MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 91 BI DNG HC SINH GII TON 9 THEO TNG DNG DNG 1: RT GN TNH GI TR CA BIU THC Bi 1:Cho biu thcP = ( ) ( )3a 122aa 1 21a 1 21+++ a) Rt gn P. b) Tm Min P. Bi 2:Cho x, y l hai s khc nhau tha mn: x2 + y = y2 + x Tnh gi tr biu thc : P = 1 - xy xy2y2x + + Bi 3:Tnh gi tr biu thc Q = y xy - x+ Bit x2 -2y2 = xy v x 0; x + y 0 Bi 4:Cho biu thcP = 3 x3 x 2x - 12 x 33 x 2 x11 x 15+++ + a) Tm cc gi tr ca x sao cho P = 21 b) Chng minh P 32 Bi 5:Cho biu thcP = a2 a2 a1 a2 a a3 9a 3a1 +++ + + a) Rt gn P. b)Tm cc gi tr nguyn ca a P nguyn. Bi 6:Cho biu thcP = 2a16a8- 14 - a 4 a 4 - a 4 a+ + + a) Rt gn P. b)Tm cc gi tr nguyn ca a (a >8) P nguyn. Bi 7:Cho biu thcP = ||.|

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\|+ 1 a21 a1:a a11 aa a) Rt gn P. www.vntoanhoc.comwww.vntoanhoc.comwww.vntoanhoc.comMATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 92 b) Tnh gi tr P khi a = 3 + 2 2c) T m cc gi tr ca a sao cho P < 0. Bi 8:Cho biu thcP = ||.|

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\| + x2x 2 x1 x:x 48xx 2x 4 a) Rt gn P. b) Tnh x P = -1 c) T m m vi mi gi tr x > 9 ta c m( x- 3)P > x + 1. Bi 9:Cho biu thcP = ||.|

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\|+++ +xyy xx xyyy xyx:y xxy - y xa) Tm x, y P c ngha. b) Rt gn P. c) Tm gi tr ca P vi x = 3, y = 4 + 23 Bi 10:Cho biu thcP = x2007 x1 x1 4x x1 x1 - x1 x1 x22+ +++|||.|

\| a) Tm x P xc nh. b) Rt gn P. c) Tm cc gi tr nguyn ca x P nguyn. Bi 11:Rt gn P. P = 22 2 42 22 22 22 2bb a a 4:b a ab a ab a ab a a + +|||.|

\| Vi | a | >| b | > 0Bi 12:Cho biu thcP = 22x 1.1 x 2 x2 x1 x2 x||.|

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\|+ ++ a)Rt gn P. b)Chng minh rng nu 0 < x < 1 th P > 0. c) Tm GTLN ca P. Bi 13:Chng minh gi tr ca biu thcP = 6 x 5 x10 x3 x 4 x1 x 52 x 3 x2x+ ++++ ++++ + Khng ph thuc vo bin s x. Bi 14:Chng minh gi tr ca biu thcP = xxx+ + + +5 2 . 5 4 93 4 7 . 3 246 3 Khng ph thuc vo bin s x. MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 93 Bi 15:Cho biu thcP = 1 x1 x xx x1 x xx x2 2+ ++ ++ + Rt gn P vi 0 x 1 . Bi 16:Cho biu thcP = 1 x) 1 2(xxx 2x1 x xx x2+++ + a)Rt gn P. b)Tm GTNN ca P c) Tm x biu thc Q = Px 2 nhn gi tr l s nguyn. Bi 17:Cho biu thcP = 1 x 2x1 x 2x1 x1 xx x1 x xx x x 2x+ + + +||.|

\| a) Tm x P c nghab) Rt gn P. c) Vi gi tr no ca x th biu thc P t GTNN v tm GTNN . Bi 18: Rt gn biu thc P = 5 3 105 35 3 105 3 ++ ++ Bi 19: Rt gn biu thc a) A =7 4 7 4 + b) B =5 2 10 4 5 2 10 4 + + + +c) C =5 3 2 15 4 15 4 + +Bi 20: Tnh gi tr biu thc P =1 2 3 4 1 2 7 24 + + + + x x x x Vi 21 x 5.Bi 21: Chng minh rng: P = 2 648 13 5 3 2++ + l mt s nguyn. Bi 22: Chng minh ng thc:MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 94 1231 1231231 1231= ++ ++ Bi 23:Cho x = 37 2 537 2 5 + Tnh gi tr ca biu thc f(x) = x3 + 3x Bi 24:Cho E = y xxy 1y xxy 1++ Tnh gi tr ca E bit: x =2 2 2 . 2 2 2 . 8 4 + + + +y = 45 27 2 18 320 12 2 8 3+ + Bi 25:Tnh P = 20082007220082200722007 1 +++Bi 26:Rt gn biu thc sau: P =5 11+ + 9 51++ ... +2005 20011+ Bi 27:Tnh gi ri ca biu thc: P = x3 + y3 - 3(x + y) + 2004 bit rng x = 32 2 332 2 3 + +y = 32 12 1732 12 17 + +Bi 28:Cho biu thc A =|.|

