CHUYÊN ĐỀ BỒI DƯỠNG HỌC SINH GIỎI

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98862646.docCHUYN B I D NG H C SINH GI I PHN I: BI 1.Chng minh 7 l s v t.2.a) Chng minh :(ac + bd)2 + (ad bc)2 = (a2 + b2)(c2 + d2) b) Chng minh bt dng thc Bunhiacpxki :(ac + bd)2 (a2 + b2)(c2 + d2)3. Cho x + y = 2. Tm gi tr nh nht ca biu thc :S = x2 + y2.4.a) Cho a 0, b 0. Chng minh bt ng thc Cauchy :a bab2+ . b) Cho a, b, c > 0. Chng minh rng :bc ca aba b ca b c+ + + + c) Cho a, b > 0 v 3a + 5b = 12. Tm gi tr ln nht ca tchP = ab.5. Cho a + b = 1. Tm gi tr nh nht ca biu thc :M = a3 + b3.6. Cho a3 + b3 = 2. Tm gi tr ln nht ca biu thc :N = a + b.7. Cho a, b, c l cc s dng. Chng minh :a3 + b3 + abc ab(a + b + c)8. Tm lin h gia cc s a v b bit rng :a b a b + > 9.a) Chng minh bt ng thc(a + 1)2 4a b) Cho a, b, c > 0 v abc = 1. Chng minh :(a + 1)(b + 1)(c + 1) 810. Chng minh cc bt ng thc :a)(a + b)2 2(a2 + b2) b)(a + b + c)2 3(a2 + b2 + c2)11. Tm cc gi tr ca x sao cho :a)| 2x 3 | = | 1 x | b)x2 4x 5 c)2x(2x 1) 2x 1.12. Tm cc s a, b, c, d bit rng :a2 + b2 + c2 + d2 = a(b + c + d)13. Cho biu thc M = a2 + ab + b2 3a 3b + 2001. Vi gi tr no ca a v b th M t gi tr nh nht ? Tm gi tr nh nht .14. Cho biu thc P = x2 + xy + y2 3(x + y) + 3. CMR gi tr nh nht ca P bng 0.15. Chng minh rng khng c gi tr no ca x, y, z tha mn ng thc sau :x2 + 4y2 + z2 2a + 8y 6z + 15 = 016. Tm gi tr ln nht ca biu thc :21Ax 4x 9 +17. So snh cc s thc sau (khng dng my tnh) :a) 7 15 v 7 + b) 17 5 1 v 45 + +c)23 2 19v 273d)3 2 v 2 318. Hy vit mt s hu t v mt s v t ln hn 2 nhng nh hn 319. Gii phng trnh :2 2 23x 6x 7 5x 10x 21 5 2x x + + + + + .20. Tm gi tr ln nht ca biu thc A = x2y vi cc iu kin x, y > 0 v 2x + xy = 4.21. Cho1 1 1 1S .... ...1.1998 2.1997 k(1998 k 1) 1998 1 + + + + + + .Hy so snh S v 19982.1999.22. Chng minh rng : Nu s t nhin a khng phi l s chnh phng th a l s v t.23. Cho cc s x v y cng du. Chng minh rng :http://kinhhoa.violet.vn 198862646.doca)x y2y x+ b) 2 22 2x y x y0y x y x _ _+ + , ,c) 4 4 2 24 4 2 2x y x y x y2y x y x y x _ _ _+ + + + , , ,.24. Chng minh rng cc s sau l s v t : a) 1 2 +b) 3mn+ vi m, n l cc s hu t, n 0.25. C hai s v t dng no m tng l s hu t khng ?26. Cho cc s x v y khc 0. Chng minh rng : 2 22 2x y x y4 3y x y x _+ + + ,.27. Cho cc s x, y, z dng. Chng minh rng : 2 2 22 2 2x y z x y zy z x y z x+ + + + .28. Chng minh rng tng ca mt s hu t vi mt s v t l mt s v t.29. Chng minh cc bt ng thc : a) (a + b)2 2(a2 + b2)b) (a + b + c)2 3(a2 + b2 + c2)c) (a1 + a2 + .. + an)2 n(a12 + a22 + .. + an2).30. Cho a3 + b3 = 2. Chng minh rnga + b 2.31. Chng minh rng :[ ] [ ] [ ]x y x y + + .32. Tm gi tr ln nht ca biu thc :21Ax 6x 17 +.33. Tm gi tr nh nht ca :x y zAy z x + +vix, y, z > 0.34. Tm gi tr nh nht ca :A = x2 + y2bitx + y = 4.35. Tm gi tr ln nht ca :A = xyz(x + y)(y + z)(z + x)vi x, y, z 0 ; x + y + z = 1.36. Xt xem cc s a v b c th l s v t khng nu : a)ab v ab l s v t.b)a + b v ab l s hu t(a + b 0)c)a + b, a2 v b2 l s hu t(a + b 0)37. Cho a, b, c > 0. Chng minh :a3 + b3 + abc ab(a + b + c)38. Cho a, b, c, d > 0. Chng minh :a b c d2b c c d d a a b+ + + + + + +39. Chng minh rng[ ]2xbng[ ]2 xhoc[ ]2 x 1 +40. Cho s nguyn dng a. Xt cc s c dng :a + 15 ; a + 30 ; a + 45 ; ; a + 15n. Chng minh rng trong cc s , tn ti hai s m hai ch s u tin l 96.41. Tm cc gi tr ca x cc biu thc sau c ngha :http://kinhhoa.violet.vn 298862646.doc22 21 1 1 2A= x 3 B C D E x 2xxx 4x 5 1 x 3x 2x 1 + + + 2G 3x 1 5x 3 x x 1 + + +42.a) Chng minh rng :| A + B | | A | + | B | . Du = xy ra khi no ? b) Tm gi tr nh nht ca biu thc sau :2 2M x 4x 4 x 6x 9 + + + +. c) Gii phng trnh :2 2 24x 20x 25 x 8x 16 x 18x 81 + + + + + +43. Gii phng trnh :2 22x 8x 3 x 4x 5 12 .44. Tm cc gi tr ca x cc biu thc sau c ngha :2 221 1A x x 2 B C 2 1 9x D1 3xx 5x 6 + + +2 221 xE G x 2 H x 2x 3 3 1 xx 42x 1 