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PAGE 45CHUYN : BI DNG HS GII V NNG KHIU

WWW.ToanTrungHocCoSo.ToanCapBa.Net

Mt s bi tp ton nng cao LP 9

PHN I: BI1. Chng minh 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 : .

b) Cho a, b, c > 0. Chng minh rng :

c) Cho a, b > 0 v 3a + 5b = 12. Tm gi tr ln nht ca tch P = 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 :

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) 8

10. 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 = 0

16. Tm gi tr ln nht ca biu thc :

17. So snh cc s thc sau (khng dng my tnh) :

a)

b)

c)

d)

18. Hy vit mt s hu t v mt s v t ln hn nhng nh hn

19. Gii phng trnh : .

20. Tm gi tr ln nht ca biu thc A = x2y vi cc iu kin x, y > 0 v 2x + xy = 4.

21. Cho .

Hy so snh S v .

22. Chng minh rng : Nu s t nhin a khng phi l s chnh phng th l s v t.

23. Cho cc s x v y cng du. Chng minh rng :

a)

b)

c) .

24. Chng minh rng cc s sau l s v t :

a)

b) 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 : .

27. Cho cc s x, y, z dng. Chng minh rng : .

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 rng a + b 2.

31. Chng minh rng : .

32. Tm gi tr ln nht ca biu thc : .

33. Tm gi tr nh nht ca : vi x, y, z > 0.

34. Tm gi tr nh nht ca : A = x2 + y2 bit x + 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 l s v t.

b) a + b v 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 :

39. Chng minh rng bng hoc

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 :

EMBED Equation.DSMT4 42. a) Chng minh rng : | A + B | | A | + | B | . Du = xy ra khi no ?

b) Tm gi tr nh nht ca biu thc sau : .

c) Gii phng trnh :

43. Gii phng trnh : .

44. Tm cc gi tr ca x cc biu thc sau c ngha :

45. Gii phng trnh :

46. Tm gi tr nh nht ca biu thc : .

47. Tm gi tr ln nht ca biu thc :

48. So snh : a) b)

c) (n l s nguyn dng)

49. Vi gi tr no ca x, biu thc sau t gi tr nh nht : .

50. Tnh :

(n 1)

51. Rt gn biu thc : .

52. Tm cc s x, y, z tha mn ng thc :

53. Tm gi tr nh nht ca biu thc : .

54. Gii cc phng trnh sau :

55. Cho hai s thc x v y tha mn cc iu kin : xy = 1 v x > y. CMR: .

56. Rt gn cc biu thc :

57. Chng minh rng .

58. Rt gn cc biu thc :

.

59. So snh :

60. Cho biu thc :

a) Tm tp xc nh ca biu thc A.

b) Rt gn biu thc A.

61. Rt gn cc biu thc sau :

62. Cho a + b + c = 0 ; a, b, c 0. Chng minh ng thc :

63. Gii bt phng trnh : .

64. Tm x sao cho : .

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: .

67. Cho biu thc : .

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 : (20 ch s 9)

69. Tm gi tr nh nht, gi tr ln nht ca : A = | x - | + | y 1 | vi | x | + | y | = 5

70. Tm gi tr nh nht ca A = x4 + y4 + z4 bit rng xy + yz + zx = 1

71. Trong hai s : (n l s nguyn dng), s no ln hn ?

72. Cho biu thc . Tnh gi tr ca A theo hai cch.

73. Tnh :

74. Chng minh cc s sau l s v t :

75. Hy so snh hai s : ;

76. So snh v s 0.

77. Rt gn biu thc : .

78. Cho . Hy biu din P di dng tng ca 3 cn thc bc hai

79. Tnh gi tr ca biu thc x2 + y2 bit rng : .

80. Tm gi tr nh nht v ln nht ca : .

81. Tm gi tr ln nht ca : vi a, b > 0 v a + b 1.

82. CMR trong cc s c t nht hai s dng (a, b, c, d > 0).

83. Rt gn biu thc : .

84. Cho , 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 : (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 di cng lp c thnh mt tam gic.