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\|+++aa aaaaa 141111 a) Rt gn A. b) Tnh A vi a = (4 +15 )( 10 - 6 ) 15 4Bi 29:Cho biu thcA = ( ) ( )( )|.|

\| + + 1111 41 4 1 42xx xx x x x a) x = ? th A c ngha. b) Rt gn A. Bi 30:Cho biu thcP = x x xxx xx+++ + ++ + + +111 11 11 11 1 a)Rt gn P. b) So snh P vi 22. MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 95 Bi 31:Cho biu thcP = 121311+ +++ x x x x x a)Rt gn P. b) Chng minh: 0 P 1. Bi 32:Cho biu thcP = aaaaa aa +++ 31 2236 59 2 a)Rt gn P. b) a = ? th P < 1 c) Vi gi tr nguyn no ca a th P nguyn. Bi 33:Cho biu thcP = xxy xy x xxy xyx + 112 222 a)Rt gn P. b)Tnh P bit 2x2 + y2 - 4x - 2xy + 4 = 0. Bi 34:Cho biu thcP = xxy xy x xxy xyx + 112 222 a)Rt gn P. b)Tnh P bit 2x2 + y2 - 4x - 2xy + 4 = 0. Bi 35:Cho biu thcP = y x xyy y x x y xy x y x y x3 33 3:1 1 2 1 1++ + +(((

+ ++||.|

\|+ a)Rt gn P. b)Cho xy = 16. Tm Min P. MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 96 DNG 2: BIN I NG NHT. Bi 1: Cho a > b > 0 tha mn: 3a2 +3b2 = 10ab. Tnh gi tr ca biu thc:P = b ab a+ Bi 2:Cho x > y > 0 v 2x2 +2y2 = 5xy Tnh gi tr biu thc E = y xy x+ Bi 3:1) Cho a + b + c = 0 CMR: a3 + b3 + c3 = 3abc 2) Cho xy + yz + zx = 0 v xyz 0 Tnh gi tr biu thc: M = 2 2 2zxyyxzxyz+ + Bi 4: Cho a3 + b3 + c3 = 3abc. Tnh gi tr ca biu thc: P =|.|

\| + |.|

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\| +accbba1 1 1 Bi 5: a) Phn tch thnh nhn t: (x + y + z)3 - x3 - y 3 -z3 b) Cho cc s x, y, z tha mn iu kin x + y + z = 1 v x3 + y3 + z3 = 1 . Tnh gi tr ca biu thc: A = x2007 + y2007 + z2007 Bi 6: Cho a + b + c = 0 v a2 + b2 + c2 = 14. Tnh gi tr ca biu thc: P =a4 + b4 + c4 Bi 7: Cho a, b l cc s thc dng tha mn: a100 + b100 = a101 + b101 = a102 + b102 Tnh gi tr ca biu thc P = a2007 + b2007 Bi 8: Cho 1 = +byax v 2 =abxy. Tnh 3333byax+ Bi 9: Cho a + b + c = 0 . Tnh gi tr ca biu thc P = 2 2 2 2 2 2 2 2 21 1 1c b a b c a a c b ++ ++ + Bi 10: Cho b a byax+= +14 4; x2 + y2 = 1. Chng minh rng: a) bx2 = ay2; MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 97 b) 1004 1004200810042008) (2b a byax+= + Bi 11: Chng minh rng nu xyz = 1 th:

xz z yz y xy x + +++ +++ + 111111 = 1 Bi 12: Cho a + b + c = 0. Tnh gi tr biu thc: A = (a b)c3 + (c a)b3 + (b c)a3 Bi 13: Cho a, b, c i mt khc nhau. Tnh gi tr ca biu thc: P = ) )( ( ) )( ( ) )( (2 2 2a c b cca b c bbc a b aa + + Bi 14: Gi a, b, c l di ba cnh mt tam gic. Cho bit (a + b)(b + c)(c + a) = 8abc Chng minh: Tam gic cho l tam gic u. Bi 15: Chng minh rng: Nu a,b,c khc nhau th: a c c b b a b c a cb aa b c bb cc a b ac b++= + + 2 2 2) )( ( ) )( ( ) )( ( Bi 16: Cho bit a + b + c = 2p Chng minh rng:) )( )( (1 1 1 1c p b p a p pabcp c p b p a p = ++ Bi 17: Cho a, b khc 0 tha mn a + b = 1. Chng minh : 3) 2 ( 21 12 2 3 3+=+ b aababba Bi 18: Cho1 = + +czbyax v0 = + +zcybxa Tnh gi tr biu thc A = 222222czbyax+ +Bi 19: Cho a, b, c i mt khc nhau v0 =++ b aca cbc ba Tnh gi tr ca P = 2 2 2) ( ) ( ) ( c aca cbc ba++ Bi 20: Phn tch cc a thc sau thnh nhn t: a)x(y2 z2) + y(z2 x2) + z(x2 y2) b)x(y + z)2 + y(z + x)2 + z(x + y)2 4xyz Bi 21: Cho ba s phn bit a, b,c. Chng minh rng biu thc A = a4(b c) + b4(c a) + c4(a b) lun khc 0. Bi 22: Cho bn s nguyn tha mn iu kin: a + b = c + d v ab + 1 = cd Chng minh: c = d. Bi 23: Cho x , y l cc s dng tha mn iu kin: 9y(y x) = 4x2. Tnh gi tr biu thc: A = y xy x+ Bi 24: Cho x, y l cc s khc khc 0 sao cho 3x2 y2 = 2xy. MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 98 Tnh gi tr ca phn thc A =2 262y xy xxy+ + Bi 25: Cho x, y, z khc 0 v a, b, c dng tho mn ax + by + cz = 0 v a + b +c = 2007. Tnh gi tr ca biu thc:P = 2 2 22 2 2) ( ) ( ) ( y x ab z x ac z y bccz by ax + + + + Bi 26: Cho x, y, z khc 0 vx + y + z = 2008. Tnh gi tr biu thc: P = ) )( ( ) )( ( ) )( (3 3 3x z y zzz y x yyz x y xx + + Bi 27:Cho = + += + += + +1113 3 32 2 2z y xz y xz y x

Tnh gi tr ca biu thc: P = x2007 + y2007 + z2007 . Bi 28: Cho a, b, c l di cc cnh ca mt tam gic. Tnh gi tr ca biu thc: P = | || |2 22 2) ( ) () ( ) (b c a c b ac b a c b a + + + + Bi 29: Cho biu thc P = (b2 + c2 a2)2 4b2c2. Chng minh rng nu a, b, c l ba cnh ca mt tam gic th P < 0. Bi 30: Cho cc s dng x, y ,z tha mn: = + += + += + +1583z x zxz y yzz y xy Tnh gi tr biu thc: P = x + y + z. Bi 31: Cho cc s x, y, ztha mn h phng trnh:

= + += + +113 3 32 2 2z y xz y x Tnh gi tr biu thc P = xyz. ( thi HSG tnh 2003) Bi 32: a) Thu gn biu thc: P = 4 3 24 8 6 3 2+ ++ + + + b)Tnh gi tr biu thc: Q = y xy x+