x + + + +45. Gii phng trnh :2x 3x0x 346. Tm gi tr nh nht ca biu thc :A x x +.47. Tm gi tr ln nht ca biu thc :B 3 x x +48. So snh :a)3 1a 2 3 v b=2+ +b)5 13 4 3 v 3 1 + c) n 2 n 1 v n+1 n + + (n l s nguyn dng)49. Vi gi tr no ca x, biu thc sau t gi tr nh nht :2 2A 1 1 6x 9x (3x 1) + + .50. Tnh :a) 4 2 3 b) 11 6 2 c) 27 10 2 + 2 2d) A m 8m 16 m 8m 16 e) B n 2 n 1 n 2 n 1 + + + + + + (n 1)51. Rt gn biu thc :8 41M45 4 41 45 4 41+ + .52. Tm cc s x, y, z tha mn ng thc :2 2 2(2x y) (y 2) (x y z) 0 + + + + 53. Tm gi tr nh nht ca biu thc :2 2P 25x 20x 4 25x 30x 9 + + +.54. Gii cc phng trnh sau :2 2 2 2 2a) x x 2 x 2 0 b) x 1 1 x c) x x x x 2 0 + + + 4 2 2d) x x 2x 1 1 e) x 4x 4 x 4 0 g) x 2 x 3 5 + + + + + 2 2 2h) x 2x 1 x 6x 9 1 i) x 5 2 x x 25 + + + + + k) x 3 4 x 1 x 8 6 x 1 1 l) 8x 1 3x 5 7x 4 2x 2 + + + + + + + 55. Cho hai s thc x v y tha mn cc iu kin :xy = 1 v x > y. CMR:2 2x y2 2x y+.56. Rt gn cc biu thc :http://kinhhoa.violet.vn 398862646.doca) 13 30 2 9 4 2 b) m 2 m 1 m 2 m 1c) 2 3. 2 2 3 . 2 2 2 3 . 2 2 2 3 d) 227 30 2 123 22 2+ + + + + + + + + + + + + + +57. Chng minh rng6 22 32 2+ + .58. Rt gn cc biu thc :( ) ( )6 2 6 3 2 6 2 6 3 29 6 2 6a) C b) D2 3+ + + + .59. So snh : a) 6 20 v 1+ 6 b) 17 12 2 v 2 1 c) 28 16 3 v 3 2 + + + 60. Cho biu thc :2A x x 4x 4 +a) Tm tp xc nh ca biu thc A.b) Rt gn biu thc A.61. Rt gn cc biu thc sau :a) 11 2 10 b) 9 2 14 3 11 6 2 5 2 6c)2 6 2 5 7 2 10+ + ++ + +62. Cho a + b + c = 0 ; a, b, c 0. Chng minh ng thc : 2 2 21 1 1 1 1 1a b c a b c+ + + +63. Gii bt phng trnh :2x 16x 60 x 6 + < .64. Tm x sao cho : 2 2x 3 3 x + .65. Tm gi tr nh nht, gi tr ln nht ca A = x2 + y2 , bit rng :x2(x2 + 2y2 3) + (y2 2)2 = 1(1)66. Tm x biu thc c ngha: 221 16 xa) A b) B x 8x 82x 1x 2x 1 + ++ .67. Cho biu thc :2 22 2x x 2x x x 2xAx x 2x x x 2x+ + .a) Tm gi tr ca x biu thc A c ngha.b) Rt gn biu thc A.c)Tm gi tr ca x A < 2.68. Tm 20 ch s thp phn u tin ca s : 0, 9999....9 (20 ch s 9)69. Tm gi tr nh nht, gi tr ln nht ca : A = | x - 2 | + | y 1 | vi | x | + | y | = 570. Tm gi tr nh nht caA = x4 + y4 + z4bit rng xy + yz + zx = 171. Trong hai s :n n 2 v 2 n+1 + + (n l s nguyn dng), s no ln hn ?72. Cho biu thcA 7 4 3 7 4 3 + + . Tnh gi tr ca A theo hai cch.73. Tnh : ( 2 3 5)( 2 3 5)( 2 3 5)( 2 3 5) + + + + + +http://kinhhoa.violet.vn 498862646.doc74. Chng minh cc s sau l s v t : 3 5 ; 3 2 ; 2 2 3 + +75. Hy so snh hai s : a 3 3 3 v b=2 2 1 ; 5 12 5 v2++76. So snh4 7 4 7 2 + v s 0.77. Rt gn biu thc :2 3 6 8 4Q2 3 4+ + + ++ +.78. ChoP 14 40 56 140 + + +. Hy biu din P di dng tng ca 3 cn thc bc hai79. Tnh gi tr ca biu thc x2 + y2 bit rng :2 2x 1 y y 1 x 1 + .80. Tm gi tr nh nht v ln nht ca :A 1 x 1 x + +.81. Tm gi tr ln nht ca :( )2M a b +vi a, b > 0 v a + b 1.82. CMR trong cc s2b c 2 ad ; 2c d 2 ab ; 2d a 2 bc ; 2a b 2 cd + + + + c t nht hai s dng(a, b, c, d > 0).83. Rt gn biu thc :N 4 6 8 3 4 2 18 + + +.84. Chox y z xy yz zx + + + + , trong x, y, z > 0. Chng minh x = y = z.85. Cho a1, a2, , an > 0 v a1a2an = 1. Chng minh:(1 + a1)(1 + a2)(1 + an) 2n.86. Chng minh :( )2a b 2 2(a b) ab + +(a, b 0).87. Chng minh rng nu cc on thng c di a, b, c lp c thnh mt tam gic th cc on thng c dia , b , ccng lp c thnh mt tam gic.88. Rt gn :a)2ab b aAb b b)2(x 2) 8xB2xx+ .89. Chng minh rng vi mi s thc a, ta u c :22a 22a 1++. Khi no c ng thc ?90. Tnh :A 3 5 3 5 + + bng hai cch.91. So snh :a)3 7 5 2v 6,9 b) 13 12 v 7 65+ 92. Tnh :2 3 2 3P2 2 3 2 2 3+ ++ + .93. Gii phng trnh :x 2 3 2x 5 x 2 2x 5 2 2 + + + .94. Chng minh rng ta lun c :n1.3.5...(2n 1) 1P2.4.6...2n 2n 1 0 th2 2a ba bb a+ +.http://kinhhoa.violet.vn 598862646.doc96. Rt gn biu thc : A = 2x 4(x 1) x 4(x 1)1. 1x 1x 4(x 1) + + _ , .97. Chng minh cc ng thc sau :a b b a 1a) : a bab a b+ (a, b > 0 ; a b)14 7 15 5 1 a a a ab) : 2 c) 1 1 1 a1 2 1 3 7 5 a 1 a 1 _ _ _ + + +