88. Rt gn : a) b) .

89. Chng minh rng vi mi s thc a, ta u c : . Khi no c ng thc ?

90. Tnh : bng hai cch.

91. So snh : a)

92. Tnh : .

93. Gii phng trnh : .

94. Chng minh rng ta lun c : ; (n ( Z+95. Chng minh rng nu a, b > 0 th .

96. Rt gn biu thc : A = .

97. Chng minh cc ng thc sau :

(a, b > 0 ; a b)

(a > 0).

98. Tnh : .

.

99. So snh :

100. Cho hng ng thc :

(a, b > 0 v a2 b > 0).

p dng kt qu rt gn :

EMBED Equation.DSMT4 101. Xc nh gi tr cc biu thc sau :

vi

(a > 1 ; b > 1)

vi .

102. Cho biu thc

a) Tm tt c cc gi tr ca x P(x) xc nh. Rt gn P(x).

b) Chng minh rng nu x > 1 th P(x).P(- x) < 0.

103. Cho biu thc .

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:

105. Rt gn biu thc : , bng ba cch ?

106. Rt gn cc biu thc sau :

.

107. Chng minh cc hng ng thc vi b 0 ; a

a) b)

108. Rt gn biu thc :

109. Tm x v y sao cho :

110. Chng minh bt ng thc : .

111. Cho a, b, c > 0. Chng minh : .

112. Cho a, b, c > 0 ; a + b + c = 1. Chng minh :

.

113. CM : vi a, b, c, d > 0.

114. Tm gi tr nh nht ca : .

115. Tm gi tr nh nht ca : .

116. Tm gi tr nh nht, gi tr ln nht ca A = 2x + 3y bit 2x2 + 3y2 5.

117. Tm gi tr ln nht ca A = x + .

118. Gii phng trnh :

119. Gii phng trnh :

120. Gii phng trnh :

121. Gii phng trnh :

122. Chng minh cc s sau l s v t :

123. Chng minh .

124. Chng minh bt ng thc sau bng phng php hnh hc :

vi a, b, c > 0.

125. Chng minh 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 di cng lp c thnh mt tam gic.

127. Chng minh vi a, b 0.

128. Chng minh vi a, b, c > 0.

129. Cho . Chng minh rng x2 + y2 = 1.

130. Tm gi tr nh nht ca

131. Tm GTNN, GTLN ca .

132. Tm gi tr nh nht ca

133. Tm gi tr nh nht ca .

134. Tm GTNN, GTLN ca :

135. Tm GTNN ca A = x + y bit x, y > 0 tha mn (a v b l hng s dng).

136. Tm GTNN ca A = (x + y)(x + z) vi x, y, z > 0 , xyz(x + y + z) = 1.

137. Tm GTNN ca vi x, y, z > 0 , x + y + z = 1.

138. Tm GTNN ca bit x, y, z > 0 , .

139. Tm gi tr ln nht ca : a) vi a, b > 0 , a + b 1

b)

vi a, b, c, d > 0 v a + b + c + d = 1.

140. Tm gi tr nh nht ca A = 3x + 3y vi x + y = 4.

141. Tm GTNN ca vi b + c a + d ; b, c > 0 ; a, d 0.

142. Gii cc phng trnh sau :

EMBED Equation.DSMT4

EMBED Equation.DSMT4

.

143. Rt gn biu thc : .

144. Chng minh rng, (n ( Z+ , ta lun c : .

145. Trc cn thc mu : .

146. Tnh :

147. Cho . Chng minh rng a l s t nhin.

148. Cho . b c phi l s t nhin khng ?

149. Gii cc phng trnh sau :

150. Tnh gi tr ca biu thc :

151. Rt gn : .

152. Cho biu thc :

a) Rt gn P.

b) P c phi l s hu t khng ?

153. Tnh : .

154. Chng minh : .

155. Cho . Hy tnh gi tr ca biu thc: A = (a5 + 2a4 17a3 a2 + 18a 17)2000.

156. Chng minh : (a 3)

157. Chng minh : (x 0)

158. Tm gi tr ln nht ca , bit x + y = 4.

159. Tnh g