Bit x2 2y2 = xy v y 0 , x + y 0. ( thi HSG tnh 2004-2005) Bi 33: Chng minh rng nu: x + y + z = 0 th: 2(x5 + y5 + z5) = 5xyz(x2 + y2 + z2) ( thi HSG tnh 2005-2006) Bi 34: Cho a, b, c l ba s dng tha mn iu kin: a2 = b2 + c2. a)So snh a v b + c. b)So snh a3 v b3 + c3. ( thi HSG tnh 2006-2007) Bi 35: 1) Gii phng trnh: x3 -6x 40 = 0 MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 99 2) Tnh A = 3 32 14 20 2 14 20 + + ( thi HSG tnh 2006-2007) DNG 3: PHNG TRNH BC HAI. Bi 1: Cho phng trnh n s x:x2 2(m 1)x 3 m = 0 (1) a)Gii phng trnh khi m = 2. b)Chng t rng phng trnh c nghim s vi mi m. c)Tm m sao cho nghim s x1, x2 ca phng trnh tha mniu kin 21x +22x > 10. Bi 2: Cho cc s a, b, c tha iu kin: ( ) + < +>ac bc ab a cc202 Chng minh rng phng trnh ax2 + bx + c= 0 lun lun c nghim. Bi 3: Cho a, b, c l cc s thc tha iu kin: a2 + ab + ac < 0.Chng minh rng phng trnh ax2 + bx + c = 0 c hai nghim phn bit. Bi 4: Cho phng trnh x2 + px + q = 0. Tm p, q bit rng phng trnh c hainghim x1, x2 tha mn: = = 35532312 1x xx x Bi 5: CMR vi mi gi tr thc a, b, c th phng trnh(x a)(x b) + (x c)(x b) + (x c)(x a) = 0 lun c nghim. Bi 6: CMR phng trnh ax2 + bx + c = 0 ( a= 0) c nghim bit rng 5a + 2c = b Bi 7: Cho a, b, c l di cc cnh ca mt tam gic. CMR phng trnh sau c nghim: (a2 + b2 c2)x2 - 4abx + (a2 + b2 c2) = 0 Bi 8: CMR phng trnh ax2 + bx + c = 0 ( a= 0) c nghim nu42+ >acab Bi 9: Cho phng trnh : 3x2 - 5x + m = 0. Xc nh m phng trnh c hai nghim tha mn: 21x -22x = 95 Bi 10: Cho phng trnh: x2 2(m + 4)x +m2 8 = 0. Xc nh m phng trnh c hai nghim x1, x2 tha mn: a)A = x1 + x2 -3x1x2 t GTLN b)B = x12 + x22 - t GTNN. c)Tm h thc lin h gia x1, x2 khng ph thuc vo m. Bi 11: Gi s x1, x2 l hai nghim ca phng trnh bc 2:3x2 - cx + 2c - 1 = 0. Tnh theo c gi tr ca biu thc: MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 910 S = 32311 1x x +Bi 12: Cho phng trnh : x2 - 2 3 x + 1 = 0. C hai nghim l x1, x2. Khng gii phng trnh trn hy tnh gi trca biu thc: A = 23132 122 2 1214 43 5 3x x x xx x x x++ + Bi 13: Cho phng trnh: x2 2(a - 1)x + 2a 5 = 0 (1) 1)CMR phng trnh (1) lun c hai nghim vi mi gi tr ca a. 2)Tm gi tr ca a pt (1) c hai nghim x1, x2 tha mn iu kin: x12 + x22 = 6. 3) Tm gi tr ca a phng trnh c hai nghim x1, x2 tha mn iu kin: x1 < 1 <

x2. Bi 14: Cho phng trnh: x2 2(m - 1)x + m 3 = 0 (1) a)CMR phng trnh (1) c nghim vi mi gi tr ca m. b)Gi x1, x2 l hai nghim ca phng trnh (1) . Tm GTNN ca M = x12 + x22 Bi 15: Cho a, b l hai s thc tha mn iu kin: 21 1 1= +b a CMR t nht mt trong hai phng trnh sau phi c nghim: x2 + ax + b = 0 v x2 + bx + a = 0. Bi 16: Cho phng trnh: x2 2(m + 1)x + 2m +10 = 0 (1) a)Gii v bin lun s nghim ca phng trnh (1) theo m. b)Tm m sao cho 10x1 x2 + x12 + x22 t GTNN. Tm GTNN . Bi 17: Chng minh rng vi mi s a, b, c khc 0, tn ti mt trong ccphng trnhsau phi c nghim: ax2 + 2bx + c = 0 (1) bx2 + 2cx + a = 0 (2) cx2 + 2ax + b = 0 (2) Bi 18: Cho phng trnh: x2 (m - 1)x + m2 + m 2 = 0 (1) a)CMR phng trnh (1) lun lun c nghim tri du vi mi gi tr ca m. b)Vi gi tr no ca m, biu thcP = x12 + x22 t GTNN. Bi 19: Cho phng trnh: x2 2(m - 1)x 3 - m = 0 (1) 1) CMR phng trnh (1) lun c hai nghim vi mi gi tr ca m. 2) Tm gi tr ca m pt (1) c hai nghim x1, x2 tha mn iu kin: x12 + x22> 10. 3) Xc nh gi tr ca m phng trnh c hai nghim x1, x2 tha mn iu kin: E = x12 + x22 t GTNN. Bi 20: Gi s phng trnh bc 2:x2 + ax + b + 1 = 0 c hai nghim nguyn dng.CMR: a2 + b2 l mt hp s. MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 911 DNG 4: PHNG TRNH BC CAO. Gii phng trnh:Bi 1:x3 + 2x2 + 2 2 x + 2 2 . Bi 2:(x + 1)4 = 2(x4 + 1) Bi 3:4(x + 5)(x + 6)(x + 10)(x + 12) = 3x2 Bi 4:3(x + 5)(x + 6)(x + 7) = 8xBi 5: (x + 2)(x + 3)(x - 7)(x - 8) = 144 Bi 6:(x + 2)4 + (x + 8)4 = 272 Bi 7:a) (x +2 )4 + (x + 1)4 = 33 + 12 2b) (x - 2)6 + (x - 4)6 = 64 Bi 8:a) x4 - 10x3 + 26x2 - 10x + 1 = 0 b) x4 + 3x3 - 14x2 - 6x + 4 = 0 c) x4 - 3x3 + 3x + 1 = 0 Bi 9: a) x4 = 24x + 32 b) x3 + 3x2 - 3x + 1 = 0 Bi 10: 1 9 83 5= + x xBi 11:12 5 372 322 2=+ ++ x xxx xx Bi 12: x2 + ( )122422=+ xx Bi 13:20 0144812512222 2=+ |.|