+ , , ,(a > 0).98. Tnh :a)5 3 29 6 20 ; b) 2 3 5 13 48 + +.c) 7 48 28 16 3 . 7 48 _+ + ,.99. So snh : a) 3 5 v 15 b) 2 15 v 12 7 + + +16c) 18 19 v 9 d) v 5. 252+100. Cho hng ng thc :

2 2a a b a a ba b2 2+ t t(a, b > 0 v a2 b > 0).p dng kt qu rt gn : 2 3 2 3 3 2 2 3 2 2a) ; b)2 2 3 2 2 3 17 12 2 17 12 2+ ++ + + +2 10 30 2 2 6 2c) :2 10 2 2 3 1+ 101. Xc nh gi tr cc biu thc sau :2 22 2xy x 1. y 1a) Axy x 1. y 1 + vi1 1 1 1x a , y b2 a 2 b _ _ + + , ,(a > 1 ; b > 1)a bx a bxb) Ba bx a bx+ + + vi ( )22amx , m 1b 1 m 1 thP(x).P(- x) < 0.103. Cho biu thc2x 2 4 x 2 x 2 4 x 2A4 41x x+ + + + +.a) Rt gn biu thc A. b) Tm cc s nguyn x biu thc A l mt s nguyn.104. Tm gi tr ln nht (nu c) hoc gi tr nh nht (nu c) ca cc biu thc sau:http://kinhhoa.violet.vn 698862646.doc2a) 9 x b) x x (x 0) c) 1 2 x d) x 5 4 > + 2 21e) 1 2 1 3x g) 2x 2x 5 h) 1 x 2x 5 i)2x x 3 + + + +105. Rt gn biu thc :A x 2x 1 x 2x 1 + , bng ba cch ?106. Rt gn cc biu thc sau :a) 5 3 5 48 10 7 4 3 + +b) 4 10 2 5 4 10 2 5 c) 94 42 5 94 42 5 + + + + +.107. Chng minh cc hng ng thc vi b 0 ; a ba)( )2a b a b 2 a a b + t t b)2 2a a b a a ba b2 2+ t t108. Rt gn biu thc :A x 2 2x 4 x 2 2x 4 + + 109. Tm x v y sao cho : x y 2 x y 2 + + 110. Chng minh bt ng thc :( ) ( )2 22 2 2 2a b c d a c b d + + + + + +.111. Cho a, b, c > 0. Chng minh :2 2 2a b c a b cb c c a a b 2+ ++ + + + +.112. Cho a, b, c > 0 ; a + b + c = 1. Chng minh :a) a 1 b 1 c 1 3, 5 b) a b b c c a 6 + + + + + < + + + + + .113. CM : ( ) ( ) ( ) ( )2 2 2 2 2 2 2 2a c b c a d b d (a b)(c d) + + + + + + + vi a, b, c, d > 0.114. Tm gi tr nh nht ca :A x x +.115. Tm gi tr nh nht ca :(x a)(x b)Ax+ + .116. Tm gi tr nh nht, gi tr ln nht caA = 2x + 3ybit2x2 + 3y2 5.117. Tm gi tr ln nht ca A = x + 2 x .118. Gii phng trnh :x 1 5x 1 3x 2 119. Gii phng trnh :x 2 x 1 x 2 x 1 2 + + 120. Gii phng trnh :2 23x 21x 18 2 x 7x 7 2 + + + + + 121. Gii phng trnh :2 2 23x 6x 7 5x 10x 14 4 2x x + + + + + 122. Chng minh cc s sau l s v t :3 2 ; 2 2 3 +123. Chng minhx 2 4 x 2 + .124. Chng minh bt ng thc sau bng phng php hnh hc :2 2 2 2a b . b c b(a c) + + +vi a, b, c > 0.125. Chng minh (a b)(c d) ac bd + + + vi a, b, c, d > 0.126. Chng minh rng nu cc on thng c di a, b, c lp c thnh mt tam gic th cc on thng c dia , b , ccng lp c thnh mt tam gic.http://kinhhoa.violet.vn 798862646.doc127. Chng minh2(a b) a ba b b a2 4+ ++ + vi a, b 0.128. Chng minha b c2b c a c a b+ + >+ + + vi a, b, c > 0.129. Cho2 2x 1 y y 1 x 1 + . Chng minh rng x2 + y2 = 1.130. Tm gi tr nh nht caA x 2 x 1 x 2 x 1 + + 131. Tm GTNN, GTLN caA 1 x 1 x + +.132. Tm gi tr nh nht ca2 2A x 1 x 2x 5 + + +133. Tm gi tr nh nht ca2 2A x 4x 12 x 2x 3 + + + +.134. Tm GTNN, GTLN ca :( )2 2a) A 2x 5 x b) A x 99 101 x + + 135. Tm GTNN caA = x + y bit x, y > 0 tha mn a b1x y+ (a v b l hng s dng).136. Tm GTNN caA = (x + y)(x + z)vi x, y, z > 0 ,xyz(x + y + z) = 1.137. Tm GTNN caxy yz zxAz x y + +vi x, y, z > 0 , x + y + z = 1.138. Tm GTNN ca2 2 2x y zAx y y z z x + ++ + + bit x, y, z > 0 , xy yz zx 1 + + .139. Tm gi tr ln nht ca : a)( )2A a b + vi a, b > 0 , a + b 1b)( ) ( ) ( ) ( ) ( ) ( )4 4 4 4 4 4B a b a c a d b c b d c d + + + + + + + + + + +via, b, c, d > 0 v a + b + c + d = 1.140. Tm gi tr nh nht caA = 3x + 3yvi x + y = 4.141. Tm GTNN cab cAc d a b ++ + vi b + c a + d;b, c > 0;a, d 0.142. Gii cc phng trnh sau :2 2a) x 5x 2 3x 12 0 b) x 4x 8 x 1 c) 4x 1 3x 4 1 + + + d) x 1 x 1 2 e) x 2 x 1 x 1 1 g) x 2x 1 x 2x 1 2 + + + h) x 2 4 x 2 x 7 6 x 2 1 i) x x 1 x 1 + + + + + 2 2 2k) 1 x x x 1 l) 2x 8x 6 x 1 2x 2 + + + +2 2m) x 6 x 2 x 1 n) x 1 x 10 x 2 x 5 + + + + + + +( ) ( )2o) x 1 x 3 2 x 1 x 3x 5 4 2x + + + + p) 2x 3 x 2 2x 2 x 2 1 2 x 2 + + + + + + + +.2 2q) 2x 9x 4 3 2x 1 2x 21x 11 + + + 143. Rt gn biu thc :( ) ( )A 2 2 5 3 2 18 20 2 2 + +.http://kinhhoa.violet.vn 898862646.doc144. Chng minh rng, n Z+ , ta lun c : ( )1 1 11 .... 2 n 1 12 3 n+ + + + > + .145. Trc cn thc mu :1 1a) b)1 2 5 x x 1 + + + +.146. Tnh : a) 5 3 29 6 20 b) 6 2 5 13 48 c) 5 3 29 12 5 + + 147. Cho ( ) ( )a 3 5. 3 5 10 2 + . Chng minh rng a l s t nhin.148. Cho 3 2 2 3 2 2b17 12 2 17 12 2 + +. b c phi l s t nhin khng ?149. Gii cc phng trnh sau :( ) ( ) ( )( ) ( )a) 3 1 x x 4 3 0 b) 3 1 x 2 3 1 x 3 35 x 5 x x 3 x 3c) 2 d) x x 5 55 x x 3 + + + + + 150. Tnh gi tr ca biu thc : M 12 5 29 25 4 21 12 5 29 25 4 21 + + + 151. Rt gn :1 1 1 1A ...1 2 2 3 3 4 n 1 n + + + ++ + + +.152. Cho biu thc :1 1 1 1P ...2 3 3 4 4 5 2n 2n 1 + + +a)Rt gn P. b)P c phi l s hu t khng ?153. Tnh :1 1 1 1A ...2 1 1 2 3 2 2 3 4 3 3 4 100 99 99 100 + + + ++ + + +.154. Chng minh :1 1 11 ... n2 3 n+ + + + >.155. Cho a 17 1 . Hy tnh gi tr ca biu thc: A = (a5 + 2a4 17a3 a2 + 18a 17)2000.156. Chng minh :a a 1 a 2 a 3 < (a 3)157. Chng minh :21x x 02 + > (x 0)158. Tm gi tr ln nht ca S x 1 y 2 + , bit x + y = 4.159. Tnh gi tr ca biu thc sau vi 3 1 2a 1 2aa : A4 1 1 2a 1 1 2a+ ++ + .160. Chng minh cc ng thc sau :( ) ( ) ( )a) 4 15 10 6 4 15 2 b) 4 2 2 6 2 3 1 + + +( ) ( ) ( )2c) 3 5 3 5 10 2 8 d) 7 48 3 1 e) 17 4 9 4 5 5 22 + + + + 161. Chng minh cc bt ng thc sau :http://kinhhoa.violet.vn 998862646.doc5 5 5 5a) 27 6 48 b) 10 05 5 5 5+ + > + < +5 1 5 1 1c) 3 4 2 0, 2 1, 01 03 1 5 3 1 3 5 _ _+ + + >

