z= + + + + ++ +

Gii: Theo gi thit( )( )( ) , , 1 1 1 1 0 1 0 x y z x y z xyz xy yz zx x y z > > + + + + + >1 xyz xy yz zx x y z > V ( ) ( )1 1 4 22 2 2 2 x y x z x y x z x y z+ > =+ + + + + + + ( ) ( )2 21 P x y z x y zx y z > + + + + + ++ +

t( )22, 3 1 t x y z t P t t f tt= + + > > + + = ( )( ) ( )( )3 23 22 22 12 2 2 23' 2 1 0 3 33t t tt tf t t t P ft t + = = = > > > = Vy 233MinP =khi1 x y z = = =