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\|+xxxxxx Bi 14: a)4171 332 2 =+ +++ x xxx xx b) 15 12415 615 102 22+ =+ + x xxx xx x c) 415 65 55 45 32222 =+ + + + x xx xx xx x Bi 15: a) x2 + ( )4098122=+ xx MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 912 b) x2 + ( )15122=+ xx Bi 16: a) 94021 12 2= |.|

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\| xxxx b)014251212222 2= |.|

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\|++xxxxxx c) x. 151818= |.|

\|xxxxx Bi 17:x2 + 21|.|

\| xx = 8( thi HSG V1 2004) Bi 18: 2 3 1 5 1 = x x xBi 19: 2 7 13 3= + + x xBi 20: 2 1 2 1 2 = + + x x x xBi 21: 3x2 + 21x + 18 + 2 2 7 72= + + x xBi 22: a) (x - 2)4 + (x - 3)4 = 1 b) x4 + 2x3 - 6x2 + 2x + 1 = 0c) x4 + 10x3 + 26x2 + 1 = 0Bi 23: (x + 2)2 + (x + 3)3 + (x + 4)4 = 2 ( thi HSG V1 2003) Bi 24: a) (x + 1)(x + 2)(x + 3)(x + 4) = 3 b) (x2 + 3x - 4)(x2 + x - 6) = 24 Bi 25: a) x3 - 6x + 4 = 0 b) x4 - 4x3 + 3x2 + 2x - 1 = 0Bi 26: a) x4 + 2x3 + 5x2 + 4x - 12 = 0b) x4 - 4x3 - 10x2 + 37x - 14 = 0Bi 27: 0431048322= |.|