+ + + , ,2 3 1 2 3 3 3 1d) 3 2 02 6 2 6 2 6 2 6 2 _+ + + + > + + ,e) 2 2 2 1 2 2 2 1 1, 9 g) 17 12 2 2 3 1 + + > + > ( )( )2 2 3 2 2h) 3 5 7 3 5 7 3 i) 0,84+ + + + + + < 0 ; a 1)187. Rt gn :( )2x 2 8x2xx+ (0 < x < 2)188. Rt gn :b ab a b a ba :a b ab b ab a ab _ + _+ + + + , ,189. Gii bt phng trnh :( )22 22 25a2 x x ax a+ + +(a 0)190. Cho( )21 a a 1 a aA 1 a : a a 11 a 1 a 1 _ _ + + + 1 + 1 , , ]a)Rt gn biu thc A.b)Tnh gi tr ca A vi a = 9.c)Vi gi tr no ca a th | A | = A.http://kinhhoa.violet.vn 1198862646.doc191. Cho biu thc :a b 1 a b b bBa ab 2 ab a ab a ab _+ + + + + ,.a) Rt gn biu thc B. b) Tnh gi tr ca B nu a 6 2 5 +.c) So snh B vi -1.192. Cho1 1 a bA : 1a a b a a b a b _+ _ + + + + , ,a) Rt gn biu thc A. b) Tm b bit | A | = -A.c) Tnh gi tr ca A khia 5 4 2 ; b 2 6 2 + + .193. Cho biu thca 1 a 1 1A 4 a aa 1 a 1 a _+ _ + + , ,a) Rt gn biu thc A.b) Tm gi tr ca A nu6a2 6+. c) Tm gi tr ca a A A >.194. Cho biu thca 1 a a a aA2 2 a a 1 a 1 _ _ +

+ , ,.a) Rt gn biu thc A. b) Tm gi tr ca A A = - 4195. Thc hin php tnh :1 a 1 a 1 a 1 aA :1 a 1 a 1 a 1 a _ _+ + + + + , ,196. Thc hin php tnh :2 3 2 3B2 2 3 2 2 3+ ++ + 197. Rt gn cc biu thc sau :( )3x y 1 1 1 2 1 1a) A : . .x y xy xy x y 2 xy x yx y 1 _ _ 1 + + + 1+ + ,+ , 1 ] vix 2 3 ; y 2 3 +.b)2 2 2 2x x y x x yB2(x y)+ vix > y > 0c)222a 1 xC1 x x++ vi1 1 a ax2 a 1 a _ , ;0 < a < 1d)( ) ( )2 22a 1 b 1D (a b)c 1+ + + + vi a, b, c > 0vab + bc + ca = 1e)x 2 x 1 x 2 x 1E . 2x 1x 2x 1 x 2x 1+ + + + http://kinhhoa.violet.vn 1298862646.doc198. Chng minh :2 2x 4 x 4 2x 4x xx x x ++ + vi x 2.199. Cho1 2 1 2a , b2 2 + .Tnh a7 + b7.200. Choa 2 1 a)Vit a2 ; a3 di dngm m 1 , trong m l s t nhin.b)Chng minh rng vi mi s nguyn dng n, s an vit c di dng trn.201. Cho bit x = 2 l mt nghim ca phng trnh x3 + ax2 + bx + c = 0 vi cc h s hu t. Tm cc nghim cn li.202. Chng minh1 1 12 n 3 ... 2 n 22 3 n < + + + < vin N ; n 2.203. Tm phn nguyn ca s6 6 ... 6 6 + + + +(c 100 du cn).204. Cho2 3a 2 3. Tnh a) a b) a11 + ] ].205. Cho 3 s x, y,x y +l s hu t. Chng minh rng mi sx , yu l s hu t206. CMR, n 1 , n N :1 1 1 1... 22 3 2 4 3 (n 1) n+ + + + 0 ; y > 0.265. Chng minh gi tr biu thc D khng ph thuc vo a:2 a a 2 a a a a 1Da 1 a 2 a 1 a _+ + + + ,vi a > 0;a 1 266. Cho biu thcc ac 1B aa c a ca cac c ac a ac _ + ++ ,+ + .a) Rt gn biu thc B.b) Tnh gi tr ca biu thc B khi c = 54 ; a = 24c) Vi gi tr no ca a v c B > 0 ; B < 0. 267. Cho biu thc :2 2 22mn 2mn 1A= m+ m 11+n 1 n n _+ + + ,vi m 0 ; n 1http://kinhhoa.violet.vn 1698862646.doca) Rt gn biu thc A. b) Tm gi tr ca A vi m 56 24 5 +.c) Tm gi tr nh nht ca A.268. Rt gn22 21 x 1 x 1 1 x xD 1x x 1 x 1 x1 x 1 x 1 x 1 x _ _+