\| +xxxx Bi 28:a) Phn tch thnh nhn t: 2(a2 + b2) -5ab b) Gii phng trnh: 2(x2 + 2) = 5 13+ x( thi HSG 1998) Bi 29: 35 3145 = + xxxBi 30: x4 - 4 3 x -5 = 0 ( thi HSG 2000) Bi 31:0 52424= +xxx ( thi HSG V2 2003) Bi 32: a) x4 - 4x3 - 19x2 + 106x - 120 = 0b) (x2 - x + 1)4 - 10(x2 - x + 1)2 +9x4 = 0Bi 33: (x + 3 x+ 2)(x + 9 x+18) = 168x ( thi HSG 2005) Bi 34: a) x2 + 4x + 5 = 2 3 2 + xb) 3 83+ x= 2x2 - 6x + 4 c)23 242 =+ + xxMATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 913 Bi 35: 0 3 2 13 3 3= + + + + + x x xBi 36: Cho phng trnh: x4 -4x3 +8x = m a) Gii phng trnh khi m = 5. b) nh m phng trnh c 4 nghim phn bit. Bi 37: Cho phng trnh (x + a)4 + (x + b)4 = c. Tm iu kin ca a, b, c phng trnh c nghim. Bi 38: Gii phng trnh: x4 + 2x3 + 5x2 + 4x - 5 = 0Bi 39: Tm nghim nguyn ca phng trnh: 4x4 + 8x2y + 3y2 - 4y - 15 = 0. Bi 40: x2 + 9x + 20 = 2 10 3 + xBi 41: x2 + 3x + 1 = (x + 3) 12+ xBi 42:x2 +2006 + x =2006 DNG 5: BT NG THC Bi 1) Vi a, b > 0 thabb a>+2. Du ng thc xy ra khi no? Bi 2) CMR vi 4 s a, b, x, y bt k ta c:> + + ) )( (2 2 2 2y x b a(ax + by)2.Du ng thc xy ra khi no? Bi 3) Cho a, b, c, d > 0. Cm:( )( ) d b c a cd ab + + s +Bi 4) CM bt ng thc: ( ) ( )2 2 2 2 2 2d b c a d c b a + + + > + + +Bi 5) Cho a, b, c l cc s dng cm bt ng thc: 22 2 2c b ab aca cbc ba + +>+++++ Bi 6) CM vi mi n nguyn dng th: 2121...2111> + ++++ n n n Bi 7) Cho a3 + b3 = 2. Cmr: a + bs 2. Bi 8) Cho a, b, c tha mn:a + b + c = -2 (1) a2 + b2 + c2 = 2(2) CMR mi s a, b, c u thuc on ((

0 ;34khi biu din trn trc s. Bi 9) Cho a, b, c tha mn h thc 2a + 3b = 5.CMR: 2a2 + 3b2> 5. Bi 10) Cho a, b l hai s tha mn iu kin: a + 4b = 1.CM: a2 + 4b2> 51. Du ng thc xy ra khi no? ( thi HSG 2003). Bi 11) Chng minh: 312 2 2 22 2 2 2 2 + + ) )( (2 2 2 2y x b a(ax + by)2

b)2 4 2 0 s + < x xBi 13) Cho a, b, c > 0. Cm: 23>+++++ b aca cbc ba Bi 14) Cho 1001...31211 + + + + = S .CMR:S khng l s t nhin. Bi 15) a) Cho x, y dng. CMR: y x y x +> +4 1 1. Du bng xy ra khi no? b) Tam gic ABC c chu vi 2c b aP+ += .Cm:|.|

\|+ + >++ c b a c p b p a p1 1 121 1 1

Du bng xy ra khi tam gic ABC c c im g? Bi 16) a) CM x > 1 ta c:21 > xx b) Cho a > 1, b > 1. Tm GTNN ca: 1 12 2+=abbaPBi 17) Cho a, b, c l di ba cnh ca mttam gic. CM: a2 + b2 + c2 < 2(ab + bc + ca) Bi 18) CMR nu a, b, c > 0 v a + b + c = 1 th 91 1 1> |.|

\|+ +c b a. Bi 19) CMR nu a, b, c l di ba cnh ca mttam gic th: ab + bc + cas a2 + b2 + c2 < 2(ab + bc + ca) Bi 20) Cho tam gic ABC c di ba cnh l a, b, c v c chu vi l 2. CMR: a2 + b2 + c2 + 2abc < 2.( thi HSG 2004-2005). Bi 21)Cho a, b l 2 s thc tha mn iu kin: (a - 1)2 + ( b - 2)2 = 5. Cm: a + 2bs 10. Bi 22) Cho a, b l cc s thc tha mn iu kina2 + b2= 4 + ab.CMR:8382 2s + s b a .Du bng xy ra khi no? Bi 23) CMR vi mi a, b > 0 tha mn ab = 1. Ta c BT:32 1 1>++ +b a b a Bi 24) CMR nu: a)5 1 s s ath10 5 4 1 3 s + a ab) a + b2 ; 0 1 ; 0 = + > + > b a bth2 2 1 1 s + + + b aBi 25) Cho biu thc 1411132 3 4 5 3 4 3 4 + + + + =x x x x x x x x x x xPCMR:9320 < < Pvi1 = x . MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 915 Bi 26) a) Cho a, b, k l cc s dng v k bk abaCmrba++< < : . 1 b) Cmr nu a, b, c l di 3 cnh ca mt tam gic th:b aca cbc ba+++++< 2. Bi 27) Cho cc s dng a, b tha mn iu kin a + b = 1. Chng minh rng:91111 > |.|