+ + + , ,

269. Cho1 2 x 2 xP : 1x 1 x 1 x x x x 1 _ _ + + , ,vi x 0 ; x 1.a) Rt gn biu thc P. b) Tm x sao cho P < 0.270. Xt biu thc2x x 2x xy 1x x 1 x+ + + +.a) Rt gn y. Tm x y = 2. b) Gi s x > 1. Chng minh rng :y - | y | = 0c) Tm gi tr nh nht ca y ?PHN II: HNG DN GIIhttp://kinhhoa.violet.vn 1798862646.doc1. Gi s 7 l s hu t m7n(ti gin). Suy ra 22 22m7 hay 7n mn (1). ng thc ny chng t 2m 7 M m 7 l s nguyn t nn m M7. t m = 7k (k Z), ta c m2 = 49k2 (2). T (1) v (2) suy ra 7n2 = 49k2 nn n2 = 7k2(3). T (3) ta li c n2 M7 v v 7 l s nguyn t nn n M7. m v n cng chia ht cho 7 nn phn s mn khng ti gin, tri gi thit. Vy 7 khng phi l s hu t; do 7 l s v t.2. Khai trin v tri v t nhn t chung, ta c v phi. T a) b) v (ad bc)2 0.3. Cch 1 : T x + y = 2 ta c y = 2 x. Do :S = x2 + (2 x)2 = 2(x 1)2 + 2 2.Vy min S = 2x = y = 1.Cch 2 : p dng bt ng thc Bunhiacopxki vi a = x, c = 1, b = y, d = 1, ta c :(x + y)2 (x2 + y2)(1 + 1)4 2(x2 + y2) = 2SS 2. mim S = 2 khi x = y = 14. b) p dng bt ng thc Cauchy cho cc cp s dngbc ca bc ab ca abv ; v ; va b a c b c, ta ln lt c: bc ca bc ca bc ab bc ab2 . 2c; 2 . 2ba b a b a c a c+ + ;ca ab ca ab2 . 2ab c b c+ cng tng v ta c bt ng thc cn chng minh. Du bng xy ra khi a = b = c.c) Vi cc s dng3a v 5b , theo bt ng thc Cauchy ta c : 3a 5b3a.5b2+ . (3a + 5b)2 4.15P(v P = a.b)122 60PP 125max P = 125.Du bng xy ra khi 3a = 5b = 12 : 2a = 2 ; b = 6/5. 5. Ta c b = 1 a, do M = a3 + (1 a)3 = 3(a )2 + . Du = xy ra khi a = .Vy min M = a = b = .6. t a = 1 + x b3 = 2 a3 = 2 (1 + x)3 = 1 3x 3x2 x3 1 3x + 3x2 x3 = (1 x)3.Suy ra :b 1 x. Ta li c a = 1 + x, nn : a + b 1 + x + 1 x = 2.Vi a = 1, b = 1 th a3 + b3 = 2 v a + b = 2. Vymax N = 2 khi a = b = 1.7. Hiu ca v tri v v phi bng(a b)2(a + b).8. V| a + b | 0 ,| a b | 0 , nn :| a + b | > | a b |a2 + 2ab + b2 a2 2ab + b2 4ab > 0ab > 0. Vy a v b l hai s cng du.9.a)Xt hiu :(a + 1)2 4a = a2 + 2a + 1 4a = a2 2a + 1 = (a 1)20.b)Ta c :(a + 1)2 4a ; (b + 1)2 4b ; (c + 1)2 4c v cc bt ng thc ny c hai v u dng, nn : [(a + 1)(b + 1)(c + 1)]264abc = 64.1 = 82. Vy (a + 1)(b + 1)(c + 1) 8.10.a) Ta c :(a + b)2 + (a b)2 = 2(a2 + b2).Do(a b)2 0, nn(a + b) 22(a2 + b2).b)Xt :(a + b + c)2 + (a b)2 + (a c)2 + (b c)2. Khai trin v rt gn, ta c : 3(a2 + b2 + c2).Vy :(a + b + c)23(a2 + b2 + c2).11.a)42x 3 1 x 3x 4 x2x 3 1 x32x 3 x 1 x 2x 2

b)x2 4x5(x 2)233| x 2 |3-3 x 2 3-1 x 5.c)2x(2x 1)2x 1(2x 1)20. Nhng(2x 1)20, nn ch c th : 2x 1 = 0Vy :x = . 12.Vit ng thc cho di dng :a2 + b2 + c2 + d2 ab ac ad = 0(1). Nhn hai v ca (1) vi 4 ri a v dng : a2 + (a 2b)2 + (a 2c)2 + (a 2d)2 = 0 (2). Do ta c :http://kinhhoa.violet.vn 1898862646.doca = a 2b = a 2c = a 2d = 0 . Suy ra :a = b = c = d = 0.13.2M = (a + b 2)2 + (a 1)2 + (b 1)2 + 2.19982.1998 M1998.Du = xy ra khi c ng thi : a b 2 0a 1 0b 1 0+ ' Vymin M = 1998a = b = 1.14.Gii tng t bi 13.15.a ng thc cho v dng :(x 1)2 + 4(y 1)2 + (x 3)2 + 1 = 0.16.( )2 21 1 1 1A . max A= x 2x 4x 9 5 5x 2 5 + +.17.a)7 15 9 16 3 4 7 + < + + .Vy7 15 + < 7b)17 5 1 16 4 1 4 2 1 7 49 45 + + > + + + + >.c)23 2 19 23 2 16 23 2.45 25 273 3 3 < < .d)Gi s ( ) ( )2 23 2 2 3 3 2 2 3 3 2 2 3 18 12 18 12 > > > > > .Bt ng thc cui cng ng, nn :3 2 2 3 >.18.Cc s c th l 1,42 v 2 32+19.Vit li phng trnh di dng :2 2 23(x 1) 4 5(x 1) 16 6 (x 1) + + + + + +.V tri ca phng trnh khng nh hn 6, cn v phi khng ln hn 6. Vy ng thc ch xy ra khi c hai v u bng 6, suy ra x = -1.20.Bt ng thc Cauchy a bab2+vit li di dng 2a bab2+ _