\| + |.|

\| +b a ( thi HSG V2 2003 - 2004) Bi 28) Chng minh bt ng thc sau y ng vi mi x, y l cc s thc bt k khc 0: ||.|

\|+ > + +xyyxxyyx3 42222 DNG 6: CC TR Bi 1)Cho hai s thc x, y tha mn iu kin: x2 + y2 = 1. Tm GTLN v GTNN ca biu thc A = x + y. Bi 2)Cho x, y > 0, x + y = 1. Tm GTNN ca P = 2 21 11 1x y| || | ||\ .\ . Bi 3) Cho P = ( )222 11x xx + ++. Tm GTNN, GTLN ca P v cc gi tr tng ng ca x. Bi 4) Tm GTLN v GTNN ca biu thc A = (x4 + 1)(y4 + 1) bit x,y> 0, x + y =10Bi 5)Tm GTLN v GTNN ca biu thc B = 2x + 3y bit 2x2 + 3y2 5. Bi 6)Tm GTLN v GTNN ca biu thc P = x2 + y2. Bit x2(x2 +2y2 3) + (y2 2)2 = 1 Bi 7)Tm GTLN v GTNN ca biu thc P = 2211x xx x ++ + Bi 8) Tm GTLN ca A = x+2 x Bi 9)Tm GTLN ca P = x y zy z x+ +vi x, y, z > 0. Bi 10) Tm GTLN ca P = 2 2( 1990) ( 1991) x x + Bi 11)Cho M =3 4 1 15 8 1 a a a a + + + a)Tm iu kin ca a M c xc nh. b)Tm GTNN ca M v gi tr ca A tng ng. Bi 12) Cho ba s dng x, y, z tha mn: 1 1 121 1 1 x y z+ + >+ + +. Tm GTNN ca P = x.y.z. Bi 13) Tm GTNN ca P = 2 11 x x+ Bi 14) Cho x, y tha mn x2 + 4y2 = 25. Tm GTLN v GTNN ca biu thc P = x + 2y. Bi 15) Cho x, y l hai s tha mn: x + 2y = 3.Tm GTNN ca E = x2 + 2y2. MATHVN.COM | www.MATHVN.com www.mathvn.com - Bi tp bi dng HSG Ton 916 Bi 16) Cho x > 0, y > 0 tha mn: x + ys 1. Tm GTNN ca biu thcP = 2 21x y + + 2xy + 4xy Bi 17) Tm GTLN v GTNN ca: P = 2211x xx+ ++ vi x bt k. Bi 18) Cho x, y l hai s dng tha mn: x + ys 1. Tm GTNN ca biu thc A = 2 21 2x y xy++ Bi 19) Cho x,y > 0; x + y = 1. Tm GTNN ca biu thc P = 221 1x yx y| || |+ + + ||\ .\ . Bi 20) Cho x,y > 0; x + y = 1. Tm GTNN ca biu thc P = 2(x4 + y4) + 14xy Bi 21) Cho x,y > 0; x + y = 1. Tm GTNN ca biu thc P = 1 11 1x y| || |+ + ||\ .\ . Bi 22) Cho x, y l hai s dng tha mn: x2 + y2 = 4. Tm GTNN ca biu thc P = 221 1x yy x| || |+ + + ||\ .\ . Bi 23) Cho ba s dng a, b, c c a + b + c = 1. Tm GTNN ca biu thc: E = 2 2 21 1 1a b ca b c| | | | | |+ + + + + |||\ . \ . \ . Bi 24) Cho a, b l hai s thc bt k c tng bng 1. Tm GTNN ca: P = a3 + b3 Bi 25) Cho a, b l hai s dng tha a + b = 1. Tm GTNN ca P = 1 11 1 a b++ + Bi 26) Cho hai s x, y tha mn xy = 2. Tm GTNN ca P = 2 2x yx y+ Bi 27) Cho hai s dng x, y c x + y = 1. Tm GTNN ca P = 8(x4 + y4) + 1xy Bi 28) Cho x, y lin h vi nhau bi h thc: x2 + 2xy + 7(x + y) + 2y2 +10 = 0 Tm GTNN, GTLN ca biu thc S = x + y + 1 Bi 29) Tm GTNN, GTLN ca biu thc S = x x+ y ybitx+y= 1 Bi 30) Tm GTNN ca biu thc P = 222 2000 x xx + www.vntoanhoc.com