,(*) (a, b 0).p dng bt dng thc Cauchy di dng (*) vi hai s dng 2x v xy ta c :22x xy2x.xy 42+ _ ,Du = xy ra khi :2x = xy = 4 : 2 tc l khi x = 1, y = 2. max A = 2x = 2, y = 2.21.Bt ng thc Cauchy vit li di dng : 1 2a b ab >+. p dng ta c S > 19982.1999.22.Chng minh nh bi 1.23.a)2 2 2x y x y 2xy (x y)2 0y x xy xy+ + .Vyx y2y x+ b)Ta c :2 2 2 22 2 2 2x y x y x y x y x yA 2y x y x y x y x y x _ _ _ _ _ + + + + + + , , , , ,. Theo cu a :222 22 2x y x y x yA 2 2 1 1 0y x y x y x _ _ _ _ + + + + , , , ,http://kinhhoa.violet.vn 1998862646.docc)T cu b suy ra : 4 4 2 24 4 2 2x y x y0y x y x _ _+ + , ,.Vx y2y x+ (cu a).Do :4 4 2 24 4 2 2x y x y x y2y x y x y x _ _ _+ + + + , , ,.24.a)Gi s 1 2 + = m(m : s hu t) 2 = m2 1 2 l s hu t (v l)b)Gi sm + 3n = a(a : s hu t) 3n = a m 3 = n(a m) 3 l s hu t, v l.25.C, chng hn 2 (5 2) 5 + 26.t2 222 2x y x ya 2 ay x y x+ + + . D dng chng minh 2 22 2x y2y x+ nn a2 4, do | a | 2(1).Bt ng thc phi chng minh tng ng vi :a2 2 + 4 3aa2 3a + 2 0(a 1)(a 2) 0(2)T(1)suy raa2 hoca-2. Nu a 2 th(2)ng. Nu a -2 th(2) cng ng. Bi ton c chng minh.27.Bt ng thc phi chng minh tng ng vi :( )4 2 4 2 4 2 2 2 22 2 2x z y x z x x z y x z y xyz0x y z+ + + +.Cn chng minh t khng m, tc l :x3z2(x y) + y3x2(y z) + z3y2(z x)0. (1)Biu thc khng i khi hon v vngx y z x nn c th gi s x l s ln nht. Xt hai trng hp :a)xyz> 0. Tch z x (1) thnh (x y + y z), (1) tng ng vi :x3z2(x y) + y3x2(y z) z3y2(x y) z3y2(y z) 0z2(x y)(x3 y2z) + y2(y z)(yx2 z3) 0D thy x y 0 , x3 y2z 0 , y z 0 , yx2 z3 0 nn bt ng thc trn ng.b)x z y > 0. Tch x y (1) thnh x z + z y , (1) tng ng vi :x3z2(x z) + x3z2(z y) y3x2(z y) z3y2(x z) 0z2(x z)(x3 zy2) + x2(xz2 y3)(z y) 0D thy bt ng thc trn dng.Cch khc : Bin i bt ng thc phi chng minh tng ng vi :22 2x y z x y z1 1 1 3y z x y z x _ _ _ _ + + + + + , , , ,.28.Chng minh bng phn chng. Gi s tng ca s hu t a vi s v t b l s hu t c. Ta c : b = c a. Ta thy, hiu ca hai s hu t c v a l s hu t, nn b l s hu t, tri vi gi thit. Vy c phi l s v t.29.a)Ta c :(a + b)2 + (a b)2 = 2(a2 + b2) (a + b)2 2(a2 + b2).b)Xt : (a + b + c)2 + (a b)2 + (a c)2 + (b c)2. Khai trin v rt gn ta c :3(a2 + b2 + c2). Vy : (a + b + c)2 3(a2 + b2 + c2)c)Tng t nh cu b30.Gi s a + b > 2 (a + b)3 > 8a3 + b3 + 3ab(a + b) > 82 + 3ab(a + b) > 8 ab(a + b) > 2 ab(a + b) > a3 + b3. Chia hai v cho s dng a + b :ab > a2 ab + b2 (a b)2 < 0, v l. Vy a + b 2.http://kinhhoa.violet.vn 2098862646.doc31.Cch 1: Ta c :[ ]xx ; [ ]yy nn [ ]x+ [ ]yx + y. Suy ra [ ]x+ [ ]yl s nguyn khng vt qu x + y (1). Theo nh ngha phn nguyn, [ ]x y +l s nguyn ln nht khng vt qu x + y(2). T (1) v (2) suy ra :[ ]x+ [ ]y[ ]x y + .Cch 2 :Theo nh ngha phn nguyn :0 x - [ ]x< 1 ;0 y - [ ]y< 1.Suy ra :0 (x + y) ([ ]x+ [ ]y ) < 2. Xt hai trng hp :- Nu0 (x + y) ([ ]x+ [ ]y ) < 1 th [ ]x y += [ ]x+ [ ]y(1)- Nu1 (x + y) ([ ]x+ [ ]y ) < 2 th0 (x + y) ([ ]x+ [ ]y+ 1) < 1 nn[ ]x y += [ ]x+ [ ]y+ 1 (2). Trong c hai trng hp ta u c : [ ]x+ [ ]y[ ]x y +32.Ta cx2 6x + 17 = (x 3)2 + 8 8 nn t v mu ca A l cc s dng , suy ra A > 0 do : A ln nht 1A nh nhtx2 6x + 17 nh nht.Vy max A = 18x = 3.33.Khng c dng php hon v vng quanh x y z x v gi sx y z.Cch 1 :p dng bt ng thc Cauchy cho 3 s dng x, y, z :3x y z x y zA 3 . . 3y z x y z x + + Do x y z x y zmin 3 x y zy z x y z x _+ + ,Cch 2 : Ta c : x y z x y y z yy z x y x z x x _ _+ + + + + , ,. Ta c x y2y x+ (do x, y > 0) nn chng minh x y z3y z x+ + ta ch cn chng minh : y z y1z x x+ (1)(1)xy + z2 yz xz(nhn hai v vi s dng xz)xy + z2 yz xz 0y(x z) z(x z) 0(x z)(y z) 0(2)(2) ng vi gi thit rng z l s nh nht trong 3 s x, y, z, do (1) ng. T tm c gi tr nh nht ca x y zy z x+ +.34.Ta c x + y = 4 x2 + 2xy + y2 = 16. Ta li c (x y)2 0 x2 2xy + y2 0. T suy ra 2(x2 + y2) 16 x2 + y2 8.min A = 8 khi v ch khi x = y = 2.35.p dng bt ng thc Cauchy cho ba s khng m : 1 = x + y + z 3.3xyz (1)2 = (x + y) + (y + z) + (z + x) 3.3(x y)(y z)(z x) + + +(2)Nhn tng v ca (1) vi (2)(do hai v u khng m) :2 9.3A A 329 _ ,max A = 329 _ , khi v ch khi x = y = z = 13.36.a)C th. b, c)Khng th.http://kinhhoa.violet.vn 2198862646.doc37.Hiu ca v tri v v phi bng (a b)2(a + b).38.p dng bt ng thc21 4xy (x y)+ vi x, y > 0 :2 2 2 22a c a ad bc c 4(a ad bc c )b c d a (b c)(a d) (a b c d)+ + + + + ++ + + + + + + +(1)Tng t2 22b d 4(b ab cd d )c d a b (a b c d)+ + ++ + + + + + (2) Cng (1) vi (2) 2 2 2 22a b c d 4(a b c d ad bc ab cd)b c c d d a a b (a b c d)+ + + + + + ++ + + + + + + + + += 4BCn chng minh B 12, bt ng thc ny tng ng vi :2B 12(a2 + b2 + c2 + d2 + ad + bc + ab + cd) (a + b + c + d)2a2 + b2 + c2 + d2 2ac 2bd 0(a c)2 + (b d)2 0 :ng.39.- Nu 0 x - [ ]x< th 0 2x - 2[ ]x< 1 nn [ ]2x =2[ ]x . - Nu x - [ ]x< 1 th 1 2x - 2[ ]x< 2 0 2x (2[ ]x+ 1) < 1[ ]2x= 2[ ]x+ 140.Ta s chng minh tn ti cc s t nhin m, p sao cho : 142 43mchso096000...00 a + 15p < 142 43mchso097000...00Tc l 96 +m ma 15p10 10 < 97(1).Gi a + 15 l s c k ch s : 10k 1 a + 15 < 10k + + + + < + .49.A = 1 - | 1 3x | + | 3x 1 |2 =( | 3x 1| - )2 + .T suy ra :min A = x = hocx = 1/651.M = 452.x = 1 ; y = 2 ; z = -3.53.P = | 5x 2 | + | 3 5x || 5x 2 + 3 5x |=1.min P = 12 3x5 5 .54.Cn nh cch gii mt s phng trnh dng sau : 2B 0 A 0 (B 0) A 0a) A B b) A B c) A B 0A B B 0 A B + ' ' ' B 0A 0d) A B e) A B 0 A BB 0A B + ' ' .a)a phng trnh v dng :A B .b)a phng trnh v dng :A B .c)Phng trnh c dng :A B 0 + .d)a phng trnh v dng :A B .e)a phng trnh v dng :| A | + | B | = 0g, h, i)Phng trnh v nghim.k)t x 1 = y 0, a phng trnh v dng : | y 2 | + | y 3 | = 1 . Xt du v tri.l)t : 8x 1 u 0 ; 3x 5 v 0 ; 7x 4 z 0 ; 2x 2 t 0 + + .Ta c h :2 2 2 2u v z tu v z t+ + ' . T suy ra : u = z tc l :8x 1 7x 4 x 3 + + .55.Cch 1 : Xt 2 2 2 2 2x y 2 2(x y) x y 2 2(x y) 2 2xy (x y 2) 0 + + + .Cch 2 : Bin i tng ng( )( )22 22 22x yx y2 2 8x yx y++ (x2 + y2)2 8(x y)2 0 (x2 + y2)2 8(x2 + y2 2) 0 (x2 + y2)2 8(x2 + y2) + 16 0 (x2 + y2 4)2 0.Cch 3 : S dng bt ng thc Cauchy : 2 2 2 2 2x y x y 2xy 2xy (x y) 2.1 2 1(x y) 2 (x y).x y x y x y x y x y+ + + + + (x > y).http://kinhhoa.violet.vn 2398862646.docDu ng thc xy ra khi6 2 6 2x ; y2 2+ hoc 6 2 6 2x ; y2 2 + 62.22 2 2 2 2 21 1 1 1 1 1 1 1 1 1 1 1 2(c ba2a b c a b c ab bc ca a b c abc+ + _ _+ + + + + + + + + + , , == 2 2 21 1 1a b c+ + . Suy ra iu phi chng minh.63.iu kin :2x 6(x6)(x 10) 0 x 16x 60 0x 10 x 10x 6 x6 0x 6 +

' ' ' .Bnh phng hai v :x2 16x + 60 < x2 12x + 36x > 6.Nghim ca bt phng trnh cho :x 10.64.iu kin x2 3. Chuyn v :2x 3 x2 3 (1)t tha chung :2x 3 .(1 - 2x 3 )022x 3x 3 0x 21 x 3 0x 2

t

Vy nghim ca bt phng trnh :x = 3 t ; x 2 ; x -2.65.Ta cx2(x2 + 2y2 3) + (y2 2)2 = 1(x2 + y2)2 4(x2 + y2) + 3 = - x2 0.Do : A2 4A + 3 0(A 1)(A 3) 01 A 3.min A = 1x = 0, khi y = 1.max A = 3x = 0, khi y = 3.66.a) x 1.b)B c ngha2224 x 44 x 416x 0x 42 212x10 (x 4) 8 x 42 22x 42 21x 8x 80x12x2

+ > <

' ' ' + + > > .67.a)A c ngha22 22x 2x 0 x(x2) 0 x 2x 0 x x 2xx x 2x ' '

< t b)A = 22 x 2x vi iu kin trn.c)A < 22x 2x < 1x2 2x < 1(x 1)2 < 2- 2 < x 1 > +( ) ( )2 23 3 2 2 2 27 8 4 8 2 15 8 2 225 128 > + > + + > > . Vy a > b l ng.b)Bnh phng hai v ln ri so snh.76.Cch 1 : t A = 4 7 4 7 + , r rng A > 0 v A2 = 2 A = 2Cch 2 : t B = 4 7 4 7 2 2.B 8 2 7 8 2 7 2 0 + + B =0.77.( ) ( )2 3 4 2 2 3 42 3 2.3 2.4 2 4Q 1 22 3 4 2 3 4+ + + + ++ + + + ++ + + +.78.Vit 40 2 2.5 ; 56 2 2.7 ; 140 2 5.7 . VyP = 2 5 7 + +.79.T gi thit ta c :2 2x 1 y 1 y 1 x . Bnh phng hai v ca ng thc ny ta c : 2y 1 x . T :x2 + y2 = 1.80.Xt A2 suy ra : 2 A2 4. Vy : min A = 2x = 1 ;max A = 2x = 0.81.Ta c :( ) ( ) ( )2 2 2M a b a b a b 2a 2b 2 + + + + .1 a bmax M 2 a b2a b 1 '+ .http://kinhhoa.violet.vn 2598862646.doc82.Xt tng ca hai s : ( ) ( ) ( ) ( )2a b 2 cd 2c d 2 ab a b 2 ab c d 2 cd a c + + + + + + + + = = ( )( ) ( )2 2a c a b c d a c 0 + + + + > .83.N 4 6 8 3 4 2 18 12 8 3 4 4 6 4 2 2 + + + + + + + + = = ( ) ( ) ( )2 22 3 2 2 2 2 3 2 2 2 3 2 2 2 3 2 2 + + + + + + + +.84.Tx y z xy yz zx + + + + ( ) ( ) ( )2 2 2x y y z z x 0 + + .Vy x = y = z.85.p dng bt ng thc Cauchy cho 1 v ai( i = 1, 2, 3, n ).86.p dng bt ng thc Cauchy vi hai s a + b 0 v 2ab 0, ta c :( )2a b 2 ab 2 2(a b) ab hay a b 2 2(a b) ab + + + + + .Du = xy ra khi a = b.87.Gi sa b c > 0. Ta cb + c > a nnb + c + 2bc > ahay ( ) ( )2 2b c a + >Do :b c a + >. Vy ba on thng a , b , clp c thnh mt tam gic.88.a)iu kin :ab 0 ; b 0. Xt hai trng hp :* Trng hp 1 :a 0 ; b > 0 : b.( a b) a a b aA 1b b b. b b .* Trng hp 2 :a 0 ; b < 0 : 22ab b a a a aA 1 1 2b b b bb + .b)iu kin :2(x 2) 8x 0x 0x 0x 22x 0x+ > > ' ' .Vi cc iu kin th :2 2x 2 . x (x 2) 8x (x 2) . xB2x 2 x 2xx + . Nu0 < x < 2 th | x 2 | = -(x 2) v B = -x . Nu x > 2 th | x 2 | = x 2 v B =x89.Ta c :( )22222 2 2a 1 1a 2 1a 1a 1 a 1 a 1+ ++ + ++ + +. p dng bt ng thc Cauchy:2 22 21 1a 1 2 a 1. 2a 1 a 1+ + + + +.Vy22a 22a 1++.ng thc xy ra khi :http://kinhhoa.violet.vn 2698862646.doc221a 1 a 0a 1+ +.93.Nhn 2 v ca pt vi 2, ta c :2x 5 3 2x 5 1 4 + + 5/2 x 3.94.Ta chng minh bng qui np ton hc : a)Vi n = 1 ta c :11 1P2 3 + >Vy ba on thng c di b , c , a lp c thnh mt tam gic.127.Ta c a, b 0. Theo bt ng thc Cauchy :2(ab) ab ab 1 1ab ab ab2 4 2 2 2+ + + _ _+ + + + + , ,http://kinhhoa.violet.vn 30c abC BA98862646.docCn chng minh :1ab ab2 _+ + , a b b a +. Xt hiu hai v :1ab ab2 _+ + , - ( )ab a b + = 1ab ab a b2 _+ + , ==2 21 1ab a b2 2 1 _ _ + 1 , , 1 ] 0Xy ra du ng thc :a = b = 14hoca = b = 0.128.Theo bt ng thc Cauchy :bc bc bca.1 1 : 2a a 2a+ + + + _ + ,.Do :a 2abc abc+ + +. Tng t :b 2b c 2c;ac abc ab abc + + + + + +Cng tng v :a b c 2(abc)2bc ca ab abc+ ++ + + + + + +.Xy ra du ng thc :a bcb c a abc 0c ab + + + + ' +, tri vi gi thit a, b, c > 0.Vy du ng thc khng xy ra.129.Cch 1 : Dng bt ng thc Bunhiacpxki. Ta c :( )( ) ( )22 2 2 2 2 2x 1 y y 1 x x y 1 y 1 x + + .tx2 + y2 = m, ta c :12 m(2 - m) (m 1)2 0 m = 1(pcm).Cch 2 : T gi thit :2 2x 1 y 1 y 1 x . Bnh phng hai v :x2(1 y2) = 1 2y21 x + y2(1 x2) x2 = 1 2y21 x + y20 = (y - 21 x )2 y = 21 x x2 + y2 = 1 .130.p dng| A | + | B | | A + B | .min A = 21 x2 .131. XtA2 = 2 + 221 x . Do0 21 x 1 2 2 + 221 x 4 2 A2 4.min A =2 vi x = 1 ,max A = 2 vi x = 0.132.p dng bt ng thc :2 2 2 2 2 2a b c d (ac) (bd) + + + + + + (bi 23)2 2 2 2 2 2A x 1 (1 x) 2 (x1 x) (1 2) 10 + + + + + + 1 x 1minA 10 2 xx 3 .133.Tp xc nh :22x 4x12 0 (x 2)(6x) 01x 3(x1)(3 x) 0x 2x30 + + + ' '+ + + (1)Xt hiu :(- x2 + 4x + 12)(- x2 + 2x + 3) = 2x + 9. Do (1) nn 2x + 9 > 0 nn A > 0.http://kinhhoa.violet.vn 3198862646.docXt : ( ) 22A (x 2)(6x) (x1)(3x) + + . Hin nhin A2 0 nhng du = khng xy ra (v A > 0). Ta bin i A2 di dng khc :A2 = (x + 2)(6 x) + (x + 1)(3 x) - 2 (x 2)(6x)(x1)(3x) + + == (x + 1)(6 x) + (6 x) + (x + 2)(3 x) (3 x) - 2 (x 2)(6x)(x1)(3x) + + = (x + 1)(6 x) + (x + 2)(3 x) - 2 (x 2)(6x)(x1)(3x) + + + 3= ( )2(x1)(6x) (x 2)(3x) 3 + + + .A2 3. Do A > 0 nnmin A = 3 vi x = 0.134.a)iu kin : x2 5.* Tm gi tr ln nht : p dng bt ng thc Bunhiacpxki :A2 = (2x + 1.25x )2 (22 + 11)(x2 + 5 x2) = 25 A2 25.22 2 222x 0x5xA 25 x 4(5x ) x 2 2x 5x 5 ' ' .Vi x = 2 th A = 5. Vymax A = 5 vi x = 2.*Tm gi tr nh nht : Ch rng tuy t A2 25, ta c 5 x 5, nhng khng xy ra A2 = - 5. Do tp xc nh ca A, ta c x2 5 -5 x 5. Do : 2x - 2 5 v25x 0. Suy ra :A = 2x + 25 x - 25.Min A = - 25 vi x = -5b)Xt biu thc ph | A | v p dng cc bt ng thc Bunhiacpxki v Cauchy :( )2 2 22 2A x 99. 991. 101 x x (99 1)(99 101 x ) x.10. 200xx 200x10. 10002 + + + 0. Ta xt biu thc : 21B 2 3 xA . Ta c :2 2 20 3 x 3 3 3 x 0 2 3 2 3 x 2 .http://kinhhoa.violet.vn 3598862646.doc2min B 2 3 3 3 x x 0 .Khi 1max A 2 32 3 + 2max B 2 3 x 0 x 3 t . Khi min A = 12181. p dng bt ng thc Cauchy, ta xt biu thc : 2x 1 xB1 x x +. Khi :2x 1 x(1) 2x 1 xB 2 . 2 2. B 2 21 x x1 x x0 x 1 (2) '< 0. Ta c 22x 4Ax+. Suy ra iu phi chng minh.209.Ta c :a + b = - 1 , ab = - 14 nn :a2 + b2 = (a + b)2 2ab = 1 + 1 32 2 .a4 + b4 = (a2 + b2)2 2a2b2 = 9 1 174 9 8 ;a3 + b3 = (a + b)3 3ab(a + b) = - 1 - 3 74 4 Do : a7 + b7 = (a3 + b3)(a4 + b4) a3b3(a + b) =( )7 17 1 239. 14 8 64 64 _ ,.210.a)2 2a ( 2 1) 3 2 2 9 8 .3 3a ( 2 1) 2 2 6 3 2 1 5 2 7 50 49 + .b)Theo khai trin Newton : (1 - 2)n = A - B2;(1 + 2)n = A + B2 vi A, B NSuy ra : A2 2B2 = (A + B2)(A - B2) = [(1 + 2)(1 - 2)]n = (- 1)n.Nu n chn th A2 2b2 = 1 (1). Nu n l th A2 2B2 = - 1 (2).By gi ta xt an. C hai trng hp :* Nu n chn th : an = (2 - 1)n = (1 - 2)n = A - B2 = 2 2A 2B . iu kin A2 2B2 = 1 c tha mn do (1).* Nu n l th : an = (2 - 1)n = - (1 - 2)n = B2 - A = 2 22B A . iu kin2B2 A2 = 1 c tha mn do (2).211.Thay a = 2 vo phng trnh cho : 22 + 2a + b2 + c = 02 (b + 2) = -(2a + c).Do a, b, c hu t nn phi c b + 2 = 0 do 2a + c = 0. Thay b = - 2 , c = - 2a vo phng trnh cho :x3 + ax2 2x 2a = 0 x(x2 2) + a(x2 2) = 0 (x2 2)(x + a) = 0.Cc nghim phng trnh cho l: 2v- a.212.t1 1 1A ...2 3 n + + + .a)Chng minh A 2 n 3 > : Lm gim mi s hng ca A :( )1 2 22 k 1 kk k k k 1 k > + + + + .Do ( ) ( ) ( )A 2 2 3 3 4 ... n n 1 1> + + + + + + + ]( )2 n 1 2 2 n 1 2 2 2 n 1 3 2 n 3 + + > + > .b)Chng minh A 2 n 2 < : Lm tri mi s hng ca A :( )1 2 22 k k 1k k k k k 1 < + + Do :( ) ( ) ( )A 2 n n 1 ... 3 2 2 1 2 n 2 1< + + + ].213.K hiu na 6 6 ... 6 6 + + + + c n du cn.Ta c : http://kinhhoa.violet.vn 3998862646.doc1 2 1 3 2 100 99a 6 3 ; a 6 a 6 3 3 ; a 6 a 6 3 3 ... a 6 a 6 3 3 < + < + + < + + < + Hin nhina100 > 6 > 2. Nh vy 2 < a100 < 3, do [ a100 ] = 2.214.a)Cch 1 (tnh trc tip) :a2 = (2 + 3)2 = 7 + 43.Ta c 4 3 48 nn 6 < 43 < 7 13 0T h phng trnh cho ta c :2y 2yx y1 y 2 y +.Tng t y z ; z x .Suy rax = y = z.Xy ra du = cc bt ng thc trn vi x = y = z = 1. Kt lun :Hai nghim (0 ; 0 ; 0),(1 ; 1 ; 1).221.a)tA = (8 + 37)7. chng minh bi ton, ch cn tm s B sao cho0 < B < 7110 v A + B l s t nhin.Chn B = (8 - 37)7. D thy B > 0 v 8 > 37. Ta c8 + 37 > 10suy ra :( )( )77 7 71 1 18 3 710 108 3 7